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Higher moments for symmetric powers of modular forms

Jiong Yang, Zhishan Yang

Abstract

Let $f$ be a cuspidal eigenform of weight $k$ on $\SL_2(\BZ)$ and let $λ_{\Sym^d f}(n)$ be the normalized Fourier coefficients of its $d$-th symmetric power lift. This paper establishes asymptotic formulas for the moments $\sum_{n\leq x}λ^l_{\Sym^d f}(n)$ for all positive integers $d$ and $l$. We also prove an asymptotic formula for the corresponding sum over the values of any positive definite binary quadratic form $Q$. Our results generalize and improve upon previous work, which was limited to small values of $d$ or $l$. The proofs rely on the decomposition of $\ell$-adic Galois representations and the analytic properties of the associated $L$-functions.

Higher moments for symmetric powers of modular forms

Abstract

Let be a cuspidal eigenform of weight on and let be the normalized Fourier coefficients of its -th symmetric power lift. This paper establishes asymptotic formulas for the moments for all positive integers and . We also prove an asymptotic formula for the corresponding sum over the values of any positive definite binary quadratic form . Our results generalize and improve upon previous work, which was limited to small values of or . The proofs rely on the decomposition of -adic Galois representations and the analytic properties of the associated -functions.
Paper Structure (13 sections, 13 theorems, 71 equations)

This paper contains 13 sections, 13 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Main theorems
  3. Proof Sketch
  4. $L$-functions associated to Weyl modules of modular forms
  5. $L$-functions of compatible systems of $\ell$-adic representations
  6. The Schur functor and Weyl modules
  7. Weyl modules associated to modular forms
  8. The calculation of Kostka numbers
  9. Binary quadratic forms
  10. The estimations via Perron's formula
  11. The Dirichlet series
  12. Subconvexity results for $L$-functions
  13. Proof of the main theorems

Key Result

Theorem 1.1

Let $f$ be a cuspidal modular form of weight $k$ on ${\mathrm{SL}}_2({\mathbb {Z}})$. For any integers $l\geq2$ and $d\geq1$ and $dl>4$, one has Here $P_{d,l}$ is a polynomial of degree $K_{0,d,l}-1$, where $K_{i,d,l}$ denote the Kostka number defined in sections section:Weylmod and section:Kostka number. The exponent $\theta_{d, l}$ is given by

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • ...and 9 more