Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The standard $L$-function attached to a vector valued modular form

Oliver Stein

Abstract

We define two $L$-functions associated to a common vector valued eigenform $f$ transforming with the ``finite'' Weil representation. The first one can be seen as a standard zeta function defined by the eigenvalues of $f$. The second one can be interpreted as standard $L$-function defined as an Euler product where each $p$-factor is a rational function in terms of two unramified characters of the $p$-adic field $\Q_p$. We show that both $L$-functions are related and prove further that they both can be continued meromorphically to the whole complex $s$-plane.

The standard $L$-function attached to a vector valued modular form

Abstract

We define two -functions associated to a common vector valued eigenform transforming with the ``finite'' Weil representation. The first one can be seen as a standard zeta function defined by the eigenvalues of . The second one can be interpreted as standard -function defined as an Euler product where each -factor is a rational function in terms of two unramified characters of the -adic field . We show that both -functions are related and prove further that they both can be continued meromorphically to the whole complex -plane.

Paper Structure

This paper contains 11 sections, 13 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Review of St3
  4. Local Weil representation $\omega_p$, St3, Sect. 3
  5. The Hecke algebras $\mathcal{H}(\mathbb{Q}_p//\mathcal{K}_p, \omega_p)$ (St3, Sect. 4
  6. Vector valued automorphic forms and the action of $\mathcal{H}(\mathcal{Q}_p//\mathcal{K}_p,\omega_p)$, Sect. 5
  7. Hecke operators $T(m(p^k,p^l))$ and standard zeta-functions
  8. Standard $L$-function of a common Hecke eigenform
  9. Standard $L$-function of a vector valued automorphic form
  10. Analytic properties of $L(s,F)$
  11. Relation to the standard zeta-function $\mathcal{Z}(s,f)$

Key Result

Lemma 4.1

Theorems & Definitions (22)

  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Corollary 4.3
  • Corollary 4.4
  • Theorem 4.5
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 12 more