claims depot shelf
Birch and Swinnerton-Dyer
Formal claims (Lean)
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Theorem 1.3 asserts that, up to a p-adic unit, $$\frac{L'(E/K,1)}{\$\Omega$^{\mathrm{cong}}_{\mathbf{f}} \operatorname{Reg}(E/K)} = \#\mathrm{Sha}(E/K)[p^\infty]\prod_u c_u(E/K),$$ with $\Omega^{\mathrm{cong}}_{\mathbf{f}} = (8\pi^2)^d (f,f)_{U_0(N)} / \eta_f$ the congruence period of the Hilbert newform $f$ attached to $E$. The proof chains three ingredients: an explicit Gross–Zagier formula expr
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The paper's central discovery is a no-go statement with a constructive complement. For a labeled quantitative (2,1)-category with an extensional height, any comparison between two realizations of one extensional object that is required to descend functorially to an inequality between extensional invariants must take the form $H\le H+\Delta$, provided the transport is free; content survives only wh
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The paper's central discovery is a no-go statement with a constructive complement. For a labeled quantitative (2,1)-category with an extensional height, any comparison between two realizations of one extensional object that is required to descend functorially to an inequality between extensional invariants must take the form $H\le H+\Delta$, provided the transport is free; content survives only wh
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Boyd's conjecture asserts $\mu_k = r_k L'(E_k,0)$ for an elliptic curve $E_k$. The paper's central discovery: for all but seven exceptional $k$ up to 250,000, $r_k$ is the reciprocal of an integer $n_k$ whose p-adic valuations are locally determined by $E_k$. For $p\ge 5$, $P(v_p(n_k)=r)=p^{-r}$. For $p=3$, valuations obey congruence and parity laws tied to bad primes $\equiv 1 \pmod{3}$. For $p=2
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For primes p ≡ 4 or 7 mod 9 with 2 not a cube in F_p, write q = p or p² accordingly. The Mordell curve E_{2q} : y² = x³ − 27 q² has analytic and algebraic rank exactly 1, so 2q is a rational cube sum; its Tate–Shafarevich group is finite and its order equals the BSD quotient up to a 2-adic unit. The companion rank-zero curve E_{2q²} satisfies the full BSD formula.
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The core claim is Theorem 1.1: under the assumptions (D,q)=1, and when q is a square also assuming some twist has root number -1, there are infinitely many coprime pairs (u,v) such that the elliptic curve E_{u,v}: Q(u,v)y^2=f(x) has analytic rank one, meaning L'(1/2,E_{u,v}) is nonzero. The engine is Theorem 1.2, a weighted average over squarefree d coprime to 2qD: the sum of r_Q(d)L'(1/2,E^(d))F(
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The central discovery is a reduction of moment bounds for analytic ranks to estimates for averaged products of Frobenius traces. Using the explicit formula, analytic rank is majorized by a smoothed sum over prime ideals attached to the curve's L-function; averaging over the height-ordered family, the odd moment contributions vanish and the even contributions factor into a leading term governed by
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There exists a non-trivial Euler system for the symmetric square of a p-adic Hida family of modular forms that interpolates the Loeffler–Zerbes Euler system for ordinary newforms; together with an algebraic functional equation for the associated dual Selmer groups and work of Büyükboduk–Ganguly, this yields a divisibility toward the Iwasawa main conjecture for that symmetric square.
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For a lisse Z_ℓ-sheaf over a global function field K of characteristic p ≠ ℓ, the Pontryagin dual of the Selmer group formed over any Z_ℓ-extension K_∞/K is a finitely generated torsion module over the Iwasawa algebra Λ with μ-invariant equal to zero. This is a positive-characteristic, prime-to-p analogue of Greenberg's μ=0 conjecture, and it yields an analogue of the weak Leopoldt conjecture over
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The central claim is that for a K3 surface X over a finite field F_q and a prime p ∤ q, the p-primary Brauer groups Br(X_n)[p^∞] in the constant Z_p-tower X_n = X ⊗ F_{q^{p^n}} are controlled by the transcendental part of the L-function. Specifically, the paper proves the Iwasawa-type formula #Br(X_n)[p^∞] = p^{µ p^n + λ n + ν} for large n, with µ = 0 and λ equal to the λ-invariant of L_tr(X, 1+T)
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The central claim is Theorem A (equation (1.1)). For a normalised cuspidal eigen-newform f of weight k ≥ 2 whose base change to K satisfies the strong Heegner hypothesis, let fα be its p-stabilisation (with a_p(f) ≠ p^{k/2} when p|N_f). Then the derivative at s = k/2 + 1 of the p-adic L-function L_p(fα/K, s) equals A/B times (1 − p^{k/2}/α)^4 · h_f(z_f, z_f)/(4|D_K|)^{k/2} if p∤N_f, and A/B times
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An element L_{A/L} in the rationalized Iwasawa algebra is built so that it interpolates the special values of twisted Hasse-Weil L-functions of an ordinary elliptic curve A over a function field K relative to any Z_p^d-extension L/K unramified outside ordinary places; after adjustment by explicit local factors it satisfies the same functional equation and specialization formulae as the characteris
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On the paper's own terms, the discovery is a systematic way to produce unconditional examples of non-trivial Tate-Shafarevich groups. A finite enumeration of congruences between weight-2 newforms with coefficient fields of degree at most 4 and levels up to 10000 yields, after passing to the corresponding abelian varieties and applying a visibility criterion, abelian surfaces A/Q such that (Z/pZ)^2
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The central result is Theorem 1.2: for h = 5, 13, the sum of 2^{s(n)} over square-free n ≤ X with n ≡ h (mod 24) and all prime factors ≡ 1 (mod 4) equals 9 times the number of such n, with an error of O(X (log X)^{-5/8} (log log X)^8). Here s(n) is the 2-Selmer rank, the exponent such that the 2-Selmer group has size 2^{2+s(n)}. The constant 9 is the core arithmetic content: it is not a power of 2
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The paper's central claim (Theorem 4.17, Corollary 4.18) is an exact identity: for an elliptic curve E/Q with associated newform f, a real quadratic field K of discriminant d_K prime to the conductor N, and a ring class character χ of conductor c, the completed central derivative Λ′(1/2, Π⊗χ) — equivalently Λ′(E/K, χ, 1) — equals −√d_K/(log ε_K h_K) times half the χ-twisted sum, over ideal classes
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The paper's central discovery is a canonical construction, for any inert RM-divisor D of strong degree zero, of a higher-weight rigid meromorphic cocycle J_{k,D} whose divisor is the prescribed divisor Div_{k,D} supported on Γ-orbits of real-quadratic points. The Green's function is defined as the period pairing G_k(D1,D2)=J_{k,D1}[y^#_{k,D2}] between this cocycle and a canonically chosen homology
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The central claim is Theorem 1.7: take σ_n, σ_{n−1} geometrically irreducible Weil local systems of ranks n and n−1 over a smooth projective curve C over F_q, set σ = (σ_n, σ_{n−1}), and assume p > n. After replacing the spectral-action σ-isotypic subspace (1.21) of the compact-support cohomology of GL_n × GL_{n−1}-shtukas by the image of the explicitly constructed injection H^r(C^I, (σ_n ⊗ σ_{n−1
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The central discovery is an exact formula for the sign in the functional equation of $E_p$: $W_{E_p}=(-1)^{(p-1)/2+((p-2)^2-1)/8}$, equal to $1$ for $p\equiv 1,7\pmod 8$ and $-1$ for $p\equiv 3,5\pmod 8$. The proof computes the local root numbers at $\infty$, $2$, $p$, and $p-2$ separately. At $p$ and $p-2$ the curve has multiplicative reduction whose split or non-split character is governed by Le
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The central discovery is that reduction type, Tamagawa number, the Birch and Swinnerton-Dyer fudge factor, and the Galois representation are all locally constant functions on the coefficient space of a family of curves defined over a discretely valued field, where the family is taken to be a set of curves obtained by perturbing the defining equations. The author states this as a theorem: within th
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On the paper's own terms, the central discovery is Theorem 1.2: under Assumption 1.1 (which includes $p$ inert in $K$, $p$ large, $f$ ordinary with big image and non-CM, $p \nmid h_K$, $N_f$ squarefree and definite, and a conductor condition on $\chi$), if $\chi_-(\Theta^{\mathrm{Heeg}}_\infty(f,\chi_t)) \neq 0$, then $\dim_E \operatorname{Sel}^{\mathrm{BK}}(K,V_{f,\chi}) = 1$. The proof chains th
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The central discovery is stated in Theorem 5.9: there exist infinitely many pairs (E1,E2) of non-isomorphic elliptic curves over Q such that j(E1)≠j(E2), BSD(E1/Q)=BSD(E2/Q), and the Kodaira symbols and minimal discriminants are the same at every prime. The proof exhibits the quadratic twists of a specific pair (the m=1 case of the family E_m: y²=x³+390n x²+1952m²n x, E′_m: y²=x³−195n x²+16·195²n
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The central assertion is that for an elliptic curve E over Q, no arithmetic invariant strictly smaller than the conductor N_E can appear in a degree-two L-function that satisfies analytic continuation, a functional equation of the shape Lambda_Phi(E,s) = Phi(E)^{s/2} (2 pi)^{-s} Gamma(s) L_Phi(E,s) = w_Phi Lambda_Phi(E,2-s), and a rank bound rank(E) << log Phi(E). By the Modularity Theorem, L(E,s)
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The central claim of the paper is Theorem 1.1. Let $E/\mathbb{Q}$ be an elliptic curve with a cyclic $\mathbb{Q}$-rational isogeny of degree $N$. Then: for $N=14,19,43,67,163$, $c(E)=2^n$ with $n\ge1$; for $N=11,27,37$, $c(E)=2^n3^m$ with $m\in\{0,1\}$; for $N=17$, $c(E)=2^n3^m17^k$ with $m,k\in\{0,1\}$ and $n\ge1$; for $N=21$, $c(E)=2^n3^m7^k$ with $m\in\{0,1,2\}$, $k\in\{0,1\}$, $n\ge1$; and for
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On the paper's own terms, the central discovery is the rank-loop correspondence: for an elliptic curve $E$ over $\mathbb{Q}$ of rank $r$, the first homology group $H_1(\Phi(E),\mathbb{Z})$ generated by $\mathbb{Q}$-rational points is claimed to be isomorphic to $\mathbb{Z}^r$, so the rank equals the number of topologically independent infinite loops in the four-dimensional embedding. The paper ext
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Write $n$ in the family as $n=m$ or $n=2m$, where $m=\prod_{i=1}^s p_i$ and every $p_i\equiv 1\pmod 4$. For the odd case, choose an integer $b$ with $b^2\equiv -1\pmod{m^2}$ and set $\tau_m=(b+i)/(2m^2)$. The paper establishes $L(E_m,1)=\pi\,|\theta_{\chi_m}(\tau_m)|^2/(4\sqrt{2m})$, where $\theta_\chi(\tau)=\sum_{k\in\mathbb{Z}}\chi(k)e^{2\pi i k^2\tau}$ with $\chi_m$ the Kronecker character. For
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Let $f$ be a cuspidal newform and $p \geq 3$ a prime such that $\rho_f$ has large image. Theorem 1.1 states: if the collection of Kurihara numbers $e^{\min,\dagger}$ for $f$ at the central critical point $s = k/2$ does not vanish, then Kato's Kolyvagin system $\kappa_{\mathrm{Kato},\dagger}$ for $T^\dagger_f$ is non-trivial, and there is an isomorphism of co-finitely generated $\mathcal{O}$-module
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For an irreducible Dedekind scheme $X$ with perfect residue fields, the paper proves two dualities: the functor sending a torsion-free $\mathbb{Z}$-constructible sheaf $F$ to the complex $[T \to \bigoplus_v i_{v,*}E_v]$ (Néron model of the Cartier dual of the generic fibre, with $\mathbb{Z}$-duals of the closed-point stalks) is quasi-inverse to $T\mapsto \mathrm{Ext}^0_{X_{\mathrm{sm}}}(T,\mathbb{
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The central discovery is a conditional proof that, under the hypotheses (irred) and (MML), the conditions rank_Z E(F)=1 and #III(E/F)_{p^\infty}<\infty imply ord_{s=1}L(E/F,s)=1. The proof moves to a carefully chosen imaginary quadratic extension K/F, proves that the central critical p-adic L-function of the associated Hida family vanishes to order exactly four at weight 2, translates this into or
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On its own terms, the central discovery is Theorem 4.1.13: for an elliptic curve $E/\mathbb{Q}$ and an abelian extension $K/\mathbb{Q}$ satisfying the stated hypotheses, the Bloch–Kato Selmer group $\mathrm{Sel}(\mathbb{Q},E[p^\infty]\otimes \mathcal{O}_d(\chi))$ is determined, as an $\mathcal{O}_d$-module, by the twisted Kurihara numbers $\delta_{n,\chi}$. For $\chi^2=1$ it is $\mathcal{O}_d^r \o
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The central discovery is Theorem 5.4: for $G=\mathrm{SL}_\ell$ with $\ell$ prime, the function $m\mapsto L(G,m)=\sum_{[\gamma]}L_S(M_{G_\gamma})$, summed over semisimple conjugacy classes in $G(\mathbb{F}_{q^m})$, is of Lefschetz type; and Theorems 6.5 and 6.6: the same holds for $G=\mathrm{Sp}_4$ and $G=\mathrm{Sp}_6$. Here $M_{G_\gamma}$ is the Artin–Tate motive of the centralizer and $L_S$ is i
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The central claim is a sandwich theorem for the $\phi$-Selmer group. For a Type I curve $E_a:y^2=x^3+a$, set $K=\mathbb{Q}(\zeta_3,m^{1/3})$, let $L=K(\sqrt{a})$ when $a\notin K^{*2}$ and $L=K\times K$ when $a\in K^{*2}$, and let $S(K)=S_a(K)\cup S_3(K)$ be the finite set of primes where the local description is exceptional. Theorem 3.9 asserts that if $a\notin K^{*2}$, then $\dim_{\mathbb{F}_3}\m
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The paper's central claim is that the vanishing order $r$ at the central point of a rational L-function is encoded in the normalized Dirichlet coefficient vector $v(L)=(a_p/(d p^{w/2}))_{p<1000}$, in the following sense: on the sub-dataset PRAT$^\star$ of 176,156 primitive rational L-functions (degree 4, motivic weight 1, mostly elliptic curves over number fields and genus-2 curves over $\mathbb{Q
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On the paper's own terms, the central discovery is that a transformer can learn the arithmetic of Frobenius traces from data alone. For the exact-value task, $a_{97}$ is predicted with test MCC $0.4711$ against a most-common-class baseline of $0.082$, and the model captures $|a_{97}|$ better than the sign (sign-agnostic MCC $0.6266$); $a_2$ and $a_3$ show the same pattern. When the target is reduc
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At its core the paper claims that for $a\notin K^{*2}$ with $3\nmid a$, the dimension of $\mathrm{Sel}_{\psi_{a,b}}(E_{a,b}/K)$ is bounded below by $\max\{h^3_{S_{1,2}}(L), h^3_{S_{1,3}}(L)+|S_3|-|S_2|-1\}$ and above by $\min\{h^3_{S_{1,2}}(L)+|S_{1,2}(L)|+|S_3|-|S_2|+1, h^3_{S_{1,3}}(L)+|S_{1,3}(L)|+2\}$. Here $L=K(\sqrt{a})$, $h^3_S(L)$ is the 3-rank of the $S$-ideal class group, and $S_1,S_2,S_
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On the paper's own terms, the discovery is that the singularities of the natural model are not merely tolerated but are structurally necessary: the exceptional divisors on the resolution are themselves sources of algebraic classes, and the singular fibers are what create the geometric monodromy that makes the period map nontrivial. The proof constructs an extended period morphism (Definition 4.3)
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The central assertion is Conjecture 1.3: for an odd prime $p$, a primitive Dirichlet character $\chi$ of squarefree order $d$ with conductor coprime to the conductor of $E$, and an embedding $\iota$ of $\mathbb{Q}(\zeta_d)$ into $\mathbb{C}$, the $p$-part of the principal ideal generated by the normalized value $\iota^{-1}(L(E,\chi))$ is predicted to be $\prod_\theta (h_\theta(\chi(\tau)^{d/d_\the
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On the paper's own terms, the central discovery is Theorem 4.1. Let $\bar{\rho}:G_{\mathbb{Q},\Sigma}\to \mathrm{GL}_2(k)$ be an odd residual Galois representation whose image contains a conjugate of $\mathrm{SL}_2(\mathbb{F}_p)$ and whose local behaviour at $p$ avoids the two degenerate character-extension cases, and assume the auxiliary primes outside the ramification set satisfy the local defor
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Indefinite analogue: under the same hypotheses, if the cohomological period class is nonzero on an irreducible tempered component, then the Selmer group has generic rank one, its dual is torsion, and the analogous divisibility for the torsion part is effective.
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Theorem 11.5 (definite case): under the paper's hypotheses, if the Bessel period is nonzero on an irreducible tempered component, then the Selmer group vanishes, the dual Selmer module is torsion, and the displayed divisibility of characteristic divisors is effective.
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The torsion-modified Conjecture 3.4 holds for all 425,713 tested pairs.
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The positive-rank consequence Conjecture 2.12 fails for 367 pairs out of the same database.
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The original Mazur--Tate Conjecture 6 (via Conjecture 0.1) fails for 886 pairs out of 425,713 tested pairs.
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The central assertion is Corollary 1.3: the 4-dimensional representation of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on $H^3_{\acute{e}t}(Y_{79}\otimes\bar{\mathbb{Q}}, \mathbb{F}_5)$ is isomorphic to the mod-5 residual representation $\rho_{F_{79},5}$ attached to the non-lift Hecke eigenform $F_{79}\in S_3(K(79))$, where $K(79)$ is the paramodular subgroup of $\mathrm{Sp}_4(\mathbb{Q})$. The a
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The central claim is that the root number determines the central vanishing order throughout the anticyclotomic tower, up to a finite set of characters. The paper proves that the average of $L^{(v)}(1,\chi)$ over Galois conjugates tends to $2L(1,\kappa)$ in the even case and grows like $2L(1,\kappa)\log(A f(\chi))$ in the odd case, where $\kappa$ is the quadratic Dirichlet character attached to $K$
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Let $E$ be an elliptic curve over a number field $k$ and let $F/k$ be a dihedral extension of degree $2n$. Fix a prime $p$ not dividing $n$ and assume the p-primary Tate–Shafarevich group $\mathrm{Sha}(E/F)[p^\infty]$ is finite. The paper establishes that for odd $n$, $|\mathrm{Sha}(E/F)[p^\infty]| / |\mathrm{Sha}(E/K)[p^\infty]| = p^{4t}$ for some integer $t$, where $K$ is the quadratic subextens
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On the paper's own terms, the central discovery is Theorem 7.12: for an abelian variety $A$ over a global function field $K$ of characteristic $p>0$, a finite Galois extension $F/K$ with group $G$, and an irreducible complex character $\chi$ of $G$ defined over a number field $E$, if the Tate–Shafarevich group $\Sha(A/F)$ is finite and the local Assumption 7.2 holds, then for any place $\lambda$ o
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On its own terms, the paper establishes Conjecture 3.8: for matching strongly regular semisimple data, $\partial\mathrm{Orb}((\gamma,u'),1_{(S_{n-1}\times V'_{n-1})(\mathcal{O}_{F_0})})=-\mathrm{Int}(g,u)\,\log q$ in the semi-Lie algebra version, together with the corresponding group version; this is Theorem 15.1, valid for odd $p\ge n$. It also establishes the Jacquet–Rallis fundamental lemma, Co
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The central claim is that in the prime-conductor family of quadratic Dirichlet $L$-functions over function fields, the second moment at the central point is now known to order $g^2$. Explicitly, Theorem 1.1 states that $$\frac{1}{|\mathcal{P}_{2g+1}|}\sum_{P\in\mathcal{P}_{2g+1}} L(1/2,\chi_P)^2 = \frac{$g^{3}$}{3\zeta_q(2)} + \left(\frac{3}{2}+\frac{1}{2q}\right)$g^{2}$ + O_\varepsilon($g^{{3/2+\
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Concretely, under the paper's Assumptions 1.1.1 and 1.1.3, the non-anomalous condition, and primitivity of the Heegner Kolyvagin system $\kappa$, Theorem A establishes the full statement of Howard's Main Conjecture: the torsion $\Lambda$-module $M$ satisfies $\operatorname{char}(M)=\operatorname{char}(M)^\iota$, the Selmer group $X$ is pseudo-isomorphic to $\Lambda\oplus M\oplus M$, and $\operator
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Theorem 1.3 is the central claim: under assumptions (QS), (Irred), with 0 and 1 not Hodge-Tate weights of M, and with π unramified and ordinary above p, the equality L(M∨(1),0)=0 implies that the rank of the Selmer group Sel_{p^∞}(M) is positive. Equivalently, a Selmer group of rank 0 forces the central L-value to be nonzero. The author establishes this by specializing an ordinary Hida family of K
Stated claims
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Assuming $\Sha(E/\mathbb{Q}(i))[p^\infty]$ is finite and the cyclotomic $p$-adic regulator $\det H$ is non-zero, Theorem 3.5 shows that the quadratic term of the two-variable Katz measure at $\psi_E$ is the explicit form $\frac14\,\iota_p(f_0\omega_E)\,\Omega_p\,e^2(1-\alpha^{-1})^2\,q_p\,[(a-b)^2\det H-(a+b)^2j^2]$, so $q_p$ is its exact constant; the same constant governs the cyclotomic, anticyclotomic and $p$-adic directions of Perrin-Riou's pencil, and with $v_p(j)=1$ the central values $M_1,\dots,M_m$ determine $q_p$ modulo $p^m$. The paper then proves that $q_p$ is the coordinate of two global classes: Rubin's elliptic-unit class $x^*$ sits in $E(\mathbb{Q})\otimes\mathbb{Q}_p$ at $q_p$ times an explicit vector, and the derivative $D=\zeta Q_v$ of the derived Heegner class over an auxiliary imaginary quadratic field $F$ satisfies the exact identity $q_p = \frac{1}{S_F}\big(2\ell_a(\sigma_a)\zeta(1-\alpha^{-1})^2\big)^2$, with $S_F$ an explicit rational number built from the rank-one twist. Consequently $q_p \in \prod_v c_v(E)\cdot(\mathbb{Q}_p^\times)^2$, so the unit $q_p/A_E$ of the $p$-adic Birch-Swinnerton-Dyer formula is a square in $\mathbb{Z}_p^\times$ whenever $\Sha(E/\mathbb{Q})$ is finite; and under the additional hypothesis that $Q_v$ has non-zero formal logarithm, the Bertolini-Darmon-Prasanna $p$-adic $L$-function vanishes to order exactly two, the conjecture of Agboola and Castella holds up to a $p$-adic unit, its exact analytic form is equivalent to $q_p=q_\infty$ (with heights read sesquilinearly), and the anticyclotomic Iwasawa-Greenberg main conjecture holds integrally.
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The central claim, Theorem 1.4, is a conditional proof of the relative $p$-parity conjecture: let $V_1$ and $V_2$ be symplectic self-dual de Rham representations of $G_{\mathbb{Q}}$ that are pure of weight $-1$, satisfy the Panchishkin condition at $p$, and satisfy $A_1[\pi] \cong A_2[\pi]$ as $G_{\mathbb{Q}}$-modules. If the dual Bloch-Kato Selmer group $S_{BK}(A_1/\mathbb{Q}_\infty)^\vee$ is torsion over the Iwasawa algebra with $\mu = 0$, and the residual isomorphism restricts to an isomorphism $A_1^-[\pi] \cong A_2^-[\pi]$ of the 'minus' Panchishkin parts at $p$, then $$W(V_1)/W(V_2) = (-1)^{\mathrm{rk}_{\mathcal{O}}\, S_{BK}(A_1/\mathbb{Q})^\vee - \mathrm{rk}_{\mathcal{O}}\, S_{BK}(A_2/\mathbb{Q})^\vee}.$$ Unconditionally (Theorem 1.3), the root-number ratio is $(-1)^{\sum_\ell (\delta(A_1,\ell) - \delta(A_2,\ell))}$, where $\delta(-,p)$ is a new local invariant built from strict local conditions over the cyclotomic $\mathbb{Z}_p$-extension and $\delta(-,\ell)$ is a local Iwasawa $\lambda$-invariant for $\ell \neq p$; both are compatible with the arithmetic local constants of [MR1] and [Ne1]. The key input, Theorem 1.5, states that the Bloch-Kato and strict local conditions at $p$ coincide over $\mathbb{Q}_{\infty,p}$ for every de Rham, Panchishkin, pure-weight $-1$ representation; its proof uses p-adic Hodge theory to control the Galois action on $H^0(\mathbb{Q}_{\infty,p}, V^-)$ and to compare its dimension with the same space over $\mathbb{Q}_p$ modulo 2.
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The paper establishes that Bloch's diamond formula, when stated for a generator of the anti-invariant subgroup $H_1(E,\mathbb{Z})^-$, holds with an extra factor $\iota(E)\in\{1,2\}$ that equals $1$ when the real curve has two components ($\Delta>0$) and $2$ when it has one component ($\Delta<0$). The natural domain of the regulator integral is the regulator quotient $Q(E)=H_1(E,\mathbb{Z})/H_1(E,\mathbb{Z})^+$, where the factor is $1$ for a generator (Theorem 4.1). The subgroup version follows by an elementary index computation: for $\Delta<0$, the class of the anti-invariant generator equals $\pm 2$ times the quotient generator in $Q(E)$ (Lemma 3.2). Consequently, the transmitted form in Lalín–Ramamonjisoa and elsewhere is correct only when $\Delta>0$; for $\Delta<0$ it is off by a factor of $2$. The paper also shows that Touafek's Theorem 2 mixes two independent normalizations and becomes consistent only after a two-part reinterpretation.
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The central claim is Theorem 4.1.1: if $E$ has ordinary reduction at $p\geq 3$ with $E[p]$ reducible over $G_{\mathbb{Q}}$, and $\chi$ is a ramified quadratic admissible twist with $H^0(\mathbb{Q}, E'[p]\otimes\chi)=0$ for all $E'$ isogenous to $E$, then the cyclotomic Iwasawa Main Conjecture holds for the twist, i.e. $\mathrm{Char}_{\Lambda}(\mathrm{Sel}(E^{K_\chi}/\mathbb{Q}_{\mathrm{cyc}})^\vee)=(L_p(E,\chi))$ in $\Lambda=\mathbb{Z}_p\llbracket T\rrbracket$. The proof splits according to the sign of the admissible twist: in both the $+$ and $-$ admissible cases the residual Selmer group of $E[p]\otimes\chi$ sits in a short exact sequence between the relaxed Selmer group of an even character and the strict Selmer group of an odd character, so Mazur–Wiles vanishing of $\mu$ and equality of $\lambda$'s for characters transfer to the curve, and Kato's one-sided divisibility is upgraded to the equality. The $\mu=0$ statement for non-admissible twists is obtained separately, without comparing $\lambda$-invariants.
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The central claim is Theorem 2.4.1. Under the hypotheses $p \ge 5$ split in $K$, $K$ imaginary quadratic of class number one, and $E/K$ an elliptic curve with complex multiplication by $\mathcal{O}_K$ and Hecke character $\psi$, for every pair of integers $(k,j) \ne (0,0),(1,1)$ with $\chi = \psi^k\bar\psi^j$ the value of the Katz $p$-adic $L$-function at $\chi$ is $$\chi(L_{\mathrm{Katz}}) = \$Omega_p^{{k-j}}$(1-\chi(p)$p^{{-1}}$)(1-\$chi^{{-1}}$(p))^{-1} \cdot \begin{cases} (k-1)!\,\exp^*_{V_\chi}(\operatorname{loc}_p z_\chi) & k>0,\\ (-1)^k(-k)!^{-1}\log_{V_\chi}(\operatorname{loc}_p z_\chi) & k\le 0, \end{cases}$$ where $z_\chi$ is the elliptic Soulé element — the cohomology class obtained from the elliptic-unit tower by twisting by $\chi^{-1}$ and dividing by $\mathrm{N}\mathfrak{a}-\chi(\mathfrak{a})$ — and $\exp^*$, $\log$ are the Bloch-Kato dual exponential and logarithm maps normalized by the period $d_\chi$. Where the classical interpolation law expresses values in the range $k>0$, $j\le 0$ through Hecke $L$-values, the new formula replaces the $L$-value by a regulator term that makes sense for every $(k,j)$; the special-value theory of the Katz function is thus governed by elliptic units throughout. The authors give two proofs, one through the two-variable Coleman map and rank-one local epsilon elements, one through the interpolation property of the two-variable Lubin-Tate regulator map, and draw three applications: the elliptic Coleman-Ihara formula, an extension of Kato's explicit reciprocity law with the Tamagawa number conjecture for critical Hecke motives, and a large part of the $p$-adic Beilinson conjecture.
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The central claim, Theorem 1.2, is that under the Heegner hypothesis, surjectivity of the mod-$2^L$ Galois representation, and a cohomological exclusion condition at level $4$, any $2$-indivisible parametrization $\varphi_0$ and any terminal support $T$ (one for which the $2$-Selmer dimensions of $E^{d_T}$ and $E^{d_T D}$ are $0$ and $1$) satisfy $\kappa^{\mathrm{std}}_{n_T,1}(\varphi_0)\ne 0$ if and only if $i(E,K)=0$, and when the index vanishes the normalized standard classes have order exactly $2^M$ for every $1\le M\le L$. The engine is Theorem 5.4, which computes the $2$-adic valuation of the index $m_T$ of the constructed point $Q_T$ in the full Mordell–Weil lattice of the terminal twist: $v_2(m_T)=v_2(c_\varphi)+i(E,K)$, a formula independent of $T$. With a $2$-indivisible parametrization supplied by the optimal-Manin-constant theorem, the criterion becomes purely local, and when $i(E,K)=0$ the minimum terminal support size is $R=\min\{d_2(E),d_2(E^D)\}$; full mod-$8$ Galois image then gives strong nonvanishing of the standard system (Theorem 1.4). The intrinsic index $i(E,K)=T_E+\delta_K$ is an explicit local quantity — the total $2$-adic Tamagawa contribution of $E$ plus half the total discrepancy between the Kummer conditions of $E$ and $E^D$ — so the paper's conclusion is that residual nonvanishing of the standard Kolyvagin class is a property of the base curve and field alone.
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The central claim is Theorem 3.3.3: there is a collection of Iwasawa cohomology classes $c_{z_{\mu_3,\chi}} \in H^1_{\mathrm{Iw}}(K_0[\mu_3 p^\infty], O_\chi)$, indexed by squarefree ideals $\mu_3$ of $\mathcal{O}_{K_3}$ whose rational norms are squarefree and split in $K_0$, such that for every auxiliary prime $\lambda_3$ of norm $\ell$ the corestriction from level $\mu_3\lambda_3$ to level $\mu_3$ sends $c_{z_{\mu_3\lambda_3,\chi}}$ to $P_{L_4}(\mathrm{Frob}_{L_4})(c_{z_{\mu_3,\chi}})$, where $P_{L_4}(X) = \det(1 - X\,\mathrm{Frob}_{L_4} \mid O_{\chi^{-1}}(1))$ is the local Euler factor at the prime $L_4$ above $\ell$. The construction proceeds in two stages: a tame norm relation is proven in weight $(2,2,2)$ by a congruence computation with explicit Hecke character values, and the general weights follow by varying the diagonal cycles inside Hida families, giving the wild norm relations over the anticyclotomic $\mathbb{Z}_p^2$-extension. The paper then shows the $\chi$-classes land in the Greenberg-type Selmer group $\mathrm{Sel}^{\emptyset,0,\emptyset,0}$ (Proposition 6.3.1), and that a non-torsion base class forces the associated Selmer groups to have rank one and yields the Iwasawa divisibility of Theorem 6.4.2. The explicit reciprocity law of Theorem 6.1.1 links the base class to the $g$-unbalanced triple product $p$-adic $L$-function, and the factorization of Proposition 6.6.6 splits that $p$-adic $L$-function into the $\psi$ and $\chi$ anticyclotomic Katz-type factors, converting the structural statements into the Bloch–Kato and Iwasawa main conjecture theorems.
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论文的核心发现是定理B：存在Lieberman类型的Enriques商S，使得f_*T(A)/2与Br(S)/Br_0(S)通过Brauer-Manin配对的配对非平凡，当且仅当Br(X)[2]/Br_1(X)[2]到Br(A)[2]/Br_1(A)[2]的映射是同构。这给出了Enriques曲面零环群非平凡性的一个判别条件，并允许构造具体例子。
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The central claim is Theorem 1.1: for any finite set $\Lambda$ of non-trivial local reduction data and any finite set $\Pi$ of primes, the limiting joint generating function of the count of contributing primes, $\Omega(\mathrm{Tam}(E))$, the valuation vector, and the per-datum counts converges, on the closed unit domain, to the absolutely convergent Euler product $\prod_p(\beta_p+\sum_{K\notin K_0}\delta_p(K)\Phi_p(K))$, where $K_0=\{I_0,I_1^{\mathrm{sp}},I_1^{\mathrm{ns}}\}$ collects the three local data with $c(K)=1$ and $\delta_p(K)$ is the Haar-measure density of a given local reduction datum. Reading off coefficients gives the limiting probabilities $\pi_0\approx0.50540113$, $\pi_1\approx0.42200272$, and $\pi_2\approx0.06841047$ for zero, one, or two primes contributing nontrivially to $\mathrm{Tam}(E)$, and the identity $\pi_0=P_{\mathrm{Tam}}(1)$ identifies the trivial-Tamagawa density with the density of curves whose Tamagawa product is exactly one. The same product yields a covariance $\mathrm{Cov}(v_\ell(\mathrm{Tam}(E)),v_{\ell'}(\mathrm{Tam}(E)))$ that is strictly negative for every distinct pair $\ell,\ell'$ (about $-0.02489$ at $\ell=2,\ell'=3$) and decays like $O((\ell\ell')^{-2})$, a correlation of about $0.9801$ between $\omega_{\mathrm{Tam}}(E)$ and $\Omega(\mathrm{Tam}(E))$, and the Euler product $\prod_p\sum_{t\ge1}\delta_p(t)t^k$ for every $k$-th moment, giving super-polynomial decay of the tail $P_{\mathrm{Tam}}(m)$.
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The central claim is that the real period has a reconstruction property beyond what was known: for elliptic curves over a number field K with a fixed real embedding $\sigma$, the quotient of the two real periods lies in $\sigma$(K^x) if and only if the curves are isogenous over K (Theorem 2.8). Over Q, this implies the real period determines the Q-isogeny class (Theorem 1.1), and the paper then sharpens this to isomorphism: an elliptic curve E/Q is uniquely determined up to Q-isomorphism by the triple consisting of c_{E/R}, the number of connected components of E(R), the real period Omega_E, and the real period Omega_{E^D} of the quadratic twist by a negative square-free D congruent to 1 mod 4 and coprime to the minimal discriminant (Theorem 3.3). The same conclusion holds when the two real periods are known only to N decimal places, where N depends on E and D. The paper also exhibits the curve $y^{2}$ = $x^{3}$ + 17x as an example where the real period together with the Tate-Shafarevich group already singles out the isomorphism class.
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Let $E/\mathbb{Q}$ be an elliptic curve and $p$ an odd good ordinary Eisenstein prime, meaning $E[p]$ fits into a short exact sequence $0\to F(\varphi)\to E[p]\to F(\psi)\to 0$ of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-modules with $\varphi\psi=\omega$. Of the two unresolved Eisenstein cases, case (i) — one of $\varphi,\psi$ unramified and odd — was settled by Greenberg–Vatsal [GV00]; this paper settles case (ii), where $\varphi$ is unramified and even, $\psi$ ramified and odd, and the extension is non-split. Over the intermediate abelian field $\mathbb{Q}(\varphi,\psi)$ the classical, classical-fine, and fine-for-$E$ $\mu$-invariants all vanish by the Ferrero–Washington theorem, and a Nakayama-based p-extension principle (Proposition 3.1.5) lifts this vanishing up to $L=\mathbb{Q}(E[p])$, where $E[p]$ splits as $F(\varphi)\oplus F(\psi)$. With these vanishings in hand, the paper tracks where the ' $\mu=1$' part of the global-to-local image of $H^1(\mathbb{Q}_\Sigma/\mathbb{Q}_\infty,E[p])$ lands: the non-splitness of the extension makes $F(\psi)$ fail to be $\mathrm{Gal}(L/\mathbb{Q})$-invariant, so restriction kills the intersection with the ramified summand, and the fine-$\mu$ vanishing kills the obstruction locally, together forcing Greenberg's $\mu$ over $\mathbb{Q}$ to vanish. Composing with the Iwasawa Main Conjecture at Eisenstein primes ([GV00], [CGS25], [KY24]) yields the analytic vanishing (Theorem 3.3.1) and confirms Stevens's conjecture. Theorem B records the vanishing of classical, classical-fine, and fine-for-$E$ $\mu$-invariants over $\mathbb{Q}(E[p])$, and Theorem C extends the reduction principle to the residually irreducible case, where vanishing of the classical fine $\mu$ over $\mathbb{Q}(E[p])$ would force Greenberg's $\mu$ over $\mathbb{Q}$ to vanish.
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On its own terms, the paper establishes that in the expansion $A_2(p) = p^2 - C_p p^{3/2} - D_p p + p(p - B_p) + O(1)$, the averages of $C_p$ and $D_p$ vanish, while $B_p$ averages to a value at least 1. The $C_p$ term is dispatched by applying the generalized Sato-Tate conjecture to the motive $H^1(\mathbb{P}^1, j_* \mathrm{Sym}^2 \mathcal{F})$ and showing that its Sato-Tate trace integral is zero; the $D_p$ term, from additive fibers, is killed by a Chebotarev argument comparing a permutation representation with a cyclotomic character; and the $d = 2$ additive case is handled by resolving the singular fiber over a quadratic extension. The residual $-B_p p$ then forces the averaged $p$-coefficient to be negative, which is the sharpened form of the bias conjecture. The argument is conditional on the non-constant $j$-invariant and on Sato-Tate, both stated explicitly.
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The discovery is that the limiting mixed Hodge structure of the third cohomology at each type-K fiber decomposes, after semisimplification, into two graded pieces, each a copy of the transcendental lattice of the same singular K3 surface, related by a Tate twist: $\mathrm{Gr}^2_W \simeq T(K_0)$ and $\mathrm{Gr}^4_W \simeq T(K_0)(-1)$. Because a singular K3 surface is modular, its transcendental lattice carries the Galois representation of a weight-three modular form $g$, so the limiting cohomology satisfies $L(H^3_{\mathrm{lim}}, s)=L(g,s)L(g,s-1)$. The same pattern holds at both degenerations, with the modular forms $32.3.d.a$ and $16.3.c.a$ respectively. The theorem is deliberately stated up to semisimplification, leaving the genuinely mixed extension between the two graded pieces unspecified.
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The paper's central claim is twofold. First, for a semi-stable elliptic curve E over Q and a good reduction prime p≥5 with E[p] irreducible, if the Z_p-corank of the Selmer group Sel(Q,E[p^∞]) is at least 2, then the finiteness of X(E/Q)[p^∞] is equivalent to the condition that certain derived Heegner points P_{n_k} lie in p^∂(E(Q) + p^k E(K[n_k])), where ∂ is a constant determined by the Heegner point Kolyvagin system. Second, the paper proves Kolyvagin's conjecture for the supersingular case: the Heegner point Kolyvagin system $κ^{{Hg}}$ is non-trivial for semi-stable elliptic curves with supersingular reduction at p≥5, under a suitable choice of imaginary quadratic field. This proof uses the signed Heegner point main conjecture of Castella-Wan and a rigidity argument for Kolyvagin systems, and is independent of the cyclotomic main conjecture.
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The paper's central claim (Theorem 1.1) is that for every prime $p \equiv 8 \pmod 9$, $\operatorname{ord}_{s=1} L(s, E_p) = \operatorname{ord}_{s=1} L(s, E_{p^2}) = 1$ for the curves $E_m : y^2 = x^3 + m^2/4$; together with the completed proof for the classes 4 and 7, this yields Sylvester's conjecture (Theorem 1.2). To get there the authors choose an auxiliary prime $q \equiv 4 \pmod 9$ whose cubic residue symbol $(p/\varpi)_3$ is non-trivial (such primes exist in density $1/9$ by Chebotarev), and factor a self-dual Rankin-Selberg convolution as $L(s, g_q \times \chi_{q,1}) = L(s, E_p) L(s, E_{qp^2})$ and $L(s, g_q \times \chi_{q,2}) = L(s, E_{p^2}) L(s, E_{qp})$. An explicit 3-isogeny descent computation together with the paper's own rank-zero 3-converse theorem for CM elliptic curves shows the complementary central values $L(1, E_{qp})$, $L(1, E_{qp^2})$ are non-zero, so the YZZ generalization of the Gross-Zagier formula reduces the theorem to proving that a cubic component of an auxiliary Heegner point is non-torsion. At that point the central new step appears: the unweighted Hecke trace of the conductor-$p$ CM orbit vanishes, so the authors take a first $\lambda$-division before tracing, $\lambda = 1 - \omega$; the resulting division boundary $\partial^c_{q,p}$ is a $\lambda$-torsion point shown to be non-zero by comparing the $p^2$-power Frobenius on Fermat co-ordinates with the CM endomorphism $[-p \alpha_q(p)]$, where the non-trivial residue symbol is exactly what is detected. Integral group-ring identities $\lambda D = N_G - \Pi$ transfer this non-zero torsion to the cubic component, and the Gross-Zagier formula turns it into $L'(1, g_q \times \chi_{q,j}) \neq 0$, completing the analytic-rank-one proof.
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On the paper's own terms the discovery is Theorem 2: for every $(m,n)$ in the complement $R_{968} = P_{100} \setminus S_{\mathrm{Pes}}$, where $P_{100}$ is the set of 2,040 coprime opposite-parity pairs with $2 \le m \le 100$, there is no positive master-tuple specialization $(a,b,m,n)$ that yields a perfect cuboid. With Peschmann's 1,072-fiber theorem this gives Corollary 3: no perfect cuboid occurs on any fiber $(m,n) \in P_{100}$. The proof is a cascade rather than a single method: 585 fibers fall to rank-zero elliptic quotients plus a uniform infinity-class computation on the CM curve $y^2 = x^3 - x$ over $\mathbb{Q}(i)$; 263 more fall to strengthened versions of the same ideas (trace curves, genus-two negative branches, bielliptic quadratic Chabauty, Gaussian and CRT covers); and the final 120 require exact higher 2-descent, torsion arithmetic, centrally bounded L-values, and a correlated p-adic height sieve on the genus-five curve $C_{U,V,W}$. Every step is recorded fiber by fiber, and the paper stresses that no bounded rational-point search serves as a completeness theorem.
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The paper proves that for a rank-two CM elliptic curve E/ℚ at a split good ordinary prime p outside an explicit finite set S_E, the following are equivalent: (1) the normalised second Taylor coefficient of the p-adic L-function of E is a p-adic unit; (2) the cyclotomic p-adic height pairing on E(ℚ) is nondegenerate with unit regulator and the p-primary part of the Shafarevich–Tate group vanishes; (3) an explicit combination of three critical Hecke L-values of weight 2p-1 (the bracket B(p)) has p-adic valuation exactly 2. This supplies a cyclotomic counterpart to the Coates–Liang–Sujatha criterion and shows that the second derivative of the p-adic L-function fully determines the arithmetic at p.
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Theorem 1.2: Under the level-aspect and specified growth conditions, the expression $\sqrt$(X) * E_{X,Y}[ ε(f) $λ^{{sym_k}}$_f(P) ] = G(P^k/X) + o_k(1) for a continuous explicit function G, showing every fixed prime power samples the same leading murmuration function when placed at the effective position P^k/X.
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On its own terms, the paper proves the following. Under Assumption 2.1, for every $\mathbb Z_p$-line $L'/K \subset L/K$, the element $L_{E/L'} = \theta_{L/L'}^{-1}\,\operatorname{pr}_{L'}(L^{\mathrm{PR}}_{E/L})$ lies in $\Lambda'$ and satisfies $$(L_{E/L'}) = \operatorname{char}_{\Lambda'}(X_{L'}),$$ where the characteristic ideal of a non-torsion module is declared to be the zero ideal. The global factor appearing in the general specialisation formula is trivial, so the entire discrepancy between the projected two-variable characteristic ideal and the one-variable characteristic ideal is the explicit principal ideal $\vartheta_{L/L'} = (\theta_{L/L'})$. Exceptional lines—those on which the projected Perrin–Riou element is zero—are exactly the lines on which $X_{L'}$ fails to be torsion. The cyclotomic line has $\theta_{L/L^+} = 1$ and is non-exceptional, recovering the product of the Mazur–Swinnerton-Dyer $p$-adic $L$-functions of $E$ and its quadratic twist, while the anticyclotomic line is exceptional because $X_{L^-}$ has rank one. The final results bound the number of exceptional lines by the cyclotomic augmentation order, show this order equals the Mordell–Weil rank under the stated finiteness and non-degeneracy assumptions, and imply uniqueness of the anticyclotomic exceptional line in rank one.
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On its own terms, the paper's substantive content is the statement of the Birch and Swinnerton-Dyer conjecture, together with the construction that makes the statement meaningful. Starting from the Weierstrass normal form $y^2 = x^3 + bx + c$, it shows that the rational points form an abelian group under the chord-and-tangent law, cites the theorem of Mordell that this group is finitely generated, and takes its free rank as the algebraic rank. It then builds $L_C(s)$ from the reduction counts $\epsilon_p = p+1-|\bar C_p|$, records the bound $|\epsilon_p| \le 2\sqrt p$, and relies on the analytic continuation and functional equation of the completed $L$-function $\Lambda_C(s)$ so that the order of vanishing at $s=1$ defines the analytic rank. The conjecture—equality of the two ranks—is supported in the exposition by the formal identity $L_C(1) = \prod_p p/|\bar C_p|$ and by the heuristic that curves with larger rational-point groups have unusually many points modulo $p$, making the product decay faster exactly when the rank is larger.
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Theorem 1.3 asserts that, up to a p-adic unit, $$\frac{L'(E/K,1)}{\$\Omega$^{\mathrm{cong}}_{\mathbf{f}} \operatorname{Reg}(E/K)} = \#\mathrm{Sha}(E/K)[p^\infty]\prod_u c_u(E/K),$$ with $\Omega^{\mathrm{cong}}_{\mathbf{f}} = (8\pi^2)^d (f,f)_{U_0(N)} / \eta_f$ the congruence period of the Hilbert newform $f$ attached to $E$. The proof chains three ingredients: an explicit Gross–Zagier formula expressing the derivative special value through the height of a Heegner point and the Tamagawa numbers at primes inert in $K$; an anticyclotomic control theorem plus the Iwasawa main conjecture that replaces the anticyclotomic Selmer group by the special value of a p-adic L-function; and the trivial-character Liu–Zhang–Zhang formula that cancels error terms. The equality is a variant of the classical BSD formula because the normalization uses the congruence period rather than a Néron period of $E$.
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The paper's central discovery is a no-go statement with a constructive complement. For a labeled quantitative (2,1)-category with an extensional height, any comparison between two realizations of one extensional object that is required to descend functorially to an inequality between extensional invariants must take the form $H\le H+\Delta$, provided the transport is free; content survives only when the transport itself carries a positive load. The concrete vehicle is the category of elliptic packages: an elliptic curve presented with a Weierstrass model, local sections, and a label, with a model defect $\mu(f)$ attached to each change of coordinates. The discriminant height obeys $h_\Delta(E')\le h_\Delta(E)+\mu(f)$, making it a transport log-volume, and a dilation tower at the bad prime shows that no isomorphism-invariant scalar can be comparable to it—so keeping the defect visible is not optional. From this the paper derives an intensional Szpiro inequality $h_\Delta(E)\le (6+\varepsilon)n(E)+\delta_{\mathrm{mod}}(E)+C_{\varepsilon,K}$, ties a sharp form to the $abc$ conjecture, and reformulates the Birch and Swinnerton-Dyer conjecture in its pre-modularity form as a statement over finite codes in $\mathsf{ACA}_0$, with the refined constant read as a transport cost separating regulator, period, Tamagawa data, and the order of the Tate–Shafarevich group, whose finiteness is reduced to a single cost hypothesis.
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Boyd's conjecture asserts μ_k = r_k L'(E_k,0) for an elliptic curve E_k. The paper's central discovery: for all but seven exceptional k up to 250,000, r_k is the reciprocal of an integer n_k whose p-adic valuations are locally determined by E_k. For p≥5, P(v_p(n_k)=r)=p^{-r}. For p=3, valuations obey congruence and parity laws tied to bad primes ≡1 mod 3. For p=2, v_2(n_k) ≈ ω_odd(k)+2ω_odd(k^2−16)−c(k)+s(k)+X_k with X_k≈NB(3,3/5). The paper conjectures \hat v(k)≤v_2(n_k) for k≠4,8 and proves it conditionally, under Bloch–Kato hypotheses, on an infinite subfamily k=4u.
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For primes p ≡ 4 or 7 mod 9 with 2 not a cube in F_p, write q = p or p² accordingly. The Mordell curve E_{2q} : y² = x³ − 27 q² has analytic and algebraic rank exactly 1, so 2q is a rational cube sum; its Tate–Shafarevich group is finite and its order equals the BSD quotient up to a 2-adic unit. The companion rank-zero curve E_{2q²} satisfies the full BSD formula.
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The core claim is Theorem 1.1: under the assumptions (D,q)=1, and when q is a square also assuming some twist has root number -1, there are infinitely many coprime pairs (u,v) such that the elliptic curve E_{u,v}: Q(u,v)y^2=f(x) has analytic rank one, meaning L'(1/2,E_{u,v}) is nonzero. The engine is Theorem 1.2, a weighted average over squarefree d coprime to 2qD: the sum of r_Q(d)L'(1/2,E^(d))F(d/X) has main term αX log X with α≠0 whenever q is not a square or the root number of E is -1. Because log X grows faster than (log X)^{1/2}(log log X)^3, the nonzero main term leaves no room for all the derivatives to vanish, so infinitely many d in the genus-represented set must give nonvanishing
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The central discovery is a reduction of moment bounds for analytic ranks to estimates for averaged products of Frobenius traces. Using the explicit formula, analytic rank is majorized by a smoothed sum over prime ideals attached to the curve's L-function; averaging over the height-ordered family, the odd moment contributions vanish and the even contributions factor into a leading term governed by a simple integral of the chosen test function. The main term yields the normal-moment expression, and the error terms are controlled by asymptotics for the number of curves with prescribed local conditions. The paper's theorem states that, conditionally, E_K[r_an^m] ≤ sum_{k=0}^{floor(m/2)} m!/((m-2
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There exists a non-trivial Euler system for the symmetric square of a p-adic Hida family of modular forms that interpolates the Loeffler–Zerbes Euler system for ordinary newforms; together with an algebraic functional equation for the associated dual Selmer groups and work of Büyükboduk–Ganguly, this yields a divisibility toward the Iwasawa main conjecture for that symmetric square.
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For a lisse Z_ℓ-sheaf over a global function field K of characteristic p ≠ ℓ, the Pontryagin dual of the Selmer group formed over any Z_ℓ-extension K_∞/K is a finitely generated torsion module over the Iwasawa algebra Λ with μ-invariant equal to zero. This is a positive-characteristic, prime-to-p analogue of Greenberg's μ=0 conjecture, and it yields an analogue of the weak Leopoldt conjecture over K_∞ together with the conclusion that the associated framed deformation ring is a formal power series ring.
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The central claim is that for a K3 surface X over a finite field F_q and a prime p ∤ q, the p-primary Brauer groups Br(X_n)[p^∞] in the constant Z_p-tower X_n = X ⊗ F_{q^{p^n}} are controlled by the transcendental part of the L-function. Specifically, the paper proves the Iwasawa-type formula #Br(X_n)[p^∞] = p^{µ p^n + λ n + ν} for large n, with µ = 0 and λ equal to the λ-invariant of L_tr(X, 1+T). It further establishes the main conjecture Char_Λ(Br(X_∞)[p^∞]^∨) = (L_tr(X, 1+T)), using an intermediate Λ-module Br(X)^{G_∞}[p^∞]^∨ whose characteristic ideal is computed from the determinant of 1 − F γ^{-1} acting on the Tate module of the Brauer group.
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The central claim is that for every prime p congruent to 4 or 7 modulo 9 the equation a^3 + b^3 = p admits solutions a, b in the rational numbers. The proof proceeds by associating to each such p a family of elliptic curves whose Manin-Stevens constants are shown to be units, using the full BSD conjecture for rank-zero curves together with the Unbounded Denominators Conjecture to guarantee that the cubic roots of certain modular functions remain invariant under appropriate congruence subgroups.
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The central claim is Theorem A (equation (1.1)). For a normalised cuspidal eigen-newform f of weight k ≥ 2 whose base change to K satisfies the strong Heegner hypothesis, let fα be its p-stabilisation (with a_p(f) ≠ p^{k/2} when p|N_f). Then the derivative at s = k/2 + 1 of the p-adic L-function L_p(fα/K, s) equals A/B times (1 − p^{k/2}/α)^4 · h_f(z_f, z_f)/(4|D_K|)^{k/2} if p∤N_f, and A/B times 4 · h_f(z_f, z_f)/(4|D_K|)^{k/2} if p||N_f. Here z_f is the Heegner cycle attached to the base change of f, h_f is the p-adic height pairing, and A, B are explicit algebraic constants; under the non-vanishing condition (NV), A/B = 1. The paper proves this by describing the p-adic L-function through
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An element L_{A/L} in the rationalized Iwasawa algebra is built so that it interpolates the special values of twisted Hasse-Weil L-functions of an ordinary elliptic curve A over a function field K relative to any Z_p^d-extension L/K unramified outside ordinary places; after adjustment by explicit local factors it satisfies the same functional equation and specialization formulae as the characteristic ideal of the dual p^infty-Selmer group, and the resulting main conjecture holds in the listed cases with a Grassmannian reduction for higher d.
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By applying the visibility theorem to elliptic curves E over Q that have additive reduction at an odd prime ℓ, one obtains quadratic twists E^D such that Sha(E^D/Q) contains a nontrivial element of order ℓ. For ℓ=3 this produces concrete pairs of non-isomorphic elliptic curves over Q that agree on Birch-Swinnerton-Dyer invariants, Kodaira symbols, and minimal discriminants, but whose Tate-Shafarevich groups are isomorphic and each contain nontrivial 3-torsion.
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On the paper's own terms, the discovery is a systematic way to produce unconditional examples of non-trivial Tate-Shafarevich groups. A finite enumeration of congruences between weight-2 newforms with coefficient fields of degree at most 4 and levels up to 10000 yields, after passing to the corresponding abelian varieties and applying a visibility criterion, abelian surfaces A/Q such that (Z/pZ)^2 embeds into X(A/Q). The key advance over previous constructions is that the congruences are proved rather than conjectured: a generalised finite-coefficient criterion checks only Fourier coefficients up to a prescribed bound and, once the mod-p representation is shown to be irreducible, upgrades to
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The central result is Theorem 1.2: for h = 5, 13, the sum of 2^{s(n)} over square-free n ≤ X with n ≡ h (mod 24) and all prime factors ≡ 1 (mod 4) equals 9 times the number of such n, with an error of O(X (log X)^{-5/8} (log log X)^8). Here s(n) is the 2-Selmer rank, the exponent such that the 2-Selmer group has size 2^{2+s(n)}. The constant 9 is the core arithmetic content: it is not a power of 2 and it reflects that, on average, the 2-Selmer condition in this family behaves like the intersection of three independent affine F2-hyperplanes. The paper also derives from this the corollaries that the density of s(n) = 0 or 2 in the h = 5 family is at least 7/16, and the density of s(n) = 1 or 3
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Theorem 1.1 asserts that for every curve in the computed set C, there exists an explicit infinite family of quadratic twists {E^d} such that each twist satisfies the full Birch–Swinnerton-Dyer conjecture. The set C contains 36,687 curves, about 0.1% of all elliptic curves of conductor below 500,000. For each accepted curve the algorithm certifies analytic rank zero, verifies the odd-prime and 2-primary parts of BSD from an aggregation of published theorems, and then outputs the admissible twist discriminants in a range. The paper further computes the unconditionally known Tate–Shafarevich orders for these twists and compares their normalized distribution with the Gaussian law predicted by ra
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The paper's central claim (Theorem 4.17, Corollary 4.18) is an exact identity: for an elliptic curve E/Q with associated newform f, a real quadratic field K of discriminant d_K prime to the conductor N, and a ring class character χ of conductor c, the completed central derivative Λ′(1/2, Π⊗χ) — equivalently Λ′(E/K, χ, 1) — equals −√d_K/(log ε_K h_K) times half the χ-twisted sum, over ideal classes A of Pic(O_c), of CT⟨⟨f⁺_{0,A}(τ), θ⁺_{L_{A,1}}⊗E_{L_{A,2}}(τ)⟩⟩ plus (vol(U_{A,2})/2) times the automorphic Green's function G_{Z(f_{0,A})} evaluated along the geodesic set G(V_{A,2}). The proof adapts the Bruinier-Yang calculation, replacing the holomorphic projection used in Gross-Zagier by the
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The paper's central discovery is a canonical construction, for any inert RM-divisor D of strong degree zero, of a higher-weight rigid meromorphic cocycle J_{k,D} whose divisor is the prescribed divisor Div_{k,D} supported on Γ-orbits of real-quadratic points. The Green's function is defined as the period pairing G_k(D1,D2)=J_{k,D1}[y^#_{k,D2}] between this cocycle and a canonically chosen homology class attached to D2. The paper proves the resulting pairing recovers the weight-two logarithm of the multiplicative cocycle when k=2, and that for k>2 the construction is governed by the classical correspondence between modular symbols and p-new eigenforms. It then conjectures that for principal D
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The central claim is Theorem 1.7: take σ_n, σ_{n−1} geometrically irreducible Weil local systems of ranks n and n−1 over a smooth projective curve C over F_q, set σ = (σ_n, σ_{n−1}), and assume p > n. After replacing the spectral-action σ-isotypic subspace (1.21) of the compact-support cohomology of GL_n × GL_{n−1}-shtukas by the image of the explicitly constructed injection H^r(C^I, (σ_n ⊗ σ_{n−1})^ϵ) → H^{2(n−1)r}_c(Sht_{GL_n×GL_{n−1},(Stdn⊠Std_{n−1})^ϵ}) (equation 1.22), the paper proves that Conjecture 1.5 and Conjecture 1.6 hold. In particular, the summed self-intersection number of the σ-isotypic parts of the Rankin–Selberg cycles equals q^{dim Bun_{GL_{n−1}}} (ln q)^{−r−2} (d/ds)^r|_{
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The central discovery is an exact formula for the sign in the functional equation of $E_p$: $W_{E_p}=(-1)^{(p-1)/2+((p-2)^2-1)/8}$, equal to $1$ for $p\equiv 1,7\pmod 8$ and $-1$ for $p\equiv 3,5\pmod 8$. The proof computes the local root numbers at $\infty$, $2$, $p$, and $p-2$ separately. At $p$ and $p-2$ the curve has multiplicative reduction whose split or non-split character is governed by Legendre symbols; at $2$ the reduction is of Kodaira type III with fixed 2-adic valuations, so the local sign is read from a published table of such signs. Multiplying the four local signs collapses to $(-1/p)(2/(p-2))$, which is the displayed exponent. Because the curve is modular, a negative root number in the $3,5\bmod 8$ cases implies $L(E_p,1)=0$, i.e. analytic rank at least one. A separate computation shows $E_{73}$ has rank zero, disproving the rank-two prediction for $p\equiv 1\pmod 8$ and leading to the corrected conjecture that the rank is zero or two in that case.
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The central discovery is that reduction type, Tamagawa number, the Birch and Swinnerton-Dyer fudge factor, and the Galois representation are all locally constant functions on the coefficient space of a family of curves defined over a discretely valued field, where the family is taken to be a set of curves obtained by perturbing the defining equations. The author states this as a theorem: within the topology induced by the valuation, every curve has a neighborhood on which these invariants do not change. The reduction type is the key invariant, and the other invariants are shown to share the same local constancy.
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On the paper's own terms, the central discovery is Theorem 1.2: under Assumption 1.1 (which includes $p$ inert in $K$, $p$ large, $f$ ordinary with big image and non-CM, $p \nmid h_K$, $N_f$ squarefree and definite, and a conductor condition on $\chi$), if $\chi_-(\Theta^{\mathrm{Heeg}}_\infty(f,\chi_t)) \neq 0$, then $\dim_E \operatorname{Sel}^{\mathrm{BK}}(K,V_{f,\chi}) = 1$. The proof chains three ingredients: the construction of a split anticyclotomic Euler system $\{\kappa_{n,f,\eta_i}\}$ from improved diagonal classes, with tame norm relations for primes splitting in $K$; the assertion (Corollary 3.29, cited to forthcoming work on split anticyclotomic Euler systems) that nonvanishing of the bottom class $\kappa_{f,\eta_i}$ forces the corresponding Bloch-Kato Selmer group to be one-dimensional; and an explicit reciprocity law showing that $\kappa_{f,\eta_1}\neq 0$ follows from the nonvanishing of a specialized triple-product $p$-adic $L$-function, which in turn factorizes as an essentially nonzero factor times $\alpha_-(\Theta^{\mathrm{Heeg}}_\infty(f,\alpha_t)) \hat\otimes \beta_-(\Theta^{\mathrm{Heeg}}_\infty(f,\beta_t))$. Choosing auxiliary characters so that $\alpha=\chi_t$ and $\beta=\delta^2$ with $L(f/K,\delta^2,k/2)\neq 0$ isolates the Heegner $\theta$ element in the statement.
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The central discovery is stated in Theorem 5.9: there exist infinitely many pairs (E1,E2) of non-isomorphic elliptic curves over Q such that j(E1)≠j(E2), BSD(E1/Q)=BSD(E2/Q), and the Kodaira symbols and minimal discriminants are the same at every prime. The proof exhibits the quadratic twists of a specific pair (the m=1 case of the family E_m: y²=x³+390n x²+1952m²n x, E′_m: y²=x³−195n x²+16·195²n x, with n=m²+64). For every square-free D with gcd(D,2·3·5·13)=1, D≡1 mod 8, and (65/p)=−1 for all p|D, the twisted pair has Sel_φ ≅ Sel_φ̂ ≅ Z/2Z, trivial 2-part of the Tate–Shafarevich groups, rank 0, and even analytic rank; the local conditions then force the BSD data, Kodaira symbols, and minimal discriminants to agree.
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The central assertion is that for an elliptic curve E over Q, no arithmetic invariant strictly smaller than the conductor N_E can appear in a degree-two L-function that satisfies analytic continuation, a functional equation of the shape Lambda_Phi(E,s) = Phi(E)^{s/2} (2 pi)^{-s} Gamma(s) L_Phi(E,s) = w_Phi Lambda_Phi(E,2-s), and a rank bound rank(E) << log Phi(E). By the Modularity Theorem, L(E,s) equals the L-function of a weight-2 newform of level N_E, and that level is minimal. The paper argues that if Phi(E) < N_E, then any such L_Phi would itself correspond to a newform of level Phi(E), contradicting the minimality of N_E. Consequently the paper concludes that the conductor cannot be replaced by a smaller invariant in the analytic framework, and that the classical bound rank(E) << log N_E is the sharpest possible bound of its type.
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The central claim of the paper is Theorem 1.1. Let $E/\mathbb{Q}$ be an elliptic curve with a cyclic $\mathbb{Q}$-rational isogeny of degree $N$. Then: for $N=14,19,43,67,163$, $c(E)=2^n$ with $n\ge1$; for $N=11,27,37$, $c(E)=2^n3^m$ with $m\in\{0,1\}$; for $N=17$, $c(E)=2^n3^m17^k$ with $m,k\in\{0,1\}$ and $n\ge1$; for $N=21$, $c(E)=2^n3^m7^k$ with $m\in\{0,1,2\}$, $k\in\{0,1\}$, $n\ge1$; and for $N=9$ or $15$, $c(E)=2^n3^m\ell^k$ with $m\in\{0,1,2\}$, $k\in\{0,1\}$, where $\ell=3$ or $5$ respectively. The proof shows that for each of these degrees every local Tamagawa factor $c_p(E)$ can only be $1,2,4$, with a single possible factor $3$, $5$, $7$, or $17$ in specified cases. It also proves that the remaining torsion orders $4,5,6,7,8,9,10,12$ admit no such prime bound, and supplies infinite families where $c(E)$ is as small as the constraints allow.
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On the paper's own terms, the central discovery is the rank-loop correspondence: for an elliptic curve $E$ over $\mathbb{Q}$ of rank $r$, the first homology group $H_1(\Phi(E),\mathbb{Z})$ generated by $\mathbb{Q}$-rational points is claimed to be isomorphic to $\mathbb{Z}^r$, so the rank equals the number of topologically independent infinite loops in the four-dimensional embedding. The paper extends this to metric claims: closed geodesics represent rational point classes, squared geodesic lengths equal canonical heights, and the squared torus volume equals the regulator. It also asserts that the average $F_{\mathrm{new}}(E,N)$ grows like $C(\log N)^r$, giving a numerical rank test from local coefficients. Together these statements would turn the first half of BSD into a geometric assertion: both the algebraic rank and the order of vanishing at $s=1$ are the same loop count.
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Write $n$ in the family as $n=m$ or $n=2m$, where $m=\prod_{i=1}^s p_i$ and every $p_i\equiv 1\pmod 4$. For the odd case, choose an integer $b$ with $b^2\equiv -1\pmod{m^2}$ and set $\tau_m=(b+i)/(2m^2)$. The paper establishes $L(E_m,1)=\pi\,|\theta_{\chi_m}(\tau_m)|^2/(4\sqrt{2m})$, where $\theta_\chi(\tau)=\sum_{k\in\mathbb{Z}}\chi(k)e^{2\pi i k^2\tau}$ with $\chi_m$ the Kronecker character. For the even case, with $\tau'_m=(b+m^2+i)/(2m^2)$ and $n=2m$, the formula is $L(E_n,1)=\pi\,|\theta_{\chi_m}(\tau'_m)|^2/\sqrt{2n}$. In particular, $L(E_n,1)=0$ exactly when the corresponding $\theta$ value vanishes.
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Let $f$ be a cuspidal newform and $p \geq 3$ a prime such that $\rho_f$ has large image. Theorem 1.1 states: if the collection of Kurihara numbers $e^{\min,\dagger}$ for $f$ at the central critical point $s = k/2$ does not vanish, then Kato's Kolyvagin system $\kappa_{\mathrm{Kato},\dagger}$ for $T^\dagger_f$ is non-trivial, and there is an isomorphism of co-finitely generated $\mathcal{O}$-modules $\mathrm{Sel}(\mathbb{Q}, W^\dagger_f) \cong (F/\mathcal{O})^{\oplus \,\mathrm{ord}(e^{\min,\dagger})} \oplus \bigoplus_{i\geq 0} (\mathcal{O}/\pi^{e_i}\mathcal{O})^{\oplus 2}$, where $e_i = \tfrac{1}{2}\big(\partial^{(\mathrm{ord}(e^{\min,\dagger})+2i)}(e^{\min,\dagger}) - \partial^{(\mathrm{ord}(e^{\min,\dagger})+2i+2)}(e^{\min,\dagger})\big)$. Consequently the Selmer corank equals $\mathrm{ord}(e^{\min,\dagger})$ and the length of the quotient by the maximal divisible submodule equals $\partial^{(\mathrm{ord})}(e^{\min,\dagger}) - \partial^{(\infty)}(e^{\min,\dagger})$. Theorem 1.2 proves that the non-vanishing of $e^{\min,\dagger}$ is equivalent to the Iwasawa main conjecture localized at the augmentation ideal, which makes the structure theorem unconditional in the good ordinary, p-distinguished, large-image case. The discovery is that discrete Kolyvagin derivatives of L-values, rather than complex analytic or p-adic derivatives, carry the exact bound of the Selmer group, so the result is insensitive to weight, to the local behaviour of $f$ at $p$, and to analytic rank.
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For an irreducible Dedekind scheme $X$ with perfect residue fields, the paper proves two dualities: the functor sending a torsion-free $\mathbb{Z}$-constructible sheaf $F$ to the complex $[T \to \bigoplus_v i_{v,*}E_v]$ (Néron model of the Cartier dual of the generic fibre, with $\mathbb{Z}$-duals of the closed-point stalks) is quasi-inverse to $T\mapsto \mathrm{Ext}^0_{X_{\mathrm{sm}}}(T,\mathbb{G}_m)$, giving a contravariant equivalence between torsion-free constructible tori and torsion-free $\mathbb{Z}$-constructible sheaves; allowing torsion extends this to a contravariant equivalence between all constructible tori and all $\mathbb{Z}$-constructible sheaves. For $X$ proper over $\mathrm{Spec}\,\mathbb{Z}$, it then defines $L_X(T^\bullet,s)$ as an Euler product of geometric-Frobenius determinants on the étale realization, and a Weil-étale fundamental line $\Delta_X(T^\bullet)$ combining the determinant of Weil-étale cohomology with the determinant of the Lie algebra. The central formula is $L^*_X(T^\bullet,0)=\pm\chi_X(T^\bullet)$, where $\chi_X$ is the resulting Weil-étale Euler characteristic; this extends the first-named author's earlier special value theorem by dropping the tame ramification hypothesis.
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The central discovery is a conditional proof that, under the hypotheses (irred) and (MML), the conditions rank_Z E(F)=1 and #III(E/F)_{p^\infty}<\infty imply ord_{s=1}L(E/F,s)=1. The proof moves to a carefully chosen imaginary quadratic extension K/F, proves that the central critical p-adic L-function of the associated Hida family vanishes to order exactly four at weight 2, translates this into order-two vanishing of the complex L-function over K, and then uses the factorization L(E/K,s)=L(E/F,s)L(E^K/F,s) together with a nonvanishing twist to conclude order-one vanishing over F. The paper also applies this to elliptic curves over Q, removing a technical hypothesis from an earlier p-converse theorem.
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On its own terms, the central discovery is Theorem 4.1.13: for an elliptic curve $E/\mathbb{Q}$ and an abelian extension $K/\mathbb{Q}$ satisfying the stated hypotheses, the Bloch–Kato Selmer group $\mathrm{Sel}(\mathbb{Q},E[p^\infty]\otimes \mathcal{O}_d(\chi))$ is determined, as an $\mathcal{O}_d$-module, by the twisted Kurihara numbers $\delta_{n,\chi}$. For $\chi^2=1$ it is $\mathcal{O}_d^r \oplus (\mathcal{O}_d/p^{a_r})^2 \oplus \cdots \oplus (\mathcal{O}_d/p^{a_{s-2}})^2$ with $a_i=(n_i-n_{i+2})/2$; for $\chi^2\neq 1$ it is $\mathcal{O}_d^r \oplus \mathcal{O}_d/p^{n_r-n_{r+1}} \oplus \cdots \oplus \mathcal{O}_d/p^{n_{s-1}-n_s}$, where $n_i$ is the exponent of the ideal $\Theta_{i,\chi}$ generated by the $\delta_{n,\chi}$ with exactly $i$ prime factors. The paper proves that these ideals are Fitting ideals of the dual Selmer group over the needed index range, and the bridge is the equality between the p-adic order of the localisation of the derived Euler-system class and the p-adic order of $\delta_{n,\chi}$. For non-self-dual twists the equality holds at every index; for self-dual (quadratic) twists the functional equation of the associated modular elements supplies the missing parity information. A companion theorem identifies the condition that some $\delta_{n,\chi}$ is a p-adic unit with the Iwasawa main conjecture for the twisted modular form.
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The central discovery is Theorem 5.4: for $G=\mathrm{SL}_\ell$ with $\ell$ prime, the function $m\mapsto L(G,m)=\sum_{[\gamma]}L_S(M_{G_\gamma})$, summed over semisimple conjugacy classes in $G(\mathbb{F}_{q^m})$, is of Lefschetz type; and Theorems 6.5 and 6.6: the same holds for $G=\mathrm{Sp}_4$ and $G=\mathrm{Sp}_6$. Here $M_{G_\gamma}$ is the Artin–Tate motive of the centralizer and $L_S$ is its $S$-modified L-function at $s=0$. In the case where the finite set $S$ has at least two places and the set $T$ is empty, the paper's Theorem 4.2 (conditional on Gross's Conjectures 1.1 and 1.2) identifies $1+(-1)^{\# S\cdot r(G)}\sum_\pi m(\pi)$ with exactly this sum, so these theorems transfer the Lefschetz property to sums of multiplicities of cuspidal representations that are Steinberg at $S$ and unramified elsewhere. Theorems 5.4 and 6.5–6.6 are unconditional statements about L-functions; the representation-theoretic conclusion carries the conjectural trace formula as a hypothesis.
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The central claim is a sandwich theorem for the $\phi$-Selmer group. For a Type I curve $E_a:y^2=x^3+a$, set $K=\mathbb{Q}(\zeta_3,m^{1/3})$, let $L=K(\sqrt{a})$ when $a\notin K^{*2}$ and $L=K\times K$ when $a\in K^{*2}$, and let $S(K)=S_a(K)\cup S_3(K)$ be the finite set of primes where the local description is exceptional. Theorem 3.9 asserts that if $a\notin K^{*2}$, then $\dim_{\mathbb{F}_3}\mathrm{Cl}_S(K)[3]\le \dim_{\mathbb{F}_3}\mathrm{Cl}_S(L)[3]\le \dim_{\mathbb{F}_3}\mathrm{Sel}_\phi(E_a/K)\le \dim_{\mathbb{F}_3}\mathrm{Cl}_S(L)[3]+|S(L)|+r_1+r_2$, and if $a\in K^{*2}$, then $\dim_{\mathbb{F}_3}\mathrm{Cl}_S(K)[3]\le \dim_{\mathbb{F}_3}\mathrm{Sel}_\phi(E_a/K)\le \dim_{\mathbb{F}_3}\mathrm{Cl}_S(K)[3]+|S(K)|+2$. For Type II curves $E_{a,b}:y^2=x^3+a(x-b)^2$, Theorem 4.5 proves the analogous statement with two prime sets $S_1,S_2$: the lower bound uses $S_1$ and the upper bound uses $S_2$. The paper's final application feeds the Type I lower bound into the known growth of the 3-rank of $\mathrm{Cl}(K)$ as $m$ varies, proving Corollary 5.4: for every fixed $a$, $\dim_{\mathbb{F}_3}\mathrm{Sel}_\phi(E_a/K)$ is arbitrarily large.
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The paper's central claim is that the vanishing order $r$ at the central point of a rational L-function is encoded in the normalized Dirichlet coefficient vector $v(L)=(a_p/(d p^{w/2}))_{p<1000}$, in the following sense: on the sub-dataset PRAT$^\star$ of 176,156 primitive rational L-functions (degree 4, motivic weight 1, mostly elliptic curves over number fields and genus-2 curves over $\mathbb{Q}$), LDA predicts $r\in\{0,1,2,3\}$ with 95.9% held-out accuracy and explained variance 0.982, and a CNN trained on the raw coefficient vector reaches over 95% accuracy on each subfamily (ECNF, BMF, HMF, G2Q). Additionally, PCA projections show visible clustering by $r$, and a CNN trained on only the first two principal components reaches about 91% accuracy. The paper interprets these accuracies as evidence that the order of vanishing is a learnable function of finitely many coefficients, not that the models have identified the underlying arithmetic mechanism.
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On the paper's own terms, the central discovery is that a transformer can learn the arithmetic of Frobenius traces from data alone. For the exact-value task, $a_{97}$ is predicted with test MCC $0.4711$ against a most-common-class baseline of $0.082$, and the model captures $|a_{97}|$ better than the sign (sign-agnostic MCC $0.6266$); $a_2$ and $a_3$ show the same pattern. When the target is reduced modulo $2$, the encoder-only transformer reaches accuracies close to $0.94$ and MCCs around $0.84$ for most primes $p<100$ (accuracy $0.9472$ at $p=83$, MCC $0.8703$ at $p=5$), with the lowest results at $p=2$. The authors read this as evidence that the model implicitly computes $a_p \bmod 2$ as an intermediate step, and they corroborate it by showing confusion matrices consistent with parity classes and PCA projections of embeddings and decoder hidden states that cluster by residues modulo $2$, $4$, and in one case $6$.
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At its core the paper claims that for $a\notin K^{*2}$ with $3\nmid a$, the dimension of $\mathrm{Sel}_{\psi_{a,b}}(E_{a,b}/K)$ is bounded below by $\max\{h^3_{S_{1,2}}(L), h^3_{S_{1,3}}(L)+|S_3|-|S_2|-1\}$ and above by $\min\{h^3_{S_{1,2}}(L)+|S_{1,2}(L)|+|S_3|-|S_2|+1, h^3_{S_{1,3}}(L)+|S_{1,3}(L)|+2\}$. Here $L=K(\sqrt{a})$, $h^3_S(L)$ is the 3-rank of the $S$-ideal class group, and $S_1,S_2,S_3$ are explicitly defined finite sets of primes of $K$ determined by the reduction types of $E_{a,b}$ and its dual curve. The argument embeds the isogeny-Selmer group into the norm-one group $L^*/L^{*3}$, compares it with class-group modules of known dimension, and then uses a Selmer-ratio identity involving local component counts to sharpen the crude inclusions. In the special case $S_1=S_2=S_3=\emptyset$, the theorem says the $\psi$-Selmer rank is either $h^3_L$ or $h^3_L+1$.
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On the paper's own terms, the discovery is that the singularities of the natural model are not merely tolerated but are structurally necessary: the exceptional divisors on the resolution are themselves sources of algebraic classes, and the singular fibers are what create the geometric monodromy that makes the period map nontrivial. The proof constructs an extended period morphism (Definition 4.3) that agrees with the ordinary Kuga–Satake period morphism on the smooth locus and extends over the discriminant locus after a simultaneous resolution of the family. It then shows, through rigidity of F-crystals and a flop decomposition of birational maps from the minimal model program, that any curve contracted by the projection to the moduli space is also contracted by the extended period morphism; this uniqueness of periods is what allows special endomorphisms of the associated abelian variety to be compared with divisor classes on the resolution, completing the reduction of the Tate conjecture to a known Tate theorem for special endomorphisms.
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The central assertion is Conjecture 1.3: for an odd prime $p$, a primitive Dirichlet character $\chi$ of squarefree order $d$ with conductor coprime to the conductor of $E$, and an embedding $\iota$ of $\mathbb{Q}(\zeta_d)$ into $\mathbb{C}$, the $p$-part of the principal ideal generated by the normalized value $\iota^{-1}(L(E,\chi))$ is predicted to be $\prod_\theta (h_\theta(\chi(\tau)^{d/d_\theta}), p)$, where $\tau$ generates $\mathrm{Gal}(K_\chi/\mathbb{Q})$, $d_\theta$ is the order of the roots of $h_\theta$, and $h_\theta$ is the minimal polynomial of the matrix giving the action of $\tau$ on the $\theta$-isotypic component of $\Sha(E/K_\chi)[p]$. The paper's supporting results include Theorem 2.2, showing that $p$ divides the normalized L-value of one of two $p$-congruent curves when the other's twisted L-value vanishes, and Corollary 2.3, transferring this to the eigenspace product under BSD and a visualization hypothesis; this verifies Conjecture 1.3 for the curve 9450du1. For the CM curves, Theorem 1.6 records the GRH-conditional verification for nine curves with $p=11$ and all primitive Dirichlet characters factoring through $\mathbb{Q}(\zeta_{11})^+$. In the worked example with the curve $y^2 = x^3 - 262395x + 51731946$, the algorithm outputs eigenspace polynomials $x-3$ and $x-4$, matching the computed factorization $\mathfrak{p}_1\overline{\mathfrak{p}}_1$ of the L-value ideal.
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On the paper's own terms, the central discovery is Theorem 4.1. Let $\bar{\rho}:G_{\mathbb{Q},\Sigma}\to \mathrm{GL}_2(k)$ be an odd residual Galois representation whose image contains a conjugate of $\mathrm{SL}_2(\mathbb{F}_p)$ and whose local behaviour at $p$ avoids the two degenerate character-extension cases, and assume the auxiliary primes outside the ramification set satisfy the local deformation-type conditions of Assumption 3.4. Then the universal zeta morphism $$z_\Sigma:\Delta_\Sigma(T_\Sigma)\otimes Q(\mathbb{T}^m_\Sigma)\xrightarrow{\sim} Q(\mathbb{T}^m_\Sigma)$$ induces an inclusion $\Delta_\Sigma(T_\Sigma)^{-1}\hookrightarrow \mathbb{T}^m_\Sigma$, and the following are equivalent: (1) the classical Iwasawa Main Conjecture holds for one motivic specialization $\lambda$; (2) it holds for every motivic specialization, and the Tamagawa Number Conjecture for the associated motive holds at $p$ whenever $L(f,\chi,r)\neq 0$; (3) the universal Iwasawa Main Conjecture holds for $T_\Sigma$. The word 'universal' means that a single zeta element and a single fundamental line over the Hecke algebra interpolate the zeta morphisms at all classical points of the deformation space.
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The central result, Theorem 11.5, is a divisibility between two codimension-one invariants on the ordinary eigenvariety. For a residual Galois representation satisfying the paper's Assumption 4.7, an interlacing Fontaine–Laffaille regular weight (ordering and $p$-smallness conditions on the weights that make the Bessel period well defined and the local representations crystalline), and $p > 2(n_0+1)$, the paper proves that on an irreducible component $E'$ of the tempered locus (an open dense subset of the normal locus containing all classical points), nonvanishing of the Bessel period $\lambda_J(V)$ implies (a) the Selmer group $H^1_f(F,R_J(\xi,V)_\gamma)$ vanishes over $E'$, (b) the dual Selmer module $X_J(\xi,V)_\gamma$ is torsion over $E'$, and (c) the divisor $2\,\mathrm{char}_{E'}(D_J(\xi,V)_\gamma^H/\lambda_J(V))-\mathrm{char}_{E'}(X_J(\xi,V)_\gamma)$ is effective. An analogous statement holds in the indefinite case: nonvanishing of the cohomological period class $\kappa_J(V)$ forces generic rank one for the Selmer group and its dual, and gives the corresponding effective divisibility for the torsion part of the dual Selmer module. The proof proceeds through two explicit reciprocity laws that relate the period to Selmer classes at auxiliary primes, organized into a bipartite Euler system over the eigenvariety, and then specializes to closed points to compare lengths and divisors.
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A feed-forward neural network classifier trained on subsets of the invariants arising in the Birch-Swinnerton-Dyer conjectural formula yields higher accuracies (>0.9) than any model previously studied. (Abstract) If correct, this means these ML models can classify curves into sha order 4 vs 9 with over 90% accuracy when given some BSD invariants.
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The paper's central claim is that Conjecture 0.1, shown in Lemma 2.9 to be equivalent to Conjecture 6 of [MT87] when the layer is a prime $p$ and $E$ has split multiplicative reduction at $p$, is not true in general. Across 425,713 computed pairs, the predicted congruence failed for 886 pairs; Conjecture 2.12, a consequence of Conjecture 4 in the positive-rank case, failed for 367 pairs. The same data showed no failure for Conjecture 2.11, which already includes the torsion inverse in its coefficient ring. When the minimal ring $R$ is enlarged to include $(\#E(\mathbb{Q})_{\mathrm{Tor}})^{-1}$, the modified Conjecture 3.4 holds for all tested pairs, and the paper proposes Conjectures 3.5 and 3.6 as corrected versions of the original Conjectures 6 and 4.
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The central assertion is Corollary 1.3: the 4-dimensional representation of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on $H^3_{\acute{e}t}(Y_{79}\otimes\bar{\mathbb{Q}}, \mathbb{F}_5)$ is isomorphic to the mod-5 residual representation $\rho_{F_{79},5}$ attached to the non-lift Hecke eigenform $F_{79}\in S_3(K(79))$, where $K(79)$ is the paramodular subgroup of $\mathrm{Sp}_4(\mathbb{Q})$. The authors do not prove full 5-adic modularity; they prove residual modularity modulo 5. The deduction runs through two theorems. Theorem 1.2 establishes a mod-$\lambda$ congruence of Hecke eigenvalues between the Hilbert newforms $f_{79}$ (weight $(2,2)$) and $h_{79}$ (weight $(2,4)$) over $\mathbb{Q}(\sqrt{5})$, and Theorem 1.1 establishes a mod-$\lambda$ congruence between $F_{79}$ and the Johnson-Leung-Roberts lift $\mathrm{JR}(h_{79})$ at paramodular level $79\cdot 5^2$. Together with the Golyshev-van Straten description of $H^3_{\acute{e}t}(Y_{79},\mathbb{F}_5)$ as the induction of the 5-torsion representation of an elliptic curve over $\mathbb{Q}(\sqrt{5})$, the congruences imply the corollary. The paper notes that the conclusion is residual in a strong sense: $F_{79}$ is not a lift of a Hilbert modular form, only congruent modulo 5 to one, and existing modularity lifting theorems do not yet upgrade the residual isomorphism to the 5-adic representation.
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The central claim is that the root number determines the central vanishing order throughout the anticyclotomic tower, up to a finite set of characters. The paper proves that the average of $L^{(v)}(1,\chi)$ over Galois conjugates tends to $2L(1,\kappa)$ in the even case and grows like $2L(1,\kappa)\log(A f(\chi))$ in the odd case, where $\kappa$ is the quadratic Dirichlet character attached to $K$ and $A$ is a constant depending only on $K$. Since a zero of any conjugate forces the average to vanish, the nonvanishing of the average for sufficiently large conductors forces the individual $L$-functions to have the minimal zero order allowed by parity.
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Let $E$ be an elliptic curve over a number field $k$ and let $F/k$ be a dihedral extension of degree $2n$. Fix a prime $p$ not dividing $n$ and assume the p-primary Tate–Shafarevich group $\mathrm{Sha}(E/F)[p^\infty]$ is finite. The paper establishes that for odd $n$, $|\mathrm{Sha}(E/F)[p^\infty]| / |\mathrm{Sha}(E/K)[p^\infty]| = p^{4t}$ for some integer $t$, where $K$ is the quadratic subextension; for even $n$, $|\mathrm{Sha}(E/F)[p^\infty]|$ equals the product $|\mathrm{Sha}(E/K_1)[p^\infty]| |\mathrm{Sha}(E/K_2)[p^\infty]| |\mathrm{Sha}(E/K_3)[p^\infty]|$ times $p^{4t}$, where $K_1, K_2, K_3$ are the three quadratic subextensions. The argument works by applying dihedral Brauer relations to obtain isogenies of Weil restrictions with degree coprime to $p$, so the two sides have identical p-primary Sha orders, and then using the fact that a finite Sha has square order to compare the extra factors. The same relations yield a Galois module statement: over the cyclic subgroup $H$ of order $n$, $\mathrm{Sha}(E/F)[p^\infty] \cong X \oplus X$ and $\mathrm{Sha}(E/K)[p^\infty] \cong X^H \oplus X^H$, and in the odd case with $p^a \equiv -1 \bmod n$ solvable, $|X|/|X^H|$ is a square.
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On the paper's own terms, the central discovery is Theorem 7.12: for an abelian variety $A$ over a global function field $K$ of characteristic $p>0$, a finite Galois extension $F/K$ with group $G$, and an irreducible complex character $\chi$ of $G$ defined over a number field $E$, if the Tate–Shafarevich group $\Sha(A/F)$ is finite and the local Assumption 7.2 holds, then for any place $\lambda$ of $E$ above a prime $\ell$ the normalized leading term $\mathcal{L}_S(A,\chi)$ satisfies the equality of fractional ideals $\mathcal{L}_S(A,\chi) O_{E,\lambda} = (\operatorname{vol}_{D_1}(A/K) \prod_{v \in D_2} |\operatorname{Lie}(A_F)(k_{\tilde v})^{G_{\tilde v}}|)^{\deg \chi} \cdot \operatorname{loc}_{D/F}(A,\chi) \cdot \frac{\operatorname{Reg}^\chi_\lambda}{|G|^{r_{\mathrm{alg}}(\chi)}} \cdot \operatorname{Char}_\lambda(X^\vee_{\chi,\lambda}(A/F))$. The paper's specific contribution is to make this explicit at $\ell=p$ by showing that the equivariant vector bundle $L=\operatorname{Lie}(A_F)(-D_F)$ satisfies the local freeness condition (3.2a) whenever $F/K$ is weakly ramified everywhere and tame at places of non-semistable reduction, so that the new equivariant Riemann–Roch theorem computes the coherent-cohomology factor directly.
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On its own terms, the paper establishes Conjecture 3.8: for matching strongly regular semisimple data, $\partial\mathrm{Orb}((\gamma,u'),1_{(S_{n-1}\times V'_{n-1})(\mathcal{O}_{F_0})})=-\mathrm{Int}(g,u)\,\log q$ in the semi-Lie algebra version, together with the corresponding group version; this is Theorem 15.1, valid for odd $p\ge n$. It also establishes the Jacquet–Rallis fundamental lemma, Conjecture 2.3, for residue field size $q\ge n$ (Theorem 13.9). The proof is global and inductive: local matching identities for orbital integrals are recovered from the equality of two holomorphic modular forms, after subtracting archimedean Green-function corrections and using linear independence of logarithms of distinct primes to separate places.
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The central claim is that in the prime-conductor family of quadratic Dirichlet $L$-functions over function fields, the second moment at the central point is now known to order $g^2$. Explicitly, Theorem 1.1 states that $$\frac{1}{|\mathcal{P}_{2g+1}|}\sum_{P\in\mathcal{P}_{2g+1}} L(1/2,\chi_P)^2 = \frac{$g^{3}$}{3\zeta_q(2)} + \left(\frac{3}{2}+\frac{1}{2q}\right)$g^{2}$ + O_\varepsilon($g^{{3/2+\varepsilon}}$),$$ where $\mathcal{P}_{2g+1}$ is the set of monic irreducible polynomials of degree $2g+1$ over $\mathbb{F}_q[x]$. The $g^3$ term was already known; the new content is the explicit $g^2$ term, which exactly matches the ratios-recipe prediction. The proof achieves this by truncating the Dirichlet series from the approximate functional equation at an interior cutoff, evaluating the tail with Perron's formula as a shifted moment over a circle, and cancelling the small-arc piece against the diagonal. The same machinery yields the leading term of the mean derivative of elliptic-curve twists, and, outside an explicit excluded case, a twist of analytic rank 1.
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Concretely, under the paper's Assumptions 1.1.1 and 1.1.3, the non-anomalous condition, and primitivity of the Heegner Kolyvagin system $\kappa$, Theorem A establishes the full statement of Howard's Main Conjecture: the torsion $\Lambda$-module $M$ satisfies $\operatorname{char}(M)=\operatorname{char}(M)^\iota$, the Selmer group $X$ is pseudo-isomorphic to $\Lambda\oplus M\oplus M$, and $\operatorname{char}(M)=\operatorname{char}(H^1_{F_\Lambda}(K,T)/\Lambda\kappa^{Hg}_1)$. In other words, the divisibility obtained in the original and supersingular formulations is upgraded to equality. The paper also proves a converse direction: assuming Howard's Main Conjecture, $\kappa$ is primitive if and only if no split prime Tamagawa factor is $p$-divisible, and an equivalence theorem states that $\kappa$ is primitive exactly when Howard's Main Conjecture and that Tamagawa condition both hold. The route combines the refined Kolyvagin-system formalism with an anticyclotomic control theorem and an explicit reciprocity law for Heegner points, which ties the BDP $p$-adic $L$-function main conjecture to the index of the Heegner classes.
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Theorem 1.3 is the central claim: under assumptions (QS), (Irred), with 0 and 1 not Hodge-Tate weights of M, and with π unramified and ordinary above p, the equality L(M∨(1),0)=0 implies that the rank of the Selmer group Sel_{p^∞}(M) is positive. Equivalently, a Selmer group of rank 0 forces the central L-value to be nonzero. The author establishes this by specializing an ordinary Hida family of Klingen Eisenstein series to the non-arithmetic point φ0 where the p-adic L-function takes the central value, showing that the specialization is a nonzero p-adic limit form, and then using an Eisenstein ideal argument to convert this nonvanishing into a lower bound on the Selmer group rank.
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With left-handed quarks and leptons in the triplet of SU(2) and the color group SU(N), anomaly cancellation leaves a cubic equation in the hypercharges. Imposing compactness of U(1) makes the hypercharges rational, and after a linear change of variables the equation becomes the Weierstrass curve $Y^2 = X^3 - 3(8N^2+1)X + 16N^4+40N^2-2$. The paper shows that every rational point on this curve gives a rational hypercharge assignment, and conversely. The rational points form the finitely generated abelian Mordell-Weil group of the curve; for N=3 the torsion subgroup is Z9, giving one nontrivial assignment, and for N=9 the torsion subgroup is Z12, giving two inequivalent nontrivial assignments. For all other N the curve has positive rank, so there are infinitely many rational assignments. The rank-zero verdicts for N=3 and N=9 follow from computing the L-value $L(E,1)$ to be nonzero at the central point, invoking the theorem that nonzero L(E,1) forces rank zero for these curves.
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The central claim is that the desingularisation $S$ of $X_{DY}$ is a K3 surface with geometric Picard lattice exactly equal to the sublattice spanned by an explicit list of $24$ divisors: the hyperplane class, seven lines, two conics, and the exceptional curves over the five singular points. The intersection matrix of these divisors has rank $19$ and discriminant $24$, and the discriminant group is $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$; the same lattice is recomputed in Section 6 from an explicit elliptic fibration via the Shioda–Tate formula and height pairings, giving an independent verification. In Section 5 the authors exhibit a quadratic base change that identifies $S$ with the Inose fibration of the Kummer surface $\mathrm{Kum}(E_1 \times E_2)$, where $E_1$ and $E_2$ are $3$-isogenous elliptic curves over $\mathbb{Q}(\sqrt{2},\sqrt{5})$. From the modularity of this pair they obtain, for every prime $p \ge 7$, explicit formulas $|S(\mathbb{F}_p)| = 1 + 17p + (1+(5/p))p + \mu(p) + p^2$ with $\mu(p) = a_p(f)^2 - (10/p)p$, and the analogous formula over $\mathbb{F}_{p^2}$. The stated consequence is that the square root in (2.10) is not rationalisable by a rational change of variables, and that the Drell–Yan surface is neither birational nor isogenous to the K3 surface coming from the two-loop Bhabha-scattering correction.
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Using the explicit relation between newforms and minimal vectors, a representation theoretical trick simplifies the computation of Waldspurger's local period integral for newforms, allowing its evaluation in new cases. As an example, this yields the value of the local integral in a special setting previously used for the 3-part full BSD conjecture.
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The central claim is Theorem 3: if K is a number field, p is odd, E/K has a cyclic p-isogeny φ defined over K and semistable reduction at all primes above p, then for every place v, w(E/K_v) = (−1, K_{v,φ}/K_v) σ_φ(E/K_v), where σ_φ counts the parity of the p-part of #coker/#ker of φ on local points and (−1, K_{v,φ}/K_v) is the local Artin symbol indicating whether −1 is a norm in the cyclic extension generated by ker φ. Because the product of these Artin symbols over all places is 1, the global root number equals the parity of the p-Selmer rank, proving Conjecture 1. For p=2, the analogous identity replaces the Artin symbol by the product of Hilbert symbols (a,−b)(−2a,δ), valid under good ordinary or multiplicative reduction at primes above 2.
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The paper's central claim is that symmetric power $L$-functions of elliptic curves behave exactly as the conjectures predict in every case that can be computed. For each curve $E/\mathbb{Q}$ and each symmetric power $m$, the paper defines $L_m(E,s)=\prod_p U_m(p;s)^{-1}$ from the local Galois representation, with the additive-prime Euler factors taken from [19]; it then completes the $L$-function, evaluates the central value or derivative with two choices of the auxiliary parameter $A$, and treats agreement between the two evaluations as the functional-equation test. The paper reports that all non-CM isogeny classes with conductor below 130,000 pass this test for odd powers at the central point and even powers at the edge of the critical strip. It also asserts specific high-order vanishings in its Table 7 and Bloch-Kato quotients in its Table 8 whose prime factorisations have the divisibility predicted by the special-value conjecture. Read sympathetically, the paper establishes that the analytic and arithmetic conjectures hold for every curve and power it could test.
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The central claim is Theorem 1. For the quartic Fermat K3 surface $S_4$ and the two degree-six weighted surfaces $S_{6A}$ and $S_{6B}$, let $M_\Omega \subset H^2(S_d)$ be the irreducible representation of $\operatorname{Gal}(\mathbb{Q}(\mu_d)/\mathbb{Q})$ determined by the orbit of the holomorphic two-form $\Omega \in H^{2,0}(S_d)$. The Mellin transforms $f_\Omega(S_d,q)$ of the associated motivic L-functions $L_\Omega(S_d,s)$ are cusp forms of weight three for $\Gamma_0(16)$, $\Gamma_0(27)$, and $\Gamma_0(48)$, given by $f_\Omega(S_4,q)=\eta^6(q^4)$, $f_\Omega(S_{6A},q)=\vartheta(q^3)\eta^2(q^3)\eta^2(q^9)$, and $f_\Omega(S_{6B},q)=\eta^3(q^2)\eta^3(q^6)\otimes\chi_3$. For $S_4$ and $S_{6A}$ the L-function factors as $L_\Omega(S_d,s)=L(\vee^2 f_d,s)$, the symmetric-square lift of a weight-two complex-multiplication cusp form coming from an elliptic curve of Brieskorn-Pham type; for $S_{6B}$ the quadratic character $\chi_3$ is required. The paper reads this as a string-theoretic interpretation of motivic L-functions: the modular forms are products of indefinite $\theta$ functions $\Theta^k_{\ell,m}$ tied to the Kac-Peterson string functions of $A_1^{(1)}$, so the arithmetic of the Calabi-Yau is explained by worldsheet data, and the construction is reversible via the twist map.