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Determination of a pair of newforms from the product of their twisted central values

Pramath Anamby, Ritwik Pal

Abstract

We show that a pair of newforms $(f,g)$ can be uniquely determined by the product of the central $L$-values of their twists. To achieve our goal, we prove an asymptotic formula for the average of the product of the central values of two twisted $L$-functions- $L(1/2, f \times χ)L(1/2, g \times χψ)$, where $(f,g)$ is a pair of newforms. The average is taken over the primitive Dirichlet characters $χ$ and $ψ$ of distinct prime moduli.

Determination of a pair of newforms from the product of their twisted central values

Abstract

We show that a pair of newforms can be uniquely determined by the product of the central -values of their twists. To achieve our goal, we prove an asymptotic formula for the average of the product of the central values of two twisted -functions- , where is a pair of newforms. The average is taken over the primitive Dirichlet characters and of distinct prime moduli.
Paper Structure (15 sections, 4 theorems, 95 equations)

This paper contains 15 sections, 4 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximate functional equation:
  4. Some character relations:
  5. The set-up:
  6. Main Term
  7. Evaluating sdn:
  8. Error Terms
  9. Estimating sdd:
  10. Evaluating $\mathcal{T}(u,v)$
  11. Estimating snd:
  12. Estimating snn:
  13. Proofs of Theorems \ref{['unipair']} and \ref{['thsimul']}
  14. Proof of Theorem \ref{['unipair']}
  15. Simultaneous non-vanishing

Key Result

Theorem 1.1

For $i=1,2$, let $f_i\in S_{k_i}(N_i)$ and $g_i\in S_{l_i}(M_i)$ be Hecke newforms such that for all primitive characters $\chi$ and $\psi$ of prime moduli. Then $k_1=k_2, \ l_1=l_2, \ N_1=N_2, \ M_1 =M_2$ and $f_1=f_2, \ g_1=g_2$.

Theorems & Definitions (6)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 5.1
  • proof
  • Remark 5.2