Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Ranking-Selberg integral on ${\bf GSp(2)}$ for square free levels

Seiji Kuga, Masao Tsuzuki

TL;DR

The paper develops an explicit global–local framework for the Rankin–Selberg integral on ${\bf GSp}_2$, expressing the adelic integral for vector-valued Siegel cusp forms of square-free level in terms of the spinor $L$-function. It combines Eisenstein series with Bessel models, employs Pitale–Schmidt newvectors, and computes a wide array of local zeta integrals across unramified and ramified cases to obtain exact local–global identities. A spectral average is derived, yielding an explicit formula involving $\widehat{L}(s+1/2,\pi,\mu)$ and local periods, with asymptotic results showing vanishing contributions from old forms and certain lifts in the level-aspect. The work also provides detailed computations of ramified Gross–Prasad-type local periods and sets up a robust framework for relative trace formula applications in this setting, linking global automorphic data to precise local invariants.

Abstract

We explicitly compute the Rankin-Selberg type integral introduced by Piatetski-Shapiro over adeles for vector-valued Siegel cusp forms of square-free levels $Γ_0(N)$. On the way, for particular test functions in the Bessel models of irreducible admissible representations, exact evaluations of the local zeta-integrals are given.

The Ranking-Selberg integral on ${\bf GSp(2)}$ for square free levels

TL;DR

The paper develops an explicit global–local framework for the Rankin–Selberg integral on , expressing the adelic integral for vector-valued Siegel cusp forms of square-free level in terms of the spinor -function. It combines Eisenstein series with Bessel models, employs Pitale–Schmidt newvectors, and computes a wide array of local zeta integrals across unramified and ramified cases to obtain exact local–global identities. A spectral average is derived, yielding an explicit formula involving and local periods, with asymptotic results showing vanishing contributions from old forms and certain lifts in the level-aspect. The work also provides detailed computations of ramified Gross–Prasad-type local periods and sets up a robust framework for relative trace formula applications in this setting, linking global automorphic data to precise local invariants.

Abstract

We explicitly compute the Rankin-Selberg type integral introduced by Piatetski-Shapiro over adeles for vector-valued Siegel cusp forms of square-free levels . On the way, for particular test functions in the Bessel models of irreducible admissible representations, exact evaluations of the local zeta-integrals are given.
Paper Structure (34 sections, 42 theorems, 242 equations)

This paper contains 34 sections, 42 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Embedding of groups
  4. Open compact subgroups at finite places
  5. Maximal compact subgroup at the archimedean place
  6. Haar measures
  7. Ray class characters
  8. Eisenstein series and Rankin-Selberg integral
  9. Eisenstein series
  10. Rankin-Selberg integral and the basic identity
  11. Automorphic forms and Fourier coefficients
  12. Sign condition
  13. Automorphic representations
  14. Basic assumptions
  15. Bessel models
  16. ...and 19 more sections

Key Result

Lemma 2.1

Let $F$ be a field with characteristic $0$.

Theorems & Definitions (76)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 66 more