claims depot shelf
Birch and Swinnerton-Dyer
Formal claims (Lean)
Stated claims
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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 abstract states: "the reduction type in these families is locally constant in the topology induced by the valuation" and "We also derive local constancy results for some related invariants, such as the Tamagawa number, the Birch and Swinnerton-Dyer 'fudge factor' and the Galois representation." If correct, the paper establishes that all these invariants are unchanged on sufficiently small p-adic neighborhoods in the coefficient parameter space.
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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 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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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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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.