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Eisenstein series of even weight $k \geq 2$ and integral binary quadratic forms

Andreas Mono

Abstract

We prove a conjecture of Matsusaka on the analytic continuationof hyperbolic Eisenstein series in weight $2$ on the full modular group $\mathrm{SL}_2(\mathbb{Z})$.

Eisenstein series of even weight $k \geq 2$ and integral binary quadratic forms

Abstract

We prove a conjecture of Matsusaka on the analytic continuationof hyperbolic Eisenstein series in weight on the full modular group .

Paper Structure

This paper contains 25 sections, 13 theorems, 54 equations.

Table of Contents

  1. Introduction and statement of results
  2. Preliminaries
  3. Fractional linear transformations
  4. Classification of motions
  5. Integral binary quadratic forms
  6. Heegner geodesics
  7. Quadratic forms associated to $\gamma \in \Gamma$
  8. Genus characters
  9. Construction of Eisenstein series
  10. Eisenstein series associated to a quadratic form
  11. Eisenstein series associated to a given discriminant
  12. Parabolic and elliptic Eisenstein series
  13. Parabolic case
  14. Modularity
  15. Analytic continuation
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\gamma \in {\text{\rm SL}}_2(\mathbb{Z})$ be hyperbolic and primitive. Then the function $\mathcal{E}_{2,\gamma}(\tau,s)$ can be analytically continued to $s=0$ and the continuation is given by for any $\tau \in \mathbb{H}$. Here, $\mathop{\mathrm{tr}}\nolimits_{d,\Delta(\gamma)}(1)$ is a twisted trace of cycle integrals given by Furthermore, if $\mathrm{Im}(\tau)$ is sufficiently large, th

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Lemma 3.5
  • proof
  • ...and 17 more