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Central Values of $L$-Functions of Twisted Modular Forms and Local Polynomials

Charlotte Dombrowsky

TL;DR

The paper provides an explicit, broadly applicable criterion for when the product of central values $L(F\otimes\chi_D,k)L(F\otimes\chi_{D_0},k)$ vanishes by relating it to a local polynomial $P_{k,N,D,D_0}(x)$ via a locally harmonic Maass form built from a Kohnen-type lift. The approach unifies Kohnen–Sakata frameworks with Shimura–Shintani theory and extends to composite levels by incorporating Hecke operators, Atkin–Lehner data, and generalized genus characters. A key contribution is the equivalence between vanishing of the $L$-value product and the equality (or projection equality) of $P_{k,N,D,D_0}$ under a tempered Hecke lift, allowing an efficient computational test by checking a finite set of rational points. The paper also provides an implementation pathway and concrete numerical examples in $S_4(9)$ and $S_4(25)$ to illustrate the practicality of the method, along with a detailed treatment of modularity errors and the local polynomial's limiting behavior. This offers a practical, general criterion for vanishing central $L$-values with potential applications to BSD-type questions and congruent-number-type problems in higher rank settings.

Abstract

In this paper we study the product of two central values of $L$-functions of a twisted modular. We show that it suffices to compute a local polynomial at a finite number of points to decide whether the product is zero. For the proof, we relate the local polynomial to the product of the $L$-functions using a locally harmonic Maass form and building on the Shimura-Shintani correspondence. This extends results from Ehlen, Guerzhoy, Kane and Rolen as well as Males, Mono, Rolen and Wagner.

Central Values of $L$-Functions of Twisted Modular Forms and Local Polynomials

TL;DR

The paper provides an explicit, broadly applicable criterion for when the product of central values vanishes by relating it to a local polynomial via a locally harmonic Maass form built from a Kohnen-type lift. The approach unifies Kohnen–Sakata frameworks with Shimura–Shintani theory and extends to composite levels by incorporating Hecke operators, Atkin–Lehner data, and generalized genus characters. A key contribution is the equivalence between vanishing of the -value product and the equality (or projection equality) of under a tempered Hecke lift, allowing an efficient computational test by checking a finite set of rational points. The paper also provides an implementation pathway and concrete numerical examples in and to illustrate the practicality of the method, along with a detailed treatment of modularity errors and the local polynomial's limiting behavior. This offers a practical, general criterion for vanishing central -values with potential applications to BSD-type questions and congruent-number-type problems in higher rank settings.

Abstract

In this paper we study the product of two central values of -functions of a twisted modular. We show that it suffices to compute a local polynomial at a finite number of points to decide whether the product is zero. For the proof, we relate the local polynomial to the product of the -functions using a locally harmonic Maass form and building on the Shimura-Shintani correspondence. This extends results from Ehlen, Guerzhoy, Kane and Rolen as well as Males, Mono, Rolen and Wagner.
Paper Structure (37 sections, 43 theorems, 271 equations, 7 tables)

This paper contains 37 sections, 43 theorems, 271 equations, 7 tables.

Table of Contents

  1. Outline.
  2. Preliminaries
  3. Modular Forms, Twists and L-functions
  4. Modular Forms of Integer Weight
  5. Modular Forms of Half-Integral Weight
  6. Binary Quadratic Forms
  7. Shimura-Shintani correspondance
  8. Central values of $L$-functions and coefficients of Modular forms of half-integral weight
  9. Newforms of Half-Integral Weight
  10. Eigenspaces of Newforms
  11. Very Newforms
  12. Eigenspaces of integer weight forms under Atkin-Lehner involutions.
  13. Eigenspaces of half-integral weight forms under twisting operators $R_p$
  14. Correspondance
  15. Central Values of $L$-functions.
  16. ...and 22 more sections

Key Result

Theorem 1

Let $k>1$ be an integer. Let $N$ be an odd integer. Let $D,D_0$ be two fundamental discriminants such that $(D,N)=(D_0,N)=1$, $DD_0$ is not a square, $D(-1)^k, D_0(-1)^k>0$ and for all $p|N$, we have $\left( \frac{D}{p}\right)=\left( \frac{D_0}{p}\right)$. Let $F \in S_{2k}(N)$ be a newform. Denote The following are equivalent

Theorems & Definitions (109)

  • Theorem 1
  • Remark 2
  • Example 3
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Proposition 1.6: Proposition 6 in Kohnen1985
  • Remark 1.7
  • ...and 99 more