Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions

Jeanine Van Order

REVIEW 4 major objections 4 minor 67 references

Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Central L-derivatives become heights of Hirzebruch–Zagier divisors

desk verdict A genuinely useful recombination of Bruinier–Yang/AGHMP with a solid sign-correction and a new real-quadratic analogue, but the advertised height formula outruns its hypotheses and the 'two proofs of Gross–Zagier' claim is overstated. read the letter →

arxiv 2510.10303 v2 pith:OOUJHRNS submitted 2025-10-11 math.NT

Jeanine Van Order This is my paper · ORCID

classification math.NT MSC 11F6711G1811F4111G50

keywords Rankin–SelbergL-functionscentralderivativevaluesHirzebruch–ZagierdivisorsarithmeticheightsHilbertmodularsurfacesregularizedthetaliftsHeegnerellipticcurves

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that the central derivative of the Rankin–Selberg L-function attached to an elliptic curve, a quadratic field, and a class-group character can be computed as a χ-weighted sum of arithmetic heights of Hirzebruch–Zagier divisors on the Hilbert modular surface Y0(N)×Y0(N) — in the imaginary-quadratic case — and as a χ-weighted sum of Green's functions along geodesic cycles in the real-quadratic case. This places rank-one leading terms of elliptic curve L-functions inside arithmetic intersection theory. It also gives two distinct derivations of the classical rank-one formula for Heegner divisors, one through the Hilbert modular surface and one through the modular curve itself, and extracts consequences for the refined leading-term conjecture and for half-integral weight Fourier coefficients.

What carries the argument

Quadratic spaces (V_A, Q_A) of signature (2,2) attached to ideal classes A of k; their spin groups identify with GL2×GL2, so the associated Shimura varieties are Hilbert modular surfaces Y0(N)×Y0(N). The argument runs through: regularized theta lifts, which are automorphic Green's functions for special (Hirzebruch–Zagier) divisors; summation formulae expressing the value of these Green's functions along a CM cycle or geodesic set as a constant term plus a Rankin–Selberg L-derivative; an L-function bridge identifying those Rankin–Selberg L-functions with Λ(s−1/2, φ×θ_A); and an arithmetic height formula — imported under hypotheses of odd discriminant and maximal CM order — converting the cons

What would settle it

Compute both sides numerically for one elliptic curve and one imaginary quadratic field with odd discriminant and the sign condition η_k(−N) = −1; a mismatch in even the first digit would refute the theorem. More narrowly, inspect a single Fourier coefficient of the identity behind the L-function bridge—any mismatch there breaks the chain before heights enter.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is an identity of the form Λ′(1/2, φ×θ(χ)) = −2π Σ_{A∈C(O_k)} χ(A) [Ẑ_A(f_{0,A}) : Z(V_{A,0})], where the bracket is an arithmetic height of a Hirzebruch–Zagier divisor on the integral model of Y0(N)×Y0(N) against a CM cycle. The same machinery, with geodesic sets replacing CM cycles, yields an analogous formula for real quadratic k in terms of Green's function sums. A corollary is that the classical rank-one formula for Heegner divisors is recovered from heights on a two-dimensional Shimura surface rather than on the modular curve.

Load-bearing premise

The load-bearing premise is the imported arithmetic height formula, established only for odd discriminants and maximal CM orders; outside those hypotheses the height interpretation of the central derivative is not yet available.

Editorial extensions

If this is right

Rank-one leading terms of elliptic curve L-functions are computed by arithmetic intersection theory on the Hilbert modular surface Y0(N)×Y0(N).
The classical rank-one Heegner-divisor formula follows from a height identity on this surface, giving a second proof.
Real-quadratic twists admit an analogous formula, with geodesic Green's function sums replacing CM-cycle heights.
Heights of Heegner divisors on X0(N) and heights of Hirzebruch–Zagier divisors on X0(N)×X0(N) are explicitly related.
The central derivative values are shown to lie in the ring of periods, conditional on the refined leading-term conjecture up to powers of 2 and 3.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the odd-discriminant/maximal-order hypotheses on the imported height formula can be lifted, the main identity would cover all quadratic twists and give a uniform arithmetic interpretation of rank-one leading terms.
The real-quadratic geodesic formula suggests a p-adic analogue: sums of Green's functions along real cycles may interpolate p-adically, connecting to anticyclotomic Iwasawa invariants.
A single numerical check on an elliptic curve and a quadratic field with even discriminant would delimit how far the height interpretation extends beyond the currently proven cases.
The quadratic-summation machinery is already sketched for higher-dimensional spin Shimura varieties, so the same reduction likely yields higher Gross–Zagier identities for CM cycles in any dimension.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version. The paper is a careful, mostly honest reorganization of Bruinier–Yang and AGHMP into a Kudla-program-shaped formula for Gross–Zagier on X_0(N)×X_0(N), plus a plausibly new real-quadratic analogue via geodesic sums. It deserves a serious referee, but only after the authors restate what is proven and what is borrowed.

What is actually good: Theorem 5.12 is a sign-corrected proof of the Bruinier–Yang summation formula, and that correction is real—useful for anyone who has struggled with the signs in [13]. Theorem 5.14, the geodesic summation, is explicitly flagged as new and is a reasonable extension of [55]; the real-quadratic derivative formula in Corollary 6.8(ii) is new if the proof holds together. The body is also honest: it says Theorem 1.2(i) reproves Bruinier–Yang/Schofer, and it flags the nonmaximal-order problem in §3.3 and Remark 7.6. That honesty is more than many papers manage.

Now the soft spots. First, the stress-test note is on target: Theorem 7.8/Corollary 7.9 is proven only for d_k odd and maximal C_0(L_0), yet the abstract and Theorem 1.4 present the height formula for arbitrary imaginary quadratic k with (d_k,N)=1. That is a direct mismatch between proof and headline claim. It is not fatal—the paper states enough for a careful reader to see the restriction—but the advertised central claim is unsupported for k=\mathbb{Q}(i) and for nonmaximal orders.

Second, Proposition 6.6 carries the L-function bridge, and one step in it is the scalar identity “It is easy to see.” Every constant in the final formulas runs through that normalization. It is probably fixable, but it needs to be written out before I trust the displayed constants.

Third, the “two distinct proofs of Gross–Zagier” language oversells. The signature-(2,2) route derives the formula from the imported Bruinier–Yang/AGHMP height theorem plus L-function bookkeeping, and the comparison in §7.2 uses the Gross–Zagier formula as an input. That is a derivation, not an independent proof. It is fine as a contribution, but the abstract should say “reformulation” or “new derivation,” not “distinct proof.”

Finally, Theorem 5.14, one of the genuinely new results, is proven largely by delegation to [55]. For a theorem the paper itself calls new, that is too thin.

Bottom line: a serious researcher in the Kudla program will want this around, especially for the sign correction and the real-quadratic formulas. It is refereeable if the claims are narrowed to the proven hypotheses and the L-function identity is filled in. I would send it to review, and I would tell the authors to revise the abstract and Theorem 1.4 first.

Referee Report

4 major / 4 minor

Summary. The paper claims a new proof and refinement of the Gross-Zagier formula via regularized theta lifts and arithmetic Hirzebruch-Zagier divisors on Hilbert modular surfaces. For an elliptic curve E/Q of conductor N and a quadratic field k with (d_k,N)=1, it asserts that the central derivative Λ'(E/k,χ,1)=Λ'(1/2,φ×θ(χ)) equals a χ-twisted average of arithmetic heights of Hirzebruch-Zagier divisors on Y_0(N)×Y_0(N) when k is imaginary quadratic, and a sum of Green-function values along geodesic cycles when k is real quadratic. The proof is built on three ingredients: a sign-corrected derivation of the Bruinier-Yang summation formula for regularized theta lifts along CM cycles and a new geodesic analogue (Theorems 5.12 and 5.14); an L-function comparison linking those Rankin-Selberg integrals to Λ(s−1/2,φ×θ(χ)) via vector-valued lifts of φ (Proposition 6.6); and the arithmetic height formula imported from Bruinier-Yang and Andreatta-Goren-Howard-Madapusi Pera (Theorem 7.5). The paper also discusses consequences for BSD constants and periods, and for Fourier coefficients of half-integral weight forms.

Significance. If the central claims hold, the paper gives a genuinely new route to Gross-Zagier through the Kudla programme on Hilbert modular surfaces, and a real-quadratic analogue expressed through geodesic Green functions. The sign-corrected proof of the Bruinier-Yang summation formula and the treatment of the geodesic case are useful contributions. However, the main arithmetic-height formula is not established by a self-contained proof: it rests on the imported Theorem 7.5, whose hypotheses are explicitly flagged in the paper as not covering the general case stated in the abstract and Theorem 1.4. The L-function bridge in Proposition 6.6 also contains an asserted coefficient identity that is load-bearing and only justified as 'easy to see'. The paper is therefore more a conditional synthesis of existing theorems than a complete proof of the advertised headline, although the components are substantial enough that a repaired version could be valuable.

major comments (4)

[Theorem 1.4 / §7.2, Theorem 7.8 and Corollary 7.9] The headline formula is stated for every imaginary quadratic field k with (d_k,N)=1 and η_k(−N)=−1. The proof passes through Theorem 7.5, which is stated only when d_{k(V_0)} is odd and C_0(L_0) is the maximal order O_{k(V_0)}. The paper itself notes in §3.3 that extension to nonmaximal orders 'remains an open problem', and Remark 7.6 says Conjecture 7.4 is not established in general. Thus for k=Q(i), or for lattices whose even Clifford algebra is a nonmaximal order, the asserted equality in Theorem 1.4/Corollary 7.9 is unsupported. The abstract and Theorem 1.4 must either include the missing hypotheses or be accompanied by a proof of the height formula in the missing cases.
[§7.1.2, Eq. (72), and §7.2 proof of Theorem 7.8] Theorem 7.5, as stated in Eq. (72), gives h[ Z(f),Z(V_0) ] = −deg(Z(V_0))/2 · (c_f^+(0,0) κ_{L_0}(0,0) + L'(0,ξ_{1−n/2}(f),θ_{L_0^⊥})). The proof of Theorem 7.8 uses only [ Z_A(f_{0,A}) : Z(V_{A,0}) ] = −h_k/w_k L'(0,g_{φ,A}×θ_{L_{A,0}^⊥}), silently omitting the c_f^+(0,0)κ term. No hypothesis on c_{f_{0,A}}^+(0,0) is stated. Unless one proves the existence of a choice of f_{0,A} with vanishing constant Fourier coefficient (or shows κ(0,0)=0 in this situation), the numerical constants in Theorem 7.8 and Corollary 7.9 are not justified.
[§6.1.5, proof of Proposition 6.6] The central equivalence L^*(2s−2,g_{φ,A}×θ_{L_{A,0}^⊥}) = Λ(s−1/2,φ×θ_A) rests on the scalar identity ⟨⟨g_{φ,A}, θ_{L_{A,0}}⊗E_{L_{A,0}}(·,s;1)⟩⟩ = φ(τ)θ_A(τ)E_A(τ,s;1), asserted just before the display with 'It is easy to see'. This identity is not proved; it requires a careful Hecke/Atkin-Lehner comparison of the Fourier coefficients c_{φ,A}(μ,m) with c_φ(m) and r_A(m), and a normalization statement for the Eisenstein series E_A. A normalization error at this point would propagate into every displayed constant in Theorems 6.7 and 7.8. Please supply a detailed proof or a precise reference for this identity.
[Theorem 7.5 and §7.2, definition of bZ_A(f_{0,A})] Theorem 7.5 and Conjecture 7.4 require the holomorphic part of f to have integral Fourier coefficients so that Z(f) is an arithmetic divisor. The form f_{0,A} produced by Corollary 6.4 is not shown to have this integrality property. If c_{f_{0,A}}^+(μ,m) are not integral, the height pairing in Theorem 7.8 is not defined on the arithmetic Chow group as stated. This is a technical but load-bearing condition: either prove integrality of the chosen f_{0,A}, or formulate the theorem for arithmetic Chow groups with rational coefficients and state the necessary compatibility.

minor comments (4)

[§6.1.3] The identity after Eq. (55) is referred to as '(??)', indicating an unresolved cross-reference. Please fix.
[Throughout] Several typos remain: 'Bruiner' for Bruinier in the proof of Theorem 4.1, 'discrimnant' in §6.1.2, and 'Strömberg' is cited inconsistently in §1 versus §6.1.4.
[§1.1.1, Theorem 1.2(ii)] The real-quadratic summation formula is claimed to be new, but the proof is only described as a 'minor generalization' of Theorem 5.12. It would help readers if the differences (especially the treatment of the nonholomorphic theta series θ_{L_W^⊥}) were spelled out more explicitly.
[§7.3, proof of Theorem 1.5] The derivation of the BSD constant relies on a long list of deep external results (Iwasawa main conjectures, Euler characteristic calculations) without precise statements of the hypotheses needed for each implication. A short appendix or a precise theorem block listing the exact inputs would improve verifiability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central formula is a recombination of external theorems and internal coefficient bookkeeping, not an identity by construction.

full rationale

The derivation of Theorem 7.8 / Corollary 7.9 is a composition of (i) Theorem 5.12, a Green-function summation formula whose proof is supplied in the paper and which is explicitly a reproof of Bruinier-Yang [13] / Schofer [48]; (ii) Proposition 6.6, which identifies the Rankin-Selberg L-function L*(2s-2, g_{\phi,A} x theta_{L^\perp}) with Lambda(s-1/2, \phi x theta_A) via coefficient counting and the Doi-Naganuma lift (57); and (iii) Theorem 7.5, the arithmetic height formula imported from Bruinier-Yang [13] and AGHMP [1]. Each ingredient is either externally established or independently argued in the paper; none defines its target in terms of the headline L-value. The paper itself flags the limits of the imported height formula: Section 3.3 says extending to nonmaximal C_0(V_0) 'remains an open problem,' and Remark 7.6 says Conjecture 7.4 'is not yet established in general.' The abstract and Theorem 1.4 state the height formula for arbitrary imaginary quadratic k with (d_k,N)=1 and eta_k(-N)=-1, while Theorem 7.8 assumes odd d_k and, via Theorem 7.5, maximal C_0(L_0). That is an overbreadth/condition mismatch, not circularity. No self-citation is load-bearing, no fitted parameter is renamed as a prediction, and the comparison with Gross-Zagier is used to derive a relation between heights, not to prove the central L-value formula circularly.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

The central formulas ride on a chain of deep cited theorems (Bruinier–Yang/AGHMP height formula, Doi–Naganuma lifting, Siegel–Weil, Iwasawa main conjectures) plus one narrative use of the target theorem (Gross–Zagier) as an input. The genuinely new local input is Theorem 5.14, whose proof is delegated to [55]. The choice of f_{0,A} is a nominally free datum whose independence is asserted rather than shown.

free parameters (1)

harmonic weak Maass form f_{0,A} with ξ_0(f_{0,A}) = g_{φ,A}
Theorems 1.3/6.7 are stated for “any” such f_{0,A}. The formulas contain CT⟨⟨f⁺_{0,A}, θ_{L^⊥}⊗E_{L_{A,0}}⟩⟩, which depends on the choice; the cancellation with L′(0, g_{φ,A}×θ) is asserted (paragraph after (58)) but not explicitly demonstrated in the text.

assumptions (8)

domain assumption Arithmetic height formula (Bruinier–Yang / AGHMP): h[Ẑ(f),Z(V_0)] = −(h_k/w_k)·L′(0, ξ_{1−n/2}(f)×θ_{L_0^⊥}), valid when d_k is odd and C_0(L_0) is the maximal order.
Theorem 7.5, attributed to Bruinier–Yang [13, Thm 4.7] and Andreatta–Goren–Howard–Madapusi Pera [1, Thm A]; load-bearing for Theorem 7.8 and Corollary 7.9. Remark 7.6 notes the underlying Conjecture 7.4 is not established in general.
ad hoc to paper Classical Gross–Zagier formula (Theorem 1.1) is invoked to compare Heegner-divisor heights and Hirzebruch–Zagier-divisor heights.
§7.2: “By comparison with the Gross–Zagier formula (Theorem 1.1) … we also derive the relation.” Since the paper frames itself as giving a distinct proof of this same theorem, using it as an input is circular in the narrative structure (the two theorems are logically compatible, but this is not a proof of the target).
domain assumption L-function bridge: L⋆(2s−2, g_{φ,A}×θ_{L_{A,0}^⊥}) = Λ(s−1/2, φ×θ_A) and its χ-twisted average (Proposition 6.6).
Relies on the Doi–Naganuma/Strömberg vector-valued lift (Cor 6.4) and the asserted identity ⟨⟨g_{φ,A}, θ⊗E⟩⟩ = φ·θ_A·E_A, justified by “It is easy to see” (§6.1.5). A normalization error here propagates into every displayed constant.
domain assumption Existence of regular flat integral models of spin Shimura varieties (Kisin; Madapusi Pera; Kim–Madapusi Pera).
§7.1.1: needed to define the arithmetic divisors Ẑ_A(µ,m) and the Faltings height pairing used in Theorem 7.8.
standard math Surjectivity of the ξ-operator (Bruinier–Funke), providing f_{0,A} with ξ_0(f_{0,A}) = g_{φ,A}.
§4.2; used throughout §6–§7 to choose the harmonic weak Maass forms appearing in the central formulas.
standard math Rankin–Selberg theory and completed functional equation Λ(s, φ×θ(χ)) = η_k(−N)|d_k N|^{1−2s}Λ(1−s, φ×θ(χ)) (Prop 6.1).
§6.1.2, citing Jacquet, Jacquet–Langlands, and Li; determines the sign condition η_k(−N) = −1 used throughout.
standard math Siegel–Weil formula converting theta averages into Eisenstein series (Theorem 5.1).
Used in Corollary 5.11 and in both summation formulae, Theorems 5.12 and 5.14.
domain assumption Cyclotomic/anticyclotomic Iwasawa main conjectures and Euler characteristic calculations (Kato; Kolyvagin; Rohrlich; Skinner–Urban; Burungale–Skinner–Tian; Castella; Jetchev–Skinner–Wan; Skinner–Zhang; Zhang).
Theorem 1.5's proof (§7.3) is a single paragraph delegating entirely to this literature; the BSD-constant identification is only “up to powers of 2 and 3.”

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Pith. "Pith review of Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions." pith.science (2026). https://pith.science/paper/OOUJHRNS

@misc{pith2026251010303,
  author       = {Pith},
  title        = {Pith review of: Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OOUJHRNS}},
  note         = {Machine review of arXiv:2510.10303}
}

abstract

Let $E$ be an elliptic curve parametrized by a newform $\phi \in S_2(\Gamma_0(N))$. Let $k$ be a quadratic field of discriminant $d_k$ prime to $N$. Given a character $\chi$ of the ideal class group of $k$ with theta series $\theta(\chi)$, we relate the central derivative values $\Lambda'(E/k, \chi, 1) = \Lambda'(1/2, \phi \times \theta(\chi))$ of the $\chi$-twisted $L$-function of $E/k$ to arithmetic heights of Hirzebruch-Zagier divisors on $X_0(N)\times X_0(N)$ when $k$ is imaginary quadratic, and to sums of Green's functions of Hirzebruch-Zagier divisors along real geodesic cycles of $X_0(N) \times X_0(N)$ determined by ideal classes when $k$ is real quadratic. More generally, we refine the higher Gross-Zagier formulae for CM cycles on spin Shimura varieties of any dimension this way. This gives two arithmetic height formulae for $\Lambda'(E/k, \chi, 1)$ when $k$ is imaginary quadratic, as well as a distinct proof of the Gross-Zagier formula, with implied relations between the arithmetic heights of Heegner divisors on $X_0(N)$ and Hirzebruch-Zagier divisors on $X_0(N) \times X_0(N)$. We also explain connections to the refined conjecture of Birch-Swinnerton-Dyer, and to Fourier coefficients of half-integral weight forms.

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Works this paper leans on

67 extracted references · 1 linked inside Pith

[13]

Bruinier and T

J.H. Bruinier and T. Yang,Faltings heights of CM cycles and derivatives ofL-functions, Invent. math.177(2009), 631-681

2009
[55]

J. Van Order,L-functions of elliptic curves in ring class extensions of real quadratic fields via arithmetic theta liftings, preprint, available athttps://www.math.uni-bielefeld.de/vanorder/realquad.pdf
[1]

Andreatta, E.Z

F. Andreatta, E.Z. Goren, B. Howard, and K. Madapusi Pera,Height pairings on orthogonal Shimura varieties, Compos. Math.153(2017), 474-534

2017
[2]

Arthur and L

J. Arthur and L. Clozel,Simple algebras, base change, and the advanced theory of the trace formula, Ann. of Math. Stud. 120Princeton Univ. Press, Princeton, NJ (1989)

1989
[3]

Borcherds,Automorphic forms with singularities on Grassmannians, Invent

R. Borcherds,Automorphic forms with singularities on Grassmannians, Invent. math.132(1998), 491-562

1998
[4]

J.B. Bost, H. Gillet, and C. Soul´ e,Heights of projective varieties and positive Green forms, J. Amer. Math. Soc.7(1994), 903-1027

1994
[5]

Breuil, B

C. Breuil, B. Conrad, F. Diamond, and R. Taylor,On the modularity of elliptic curves overQ: wild3-adic exercises, J. Amer. Math. Soc.14(2001), 843-939

2001
[6]

Bruinier,Borcherds products onO(2, l)and Chern classes of Heegner divisors, Springer Lecture Notes in Math.1780, Springer New York (2002)

J.H. Bruinier,Borcherds products onO(2, l)and Chern classes of Heegner divisors, Springer Lecture Notes in Math.1780, Springer New York (2002)

2002

Show all 67 references

[7]

The 1-2-3 of Modular Forms

J.H. Bruiner,Hilbert Modular Forms and Their Applications, in “The 1-2-3 of Modular Forms”, Lectures at a Summer School in Nordfjordeid, Norway, Ed. Ranestad, Springer Universitext 2007

2007
[8]

Bruinier,On Borcherds products associated with lattices of prime discriminant, Ramanujan J.7(2003), 49-61

J.H. Bruinier,On Borcherds products associated with lattices of prime discriminant, Ramanujan J.7(2003), 49-61

2003
[9]

Bruinier, S

J.H. Bruinier, S. Ehlen, and T. Yang,CM values of higher automorphic Green functions for orthogonal groups, Invent. math.225(2021), 693-785

2021
[10]

Bruinier and J

J.H. Bruinier and J. Funke,On two geometric theta lifts, Duke Math. J.125(2004), 45-90

2004
[11]

J. H. Bruinier, J. Funke, and ¨O. Imamo˘ glu,Regularized theta liftings and periods of modular functionsJ. Reine Angew. Math.703(2015), 43-93

2015
[12]

Bruinier and K

J. Bruinier and K. Ono,Heegner divisors,L-functions and harmonic weak Maass forms, Ann. of Math.172(2010), 2135-2181

2010
[14]

D. Bump, S. Friedberg, and J. Hoffstein,Eisenstein series on the Metaplectic Group and Nonvanishing Theorems for AutomorphicL-Functions and their Derivatives, Ann. of Math.131no. 1 (1990), 53-127

1990
[15]

Burungale, C

A. Burungale, C. Skinner, and Y. Tian,Elliptic curves and Beilinson-Kato elements: rank one aspects(preprint), 2020

2020
[16]

Burungale, C

A. Burungale, C. Skinner, and Y. Tian,The Birch and Swinnerton-Dyer Conjecture: a brief survey(preprint), 2023

2023
[17]

Burgos, J

J. Burgos, J. Kramer, and J. K¨ uhn,Cohomological arithmetic Chow groups, J. Inst. Math. Jussieu6(2007), 1-178

2007
[18]

Castella,On thep-part of the Birch-Swinnerton-Dyer formula for multiplicative primes, Camb

F. Castella,On thep-part of the Birch-Swinnerton-Dyer formula for multiplicative primes, Camb. J. Math6no. 1 (2018), 1-23

2018
[19]

Cox,Primes of the formx 2 +ny 2: Fermat, Class Field Theory, and Complex Multiplication, John Wiley & Sons (1989)

D.A. Cox,Primes of the formx 2 +ny 2: Fermat, Class Field Theory, and Complex Multiplication, John Wiley & Sons (1989)

1989
[20]

Doi and H

K. Doi and H. Naganuma,On the functional equation of certain Dirichlet series, Invent. math.9(1969), 1-14

1969
[21]

Erd´ elyi, W

A. Erd´ elyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi,Tabls of Integral Transforms, vol. I, McGraw-Hill (1954)

1954
[22]

Ehlen,CM values of regularized theta lifts and harmonic Maass forms of weight one, Duke Math

S. Ehlen,CM values of regularized theta lifts and harmonic Maass forms of weight one, Duke Math. J. (13)166(2017), 2447-2519

2017
[23]

Eichler and D

M. Eichler and D. Zagier,The Theory of Jacobi Forms, Progress in Mathematics55, Birkh¨ auser Boston (1985)

1985
[24]

Gelbart,Weil’s representation and the spectrum of the metaplectic group, Lecture Notes in Math.530, Springer (1976)

S.S. Gelbart,Weil’s representation and the spectrum of the metaplectic group, Lecture Notes in Math.530, Springer (1976)

1976
[25]

Automorphic forms, representations and L-functions

P. G´ erardin and J.-P. Labesse,The solution of a base change problem forGL(2) , in “Automorphic forms, representations and L-functions”, Pt. 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., (1979), 115-133

1979
[26]

Gross and D

B. Gross and D. Zagier,Heegner points and derivatives ofL-series, Invent. math.84(1986), 225-320

1986
[27]

Gross, W

B. Gross, W. Kohnen, and D. Zagier,Heegner points and derivatives ofL-series. II, Math. Ann.278(1987), 497-562

1987
[28]

Harris,Arithmetic vector bundles and automorphic forms on Shimura varieties, Invent

M. Harris,Arithmetic vector bundles and automorphic forms on Shimura varieties, Invent. math.82(1985), 151-189

1985
[29]

Howard and K

B. Howard and K. Madapusi Pera,Arithmetic of Borcherds Products, Ast´ erisque421(2020), 187-297

2020
[30]

Howard and T

B. Howard and T. Yang,Intersections of Hirzebruch-Zagier Divisors and CM Cycles, Lecture Notes in Math.2041, Springer (2012)
[31]

Jacquet,Automorphic forms onGL(2), Part II, Springer Lecture Notes in Math., New York, 1972

H. Jacquet,Automorphic forms onGL(2), Part II, Springer Lecture Notes in Math., New York, 1972

1972
[32]

Jacquet and R

H. Jacquet and R. Langlands,Automorphic forms onGL(2), Springer Lecture Notes in Math.278, New York, 1970

1970
[33]

Jetchev, C

D. Jetchev, C. Skinner, and X. Wan,The Birch and Swinnerton-Dyer formula for elliptic curves of analytic rank one, Camb. J. Math.5no. 3 (2017), 369-434

2017
[34]

Katop-adic Hodge theory and values of zeta functions of modular forms, Ast´ erisque295(2004), 117-290

K. Katop-adic Hodge theory and values of zeta functions of modular forms, Ast´ erisque295(2004), 117-290

2004
[35]

Katok and P

S. Katok and P. Sarnak,Heegner Points, Cycles, and Maass Forms, Israel J. Math.84(1993), 193-227

1993
[36]

Kim and K

W. Kim and K. Madapusi Pera, 2-adic integral canonical models and the Tate conjecture in characteristic2, Forum Math. Sigma4(2016), e28

2016
[37]

Kisin,Integral models for Shimura varieties of abelian type, J

M. Kisin,Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc.23(2010), 967-1012

2010
[38]

Kolyvagin,Finiteness ofE(Q)andX(E,Q)for a class of Weil curves, Math

V. Kolyvagin,Finiteness ofE(Q)andX(E,Q)for a class of Weil curves, Math. USSR Izv.32(3) (1989), 523-541

1989
[39]

Kontsevich and D.B

M. Kontsevich and D.B. Zagier,Periods, In Mathematics Unlimited–2001 and Beyond (B. Engquist and W. Schmid, eds.), Springer, Berlin-Heidelberg-New York (2001), 771-808

2001
[40]

Kudla,Algebraic cycles on Shimura varieties of orthogonal type, Duke Math

S.S. Kudla,Algebraic cycles on Shimura varieties of orthogonal type, Duke Math. J.86(1997), 39-78

1997
[41]

Kudla,Integrals of Borcherds Forms, Compos

S.S. Kudla,Integrals of Borcherds Forms, Compos. Math.137(2003), 293-349

2003
[42]

Madapusi Pera,Integral canonical models for spin Shimura varieties, Compositio Math.152(2016), 769-824

K. Madapusi Pera,Integral canonical models for spin Shimura varieties, Compositio Math.152(2016), 769-824

2016
[43]

Langlands,Base change forGL(2), Annals of Math

R. Langlands,Base change forGL(2), Annals of Math. Stud.96, Princeton University Press (1980). 82

1980
[44]

W.-C. W. Li,L-series of Rankin Type and Their Functional Equations, Math. Ann.224(1979), 135-166

1979
[45]

Naganuma,On the coincidence of two Dirichlet series associated with cusp forms of Hecke’s “Neben”-type and Hilbert modular forms over a real quadratic field, J

H. Naganuma,On the coincidence of two Dirichlet series associated with cusp forms of Hecke’s “Neben”-type and Hilbert modular forms over a real quadratic field, J. Math. Soc. Japan25(1973), 547-555

1973
[46]

Rohrlich,OnL-functions of elliptic curves and cyclotomic towers, Invent

D. Rohrlich,OnL-functions of elliptic curves and cyclotomic towers, Invent. math.75(1984), 409-423

1984
[47]

N. R. Scheithauer,The Weil representation ofSL 2(Z)and some applications, Int. Math. Res. Not. IMRN2009(2009), no. 8, 1488–1545

2009
[48]

Schofer,Borcherds forms and generalizations of singular moduli, J

J. Schofer,Borcherds forms and generalizations of singular moduli, J. reine angew. Math.69(2009), 1-36

2009
[49]

Skinner and E

C. Skinner and E. Urban,The Iwasawa main conjecture forGL(2), Invent. math.195(2014), 1-277

2014
[50]

Skoruppa and D

N. Skoruppa and D. Zagier,Jacobi forms and a certain space of modular forms, Invent. math.94(1988), 113-146

1988
[51]

Skinner and W

C. Skinner and W. Zhang,Indivisibility of Heegner points in the multiplicative case,arxiv:1407.1099

arXiv
[52]

Str¨ omberg,On liftings of modular forms and Weil representations, Forum Math.36(1) (2024), 33-52

F. Str¨ omberg,On liftings of modular forms and Weil representations, Forum Math.36(1) (2024), 33-52

2024
[53]

Taylor and A

R. Taylor and A. Wiles,Ring theoretic properties of certain Hecke algebras, Ann. of Math.141(1995), 553-572

1995
[54]

van der Geer,Hilbert Modular Surfaces, Ergeb

G. van der Geer,Hilbert Modular Surfaces, Ergeb. Math. Grenzgeb., Springer (1988)

1988
[56]

Waldspurger,Correspondence de Shimura, J

J.L. Waldspurger,Correspondence de Shimura, J. Math. Pure Appl.59(1980), 1-113

1980
[57]

Waldspurger,Sur les coefficients de Fourier des formes modulaires do poids demi-entiers, J

J.L. Waldspurger,Sur les coefficients de Fourier des formes modulaires do poids demi-entiers, J. Math. Pures Appl.60 (9) no. 4 (1981), 375-484

1981
[58]

Waldspurger,Sur les valeurs de certaines fonctionsLautomorphes en leur centre de sym´ etrie, Compos

J.-L. Waldspurger,Sur les valeurs de certaines fonctionsLautomorphes en leur centre de sym´ etrie, Compos. Math.54 (1985), 173-242

1985
[59]

Wiles,Modular elliptic curves and Fermat’s last theorem, Ann

A. Wiles,Modular elliptic curves and Fermat’s last theorem, Ann. of Math.141(1995), 443-551

1995
[60]

Yuan, S.-W

X. Yuan, S.-W. Zhang, and W. Zhang,The Gross-Zagier Formula for Shimura Curves, Ann. of Math. Stud.184, Princeton University Press (2013)

2013
[61]

Zhang,An isomorphism between scalar-valued modular forms and modular forms for Weil representations, Ramanujan J.37(2015), 181-201

Y. Zhang,An isomorphism between scalar-valued modular forms and modular forms for Weil representations, Ramanujan J.37(2015), 181-201

2015
[62]

Zagier,Modular Forms Associated to Real Quadratic Fields, Invent

D. Zagier,Modular Forms Associated to Real Quadratic Fields, Invent. math.30(1975), 1-46

1975
[63]

Zhang,Gross-Zagier formula forGL 2, Asian J

S.-W. Zhang,Gross-Zagier formula forGL 2, Asian J. Math.5no. 2 (2001), 183-290

2001
[64]

Heegner points and RankinL-series

S.-W. Zhang,Gross-Zagier formula forGL 2. II, in “Heegner points and RankinL-series”, MSRI Publications49, 191-214
[65]

Zhang,Heights of Heegner cycles and derivatives ofL-series, Invent

S.-W. Zhang,Heights of Heegner cycles and derivatives ofL-series, Invent. math.130(1997), 99-152

1997
[66]

Zhang,Heights of Heegner points on Shimura curves, Ann

S.-W. Zhang,Heights of Heegner points on Shimura curves, Ann. of Math.153(2001), 27-147

2001
[67]

Zhang,Selmer groups and the indivisibility of Heegner points, Camb

W. Zhang,Selmer groups and the indivisibility of Heegner points, Camb. J. Math.2no. 2 (2014), 191-253. 83

2014

Pith tools

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[1] [13]

Bruinier and T

J.H. Bruinier and T. Yang,Faltings heights of CM cycles and derivatives ofL-functions, Invent. math.177(2009), 631-681

2009

[2] [55]

J. Van Order,L-functions of elliptic curves in ring class extensions of real quadratic fields via arithmetic theta liftings, preprint, available athttps://www.math.uni-bielefeld.de/vanorder/realquad.pdf

[3] [1]

Andreatta, E.Z

F. Andreatta, E.Z. Goren, B. Howard, and K. Madapusi Pera,Height pairings on orthogonal Shimura varieties, Compos. Math.153(2017), 474-534

2017

[4] [2]

Arthur and L

J. Arthur and L. Clozel,Simple algebras, base change, and the advanced theory of the trace formula, Ann. of Math. Stud. 120Princeton Univ. Press, Princeton, NJ (1989)

1989

[5] [3]

Borcherds,Automorphic forms with singularities on Grassmannians, Invent

R. Borcherds,Automorphic forms with singularities on Grassmannians, Invent. math.132(1998), 491-562

1998

[6] [4]

J.B. Bost, H. Gillet, and C. Soul´ e,Heights of projective varieties and positive Green forms, J. Amer. Math. Soc.7(1994), 903-1027

1994

[7] [5]

Breuil, B

C. Breuil, B. Conrad, F. Diamond, and R. Taylor,On the modularity of elliptic curves overQ: wild3-adic exercises, J. Amer. Math. Soc.14(2001), 843-939

2001

[8] [6]

Bruinier,Borcherds products onO(2, l)and Chern classes of Heegner divisors, Springer Lecture Notes in Math.1780, Springer New York (2002)

J.H. Bruinier,Borcherds products onO(2, l)and Chern classes of Heegner divisors, Springer Lecture Notes in Math.1780, Springer New York (2002)

2002