Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Shimura lift of Rankin-Cohen brackets of eigenforms and theta series

Wei Wang

TL;DR

The paper addresses the problem of determining the Shimura lift of Rankin–Cohen brackets between a normalized eigenform and a theta series. It generalizes previous product-based results by proving that the $t$-Shimura lift of $[f(4rz),\theta_{\psi}(tz)]_w$ is expressible as a scaled Rankin–Cohen bracket involving the eigenform with itself, via an auxiliary form $g_{\psi}(z)$ built from divisor data. The proofs rely on a detailed Fourier-coefficient analysis, divisor-sum decompositions, and Dirichlet-series techniques with Euler-product factors to derive explicit lift formulas, including extensions to Kohnen's plus space and the $t$-lift for square-free $t$. The results provide explicit lifting formulas and level-tracking insights, with potential generalizations to broader automorphic contexts.

Abstract

The Shimura lift of a Hekce eigenform multiplied by a theta series is the square of the form. We extend this result by generalizing the product map to the Rankin-Cohen bracket. We prove that the Shimura lift of Rankin-Cohen bracket of an eigenform and a theta series is given by Rankin-Cohen bracket of the eigenform and itself.

Shimura lift of Rankin-Cohen brackets of eigenforms and theta series

TL;DR

The paper addresses the problem of determining the Shimura lift of Rankin–Cohen brackets between a normalized eigenform and a theta series. It generalizes previous product-based results by proving that the -Shimura lift of is expressible as a scaled Rankin–Cohen bracket involving the eigenform with itself, via an auxiliary form built from divisor data. The proofs rely on a detailed Fourier-coefficient analysis, divisor-sum decompositions, and Dirichlet-series techniques with Euler-product factors to derive explicit lift formulas, including extensions to Kohnen's plus space and the -lift for square-free . The results provide explicit lifting formulas and level-tracking insights, with potential generalizations to broader automorphic contexts.

Abstract

The Shimura lift of a Hekce eigenform multiplied by a theta series is the square of the form. We extend this result by generalizing the product map to the Rankin-Cohen bracket. We prove that the Shimura lift of Rankin-Cohen bracket of an eigenform and a theta series is given by Rankin-Cohen bracket of the eigenform and itself.
Paper Structure (2 sections, 5 theorems, 42 equations)

This paper contains 2 sections, 5 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. proof of theorems

Key Result

Theorem 1.1

Let $f$, $g$ be two modular forms of weights $k$, $l$ and characters $\chi_1$, $\chi_2$ respectively, on a subgroup $\Gamma$ of $\operatorname{SL}_2(\mathbb{Z})$. Then $[f,g]_w\in M_{k+l+2w}(\Gamma, \chi_1\chi_2\chi)$ where $\chi=1$ if $k_1$ and $k_2\in \mathbb{Z}$, $\chi=\chi_{-4}^{k_i}$ if $k_i\in

Theorems & Definitions (6)

  • Theorem 1.1: Cohen
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof