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Symmetric square type $L$-series

Ingmar Metzler

TL;DR

The paper extends Shimura's symmetric square L-series to vector-valued modular forms transforming under the Weil representation, defining L_{(λ,t)}(f,s) via Fourier data of f and establishing its analytic properties. It develops a compatible Hecke theory for vector-valued forms, derives explicit formulas for Fourier coefficients under Hecke operators, and uses Rankin–Selberg integrals with theta- and Eisenstein-series to obtain meromorphic continuation and functional equations. A central achievement is the construction of Euler product expansions for L-series attached to Hecke eigenforms, with detailed local factors incorporating Gauss sums and Weil representation data. The work thus furnishes analytic tools and product expansions to study automorphic lifts, representation numbers, and higher-rank generalizations within the Weil-representation framework.

Abstract

We construct symmetric square type $L$-series for vector-valued modular forms transforming under the Weil representation associated to a discriminant form. We study Hecke operators and integral representations to investigate their properties, deriving functional equations and infinite product expansions.

Symmetric square type $L$-series

TL;DR

The paper extends Shimura's symmetric square L-series to vector-valued modular forms transforming under the Weil representation, defining L_{(λ,t)}(f,s) via Fourier data of f and establishing its analytic properties. It develops a compatible Hecke theory for vector-valued forms, derives explicit formulas for Fourier coefficients under Hecke operators, and uses Rankin–Selberg integrals with theta- and Eisenstein-series to obtain meromorphic continuation and functional equations. A central achievement is the construction of Euler product expansions for L-series attached to Hecke eigenforms, with detailed local factors incorporating Gauss sums and Weil representation data. The work thus furnishes analytic tools and product expansions to study automorphic lifts, representation numbers, and higher-rank generalizations within the Weil-representation framework.

Abstract

We construct symmetric square type -series for vector-valued modular forms transforming under the Weil representation associated to a discriminant form. We study Hecke operators and integral representations to investigate their properties, deriving functional equations and infinite product expansions.

Paper Structure

This paper contains 11 sections, 40 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Gauss sums
  4. Modular forms
  5. Eisenstein series
  6. Hecke theory
  7. Hecke action on Fourier coefficients
  8. $L$-series
  9. Definition and convergence
  10. Rankin Selberg realisation
  11. Product expansions

Key Result

Theorem 1.1

Assume $k \geq 2$, $L = \mathbb{Z}\ell \oplus L_2$ for some lattice $L_{2}$ and $\ell \in L$ with $\mathop{\mathrm{q}}\nolimits(\ell) > 0$. Let $E_{L_2,k_2}(\,\cdot\,,s)$ be the Eisenstein series from Definition de:VVEisensteinseriesnonholomorphic, assume $k_2 = k - 1/2$ and write ${\ell_0} = \tfrac of holomorphic functions in $\overline{s}$ for $\Re(s) > \tfrac{5 - 2k }{4}$. Consequently, $L_{(\o

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 50 more