Scientific

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 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.

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.

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.

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.