In his thesis, Venkatesh gave a new proof of the classical converse theorem for modular forms of level~$1$ in the context of Langlands' ``Beyond Endoscopy". We extend his approach to arbitrary levels and characters. The method of proof, via the Petersson trace formula, allows us to treat arbitrary degree~$2$ gamma factors of Selberg class type.
This is joint work with Andrew R. Booker and Michael Farmer.
We study the moments of $L$-functions associated with primitive cusp forms, in the weight aspect. In particular, we present recent joint work with Brian Conrey, where we obtain an asymptotic formula for the twisted $r$-th moment of a long Dirichlet polynomial approximation of such $L$-functions. This result, which is conditional on the Generalized Lindel\"of Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.
We discuss the local statistics of zeros of $L$-functions attached to Artin--Scheier curves over finite fields, that is, curves defined by equations of the form $y^p-y=f(x)$, where $f$ is a rational function with coefficients in $F_q$ ($q$ a power of~$p$).
We consider three families of Artin--Schreier $L$-functions: the ordinary, polynomial (the $p$-rank $0$ stratum) and odd-polynomial families.
We present recent results on the $1$-level zero-density of the first and third families and the $2$-level density of the second family, for test functions with Fourier transform supported in suitable intervals. In each case we obtain agreement with a unitary or symplectic random matrix model.
We discuss some appearances of $L$-function moments in number field counting problems, with a particular focus on counting abelian extensions of number fields with restricted ramification.
I will talk about recent work joint with Nathan Ng and Peng-Jie Wong. We established an asymptotic formula for the eighth moment of the Riemann zeta function, assuming the Riemann hypothesis and a quaternary additive divisor conjecture.
How large are the $L^2$-restrictions of automorphic forms to closed geodesics? I will discuss how this problem can be shown to be equivalent to proving bounds for certain weighted moments of Hecke $L$-functions, and how the lattice structure of the ring of integers of real quadratic numbers fields can be exploited to obtain essentially optimal upper bounds for these weighted moments.
We compute a first moment of $GL(3)\times GL(2)$ $L$-functions twisted by a $GL(2)$ Hecke eigenvalue at a prime. We talk about the ideas behind the proof, ways in which it can be generalised or extended, and obstacles for doing so in other directions. We also talk a bit about why such moments are interesting, briefly discussing some applications.
The celebrated Motohashi phenomenon concerns the duality between the fourth moment of the Riemann zeta function and the cubic moment of automorphic $L$-functions of $GL(2)$. Apart from its structural elegance, such a duality plays a very important role in various moment problems. In this talk, we will discuss the generalized Motohashi phenomena for the group $GL(3)$ through the lenses of period integrals and the method of unfolding. As a consequence, the Kuznetsov and the Voronoi formulae are not needed in our argument.
Let $f$ and $g$ be holomorphic cusp forms for the modular group $SL_2(\mathbb Z)$ of weight $k_1$ and $k_2$ with
Fourier coefficients $\lambda_f(n)$ and $\lambda_g(n)$, respectively. For real $\alpha\neq0$ and $0<\beta\leq1$, consider a smooth resonance sum $S_X(f,g;\alpha,\beta)$ of $\lambda_f(n)\lambda_g(n)$ against $e(\alpha n^\beta)$ over $X\leq n\leq2X$. Double square moments of $S_X(f,g;\alpha,\beta)$ over both $f$ and $g$ are nontrivially bounded when their weights $k_1$ and $k_2$ tend to infinity together. By allowing both $f$ and $g$ to move, these double moments are indeed square moments associated with automorphic forms for $GL(4)$. These bounds reveal insights into the size and oscillation of the resonance sums and their potential resonance for $GL(4)$ forms when $k_1$ and $k_2$ are large.
A fundamental problem in analysis is understanding the distribution of mass of Laplacian eigenfunctions via bounds for their $L^p$ norms in terms of the size of their Laplacian eigenvalue. Number theorists are interested in the Laplacian eigenfunctions on the modular surface that are additionally joint eigenfunctions of every Hecke operator---namely the Hecke--Maass cusp forms. In this talk, I will describe joint work with Peter Humphries in which we prove new bounds for $L^p$ norms in this situation. This is achieved by using $L$-functions and their reciprocity formulae: certain special identities between two different moments of central values of $L$-functions.