Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the trace formula

Eran Assaf

Abstract

In this mostly expository note, we prove explicit formulas for the traces of Hecke operators on spaces of cusp forms fixed by Atkin-Lehner involutions, which are suitable for efficient implementation. In addition, we correct a couple of errors in previously published formulas.

A note on the trace formula

Abstract

In this mostly expository note, we prove explicit formulas for the traces of Hecke operators on spaces of cusp forms fixed by Atkin-Lehner involutions, which are suitable for efficient implementation. In addition, we correct a couple of errors in previously published formulas.
Paper Structure (13 sections, 20 theorems, 121 equations)

This paper contains 13 sections, 20 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Results
  3. Acknowledgements
  4. Modular Forms
  5. Level raising and Hecke operators
  6. Trace Formulas
  7. Trace Formula for $\mathop{\mathrm{SL}}\nolimits_2(\mathbb{Z})$
  8. Trace formula for higher levels
  9. Trace formula for newforms
  10. Arithmetic functions and Möbius inversion
  11. Correction of previous formulas
  12. Trace formula in SZ
  13. Trace formula in P

Key Result

Theorem 1.2

Let $n, N$ be positive integers, and let $Q \parallel N$. Denote where $N'$ ranges over the set $\mathcal{N}_{Q,N,n}(d, n')$ defined in eq: N(d,n) sets. Then

Theorems & Definitions (36)

  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.1
  • proof
  • Corollary 3.4
  • Remark 3.6
  • Lemma 3.7: DS*Ex.5.6.3
  • Lemma 3.8
  • proof
  • Corollary 3.10
  • ...and 26 more