Extreme values of $L$-functions of newforms

TL;DR

This paper investigates the extreme values of L-functions associated with newforms, establishing lower bounds on the number of forms with large L-values and exploring the behavior of twisted L-functions for forms with certain characters.

Contribution

It extends previous results by showing that a large proportion of newforms have L-values exceeding a specific exponential threshold, and analyzes extremal values of twisted L-functions for forms with particular characters.

Findings

Many newforms have L(1/2, f) exceeding a specified exponential bound.

The number of such forms is proportional to a power of the level k.

Results apply to twisted L-functions with certain Dirichlet characters.

Abstract

In 2008, Soundararajan showed that there exists a normalized Hecke eigenform $f$ of weight $k$ and level one such that $$ L(1/2, f ) ~\geq~ \exp\Bigg( (1 + o(1)) \sqrt{\frac{2\log k}{\log\log k} }\Bigg) $$ for sufficiently large $k \equiv 0 \pmod{4}$. In this note, we show that for any $\epsilon>0$ and for all sufficiently large $k \equiv 0 \pmod{4}$, the number of normalized Hecke eigenforms of weight $k$ and level one for which $$ L(1/2, f ) ~\geq~ \exp\left(1.41\sqrt{ \frac{ \log k }{\log\log k} }\right) $$ is $\gg_{\epsilon} k^{1-\epsilon}$. For an odd fundamental discriminant $D$, let $B_{k}(|D|)$ be the set of all cuspidal normalized Hecke eigenforms of weight $k$ and level dividing $|D|$. When the real primitive Dirichlet character $\chi_D$ satisfies $\chi_D(-1)= i^k$, we investigate the number of $f \in B_{k}(|D|)$ for which $L(1/2, f \otimes \chi_D)$ takes extremal values.