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

Jeanine Van Order

TL;DR

This work develops two distinct proofs of the Gross–Zagier formula using sums of automorphic Green's functions realized as regularized theta lifts, including a construction on the Hilbert modular surface $X_0(N)\times X_0(N)$ via arithmetic Hirzebruch–Zagier divisors. It presents a comprehensive framework for rational quadratic spaces of signature $(n,2)$ and the associated $\operatorname{GSpin}$ Shimura varieties, producing explicit summation formulas along CM cycles and geodesic sets that relate Green's function sums to derivatives of Rankin–Selberg $L$-functions. A key part of the paper identifies spaces of signature $(2,2)$ tied to quadratic fields with exceptional isomorphisms $\operatorname{GSpin}(V_A)\cong GL_2^2$, enabling integral presentations of standard Rankin–Selberg $L$-functions via lifts like the Doi–Naganuma correspondence and the Shimura correspondence. The authors connect these analytic constructions to arithmetic geometry through arithmetic Hirzebruch–Zagier divisors on Hilbert modular surfaces, deriving BSD-type height formulas and linking central derivatives to arithmetic heights and periods, with extensions to the (1,2) signature and conjectural relations to half-integral weight Fourier coefficients. They also spell out a robust web of equivalences among $L$-functions arising from different realizations and lifts, providing a unifying perspective that incorporates Langlands–Siegel–Weil theory, base change, and vector-valued lifts, with implications for understanding special values and derivatives in the BSD program. Overall, the work deepens the interaction between automorphic forms, arithmetic geometry, and $L$-functions, offering new tools to express central derivatives as geometric/arithmetic data on modular surfaces and Hilbert modular varieties.

Abstract

We give two distinct proofs of the Gross-Zagier formula in terms of sums of automorphic Green's functions realized as regularized theta lifts, including one involving arithmetic Hirzebruch-Zagier divisors on the Hilbert modular surface $X_0(N) \times X_0(N)$. We then describe applications to the refined conjecture of Birch and Swinnerton-Dyer. Through these calculations, we also describe known and conjectural relations of the central derivative values of Rankin-Selberg $L$-functions that appear to Fourier coefficients of certain half-integral weight forms.

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

TL;DR

This work develops two distinct proofs of the Gross–Zagier formula using sums of automorphic Green's functions realized as regularized theta lifts, including a construction on the Hilbert modular surface via arithmetic Hirzebruch–Zagier divisors. It presents a comprehensive framework for rational quadratic spaces of signature and the associated Shimura varieties, producing explicit summation formulas along CM cycles and geodesic sets that relate Green's function sums to derivatives of Rankin–Selberg -functions. A key part of the paper identifies spaces of signature tied to quadratic fields with exceptional isomorphisms , enabling integral presentations of standard Rankin–Selberg -functions via lifts like the Doi–Naganuma correspondence and the Shimura correspondence. The authors connect these analytic constructions to arithmetic geometry through arithmetic Hirzebruch–Zagier divisors on Hilbert modular surfaces, deriving BSD-type height formulas and linking central derivatives to arithmetic heights and periods, with extensions to the (1,2) signature and conjectural relations to half-integral weight Fourier coefficients. They also spell out a robust web of equivalences among -functions arising from different realizations and lifts, providing a unifying perspective that incorporates Langlands–Siegel–Weil theory, base change, and vector-valued lifts, with implications for understanding special values and derivatives in the BSD program. Overall, the work deepens the interaction between automorphic forms, arithmetic geometry, and -functions, offering new tools to express central derivatives as geometric/arithmetic data on modular surfaces and Hilbert modular varieties.

Abstract

We give two distinct proofs of the Gross-Zagier formula in terms of sums of automorphic Green's functions realized as regularized theta lifts, including one involving arithmetic Hirzebruch-Zagier divisors on the Hilbert modular surface . We then describe applications to the refined conjecture of Birch and Swinnerton-Dyer. Through these calculations, we also describe known and conjectural relations of the central derivative values of Rankin-Selberg -functions that appear to Fourier coefficients of certain half-integral weight forms.

Paper Structure

This paper contains 81 sections, 51 theorems, 573 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Quadratic summation formulae
  4. Spaces of signature $(2,2)$ associated to quadratic fields and relations to Rankin-Selberg $L$-functions
  5. Relations to arithmetic Hirzebruch-Zagier divisors, Birch-Swinnerton-Dyer constants, and periods
  6. Relation to quadratic spaces of signature $(1,2)$ and Fourier coefficients of half-integral weight forms
  7. Quadratic spaces and spin groups
  8. Quadratic spaces associated to class groups of quadratic fields
  9. Anisotopic subspaces
  10. Spin groups
  11. General characterization
  12. Exceptional isomorphisms
  13. $\operatorname{GSpin}$ Shimura varieties
  14. Complex Shimura varieties
  15. Special divisors
  16. ...and 66 more sections

Key Result

Theorem 1.1

Let $E$ be an elliptic curve of conductor $N$, parametrized by a cuspidal newform $\phi \in S_2^{\operatorname{new}}(\Gamma_0(N))$. Let $k$ be an imaginary quadratic field of discriminant $d_k$ prime to $2 N$ and odd Dirichlet character $\eta_k(\cdot) = (\frac{d_k}{\cdot})$. Assume the Heegner hypot Here, $\vert \vert \phi \vert \vert^2 = \langle\phi, \phi \rangle$ denoted the Petersson inner prod

Theorems & Definitions (110)

  • Theorem 1.1: Gross-Zagier
  • Theorem 1.2: Theorem \ref{['BY4.7']} and \ref{['realquad']}
  • Theorem 1.3: Theorem \ref{['RSIP']} and Corollary \ref{['simplified']}
  • Theorem 1.4: Theorem \ref{['main']}, Corollary \ref{['ec']}, Corollary \ref{['heightrelation']}
  • Theorem 1.5
  • Theorem 1.6: Theorem \ref{['metatrace']}
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 100 more