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$L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings

Jeanine Van Order

TL;DR

The paper develops an integral framework to extract central-derivative values $\Lambda'(E/K,χ,1)$ for an elliptic curve $E/{\bf Q}$ base-changed to a real quadratic field ${K}$ and twisted by ring class characters $χ$. By constructing regularized theta lifts on spin Shimura varieties and evaluating them along anisotropic geodesic spaces, the authors express $\Lambda'(1/2,Π\otimesχ)$ as a χ-twisted sum of automorphic Green’s functions and holomorphic-constant terms, connecting these derivatives to Hirzebruch–Zagier-type divisors and to Birch–Swinnerton-Dyer–type invariants (tate–Shafarevich groups, regulators, and BSD constants). The approach relies on Siegel–Weil, Kudla–Bruinier–Yang, Borcherds products, and the Langlands base-change theory to relate geometric-geodesic data to analytic L-function derivatives. Under an ersatz Heegner hypothesis, the central value vanishes and the derivative captures arithmetic height information, yielding both conditional and unconditional BSD-type consequences and providing a novel real-quadratic analogue of Gross–Zagier–Popa type formulas. The framework opens pathways to interpret real-quadratic Heegner-like phenomena via boundary components of Siegel threefolds and to connect automorphic Green’s functions with arithmetic invariants in rank-one BSD-type scenarios.

Abstract

We derive new integral presentations for central derivative values of $L$-functions of elliptic curves defined over the rationals, basechanged to a real quadratic field $K$, twisted by ring class characters of $K$ in terms of sums along ``geodesics" corresponding to the class group of $K$ of automorphic Green's functions for certain Hirzebruch-Zagier-like arithmetic divisors on Hilbert modular surfaces. We also relate these sums to Birch-Swinnerton-Dyer constants and periods.

$L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings

TL;DR

The paper develops an integral framework to extract central-derivative values for an elliptic curve base-changed to a real quadratic field and twisted by ring class characters . By constructing regularized theta lifts on spin Shimura varieties and evaluating them along anisotropic geodesic spaces, the authors express as a χ-twisted sum of automorphic Green’s functions and holomorphic-constant terms, connecting these derivatives to Hirzebruch–Zagier-type divisors and to Birch–Swinnerton-Dyer–type invariants (tate–Shafarevich groups, regulators, and BSD constants). The approach relies on Siegel–Weil, Kudla–Bruinier–Yang, Borcherds products, and the Langlands base-change theory to relate geometric-geodesic data to analytic L-function derivatives. Under an ersatz Heegner hypothesis, the central value vanishes and the derivative captures arithmetic height information, yielding both conditional and unconditional BSD-type consequences and providing a novel real-quadratic analogue of Gross–Zagier–Popa type formulas. The framework opens pathways to interpret real-quadratic Heegner-like phenomena via boundary components of Siegel threefolds and to connect automorphic Green’s functions with arithmetic invariants in rank-one BSD-type scenarios.

Abstract

We derive new integral presentations for central derivative values of -functions of elliptic curves defined over the rationals, basechanged to a real quadratic field , twisted by ring class characters of in terms of sums along ``geodesics" corresponding to the class group of of automorphic Green's functions for certain Hirzebruch-Zagier-like arithmetic divisors on Hilbert modular surfaces. We also relate these sums to Birch-Swinnerton-Dyer constants and periods.

Paper Structure

This paper contains 41 sections, 23 theorems, 282 equations.

Table of Contents

  1. Introduction
  2. A geometric interpretation
  3. Other remarks
  4. Applications towards Birch-Swinnerton-Dyer
  5. Background on $L$-functions
  6. Equivalences of $L$-functions and symmetric functional equations
  7. The basechange representation
  8. Spin groups and orthogonal groups
  9. Rational quadratic spaces of signature $(2, 2)$
  10. Spin groups and exceptional isomorphisms
  11. Regularized theta lifts and automorphic Green's functions
  12. Setup
  13. Exceptional isomorphisms
  14. Weil representations
  15. Subspaces of Schwartz functions
  16. ...and 26 more sections

Key Result

Lemma 1.1

Let $E$ be an elliptic curve of conductor $N$ defined over ${\bf{Q}}$, and $\pi = \pi(f)$ the cuspidal automorphic representation of $\operatorname{GL}_2({\bf{A}})$ associated to the eigenform $f \in S_2^{\operatorname{new}}(\Gamma_0(N))$ parametrizing $E$. Let $K$ be a real quadratic field of discr for any ring class character $\chi$ of $K$ of conductor $c$ prime to $d_K N$.

Theorems & Definitions (47)

  • Lemma 1.1
  • Theorem 1.2: Theorem \ref{['MAIN']}, Corollary \ref{['Green']}
  • Corollary 1.3: Theorem \ref{['URBSD']}
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • ...and 37 more