[Paper Review] On M-functions associated with modular forms

Philippe Lebacque, Alexey Zykin|arXiv (Cornell University)|Feb 24, 2017

Analytic Number Theory Research12 references4 citations

TL;DR

This paper studies the value distribution of $\mathfrak{L}(f\otimes\chi,s)$, where $\mathfrak{L}$ is either $\log L(f\otimes\chi,s)$ or $(L'/L)(f\otimes\chi,s)$, for primitive cusp forms $f$ of weight $k$ and level $N$, and Dirichlet characters $\chi$ of prime conductor. Under the Generalized Riemann Hypothesis, it establishes equidistribution of these values via limit functions and harmonic averages, proving the existence of limiting density functions $M_\sigma(w)$ and $\tilde{M}_\sigma(z_1,z_2)$ with analytic and Euler product properties.

ABSTRACT

Let $f$ be a primitive cusp form of weight $k$ and level $N,$ let $χ$ be a Dirichlet character of conductor coprime with $N,$ and let $\mathfrak{L}(f\otimes χ, s)$ denote either $\log L(f\otimes χ, s)$ or $(L'/L)(f\otimes χ, s).$ In this article we study the distribution of the values of $\mathfrak{L}$ when either $χ$ or $f$ vary. First, for a quasi-character $ψ\colon \mathbb{C} o \mathbb{C}^ imes$ we find the limit for the average $\mathrm{Avg}\_χψ(L(f\otimesχ, s)),$ when $f$ is fixed and $χ$ varies through the set of characters with prime conductor that tends to infinity. Second, we prove an equidistribution result for the values of $\mathfrak{L}(f\otimes χ,s)$ by establishing analytic properties of the above limit function. Third, we study the limit of the harmonic average $\mathrm{Avg}^h\_f ψ(L(f, s)),$ when $f$ runs through the set of primitive cusp forms of given weight $k$ and level $N o \infty.$ Most of the results are obtained conditionally on the Generalized Riemann Hypothesis for $L(f\otimesχ, s).$

Motivation & Objective

• To analyze the distribution of $\mathfrak{L}(f\otimes\chi,s) = \log L(f\otimes\chi,s)$ or $(L'/L)(f\otimes\chi,s)$ as $\chi$ varies over Dirichlet characters with prime conductor tending to infinity.
• To establish equidistribution results for these $\mathfrak{L}$-values by analyzing the limit of the average $\mathrm{Avg}_\chi \psi(L(f\otimes\chi,s))$ for quasi-characters $\psi$.
• To extend the analysis to harmonic averages over primitive cusp forms of fixed weight $k$ and level $N \to \infty$, and to study the resulting limiting density functions.
• To explore the analytic and arithmetic properties of the limiting functions $M_\sigma(w)$ and $\tilde{M}_\sigma(z_1,z_2)$, including their Euler product structure and growth behavior.

Proposed method

• The authors use the theory of $L$-functions associated with modular forms and their twists by Dirichlet characters, focusing on the logarithmic derivative and logarithm of these $L$-functions.
• They define a limit function $\mathcal{M}_\sigma(w)$ as the limit of the average $\mathrm{Avg}_\chi \psi(L(f\otimes\chi,s))$ for quasi-characters $\psi$, under the GRH.
• The equidistribution of $\mathfrak{L}(f\otimes\chi,s)$ is established by showing that the limit measure corresponds to a density function $M_\sigma(w)$, with $\tilde{M}_\sigma(z_1,z_2)$ as its Fourier transform.
• The method relies on analytic continuation and Euler product expansions of the limiting functions $\tilde{M}_\sigma(z_1,z_2)$, derived from local $L$-factors and the structure of Hecke eigenforms.
• For harmonic averages over $f$, the authors use the Petersson trace formula and relate the weights to $L(\mathrm{Sym}^2 f, 1)$, aiming to remove harmonic weights via trace formula techniques.
• The analysis is conditional on the Generalized Riemann Hypothesis for $L(f\otimes\chi,s)$, which ensures convergence and analytic behavior of the $L$-functions and their logarithmic derivatives.

Experimental results

Research questions

• RQ1 Can the limit of the average $\mathrm{Avg}_\chi \psi(L(f\otimes\chi,s))$ be computed as the conductor of $\chi$ tends to infinity, for fixed $f$?
• RQ2 Does the distribution of $\mathfrak{L}(f\otimes\chi,s)$ equidistribute in the complex plane, and what is the associated limiting density function $M_\sigma(w)$?
• RQ3 Can the harmonic average $\mathrm{Avg}^h_f \psi(L(f,s))$ over primitive cusp forms of level $N \to \infty$ be analyzed, and does it yield a limiting density function?
• RQ4 What are the analytic properties of the function $\tilde{M}_\sigma(z_1,z_2)$, including its Euler product, analytic continuation, and growth?
• RQ5 Can the results be extended unconditionally or to other families of automorphic forms, and what is the function field analog?

Key findings

• The limit of the average $\mathrm{Avg}_\chi \psi(L(f\otimes\chi,s))$ exists and defines a function $\tilde{M}_\sigma(z_1,z_2)$ that is entire in $z_1, z_2$ and admits an Euler product expansion under GRH.
• The distribution of $\mathfrak{L}(f\otimes\chi,s)$ equidistributes with respect to a limiting density $M_\sigma(w)$, satisfying $\int_{\mathbb{C}} M_\sigma(w) \Phi(w) |dw| = \lim_{\chi} \mathrm{Avg}_\chi \Phi(\mathfrak{L}(f\otimes\chi,s))$ for test functions $\Phi$ of at most exponential growth.
• The function $\tilde{M}_\sigma(z_1,z_2)$ is shown to be the inverse Fourier transform of $M_\sigma(w)$, and its Plancherel volume $\int_{\mathbb{C}} |\tilde{M}_\sigma(z,\bar{z})|^2 |dz|$ is a well-defined and interesting object.
• For harmonic averages over primitive cusp forms of level $N \to \infty$, the paper establishes the existence of a limiting density function $\tilde{M}_\sigma(z_1,z_2)$, though it is not holomorphic in $s$ due to dependence on both $s$ and $\bar{s}$.
• The results are conditional on the Generalized Riemann Hypothesis, and the authors identify key challenges in removing this assumption, particularly in the harmonic average case.
• The paper outlines a program to remove harmonic weights and extend results to other automorphic forms, suggesting that trace formulas and local theory may be essential tools.