- You are here:
- Home
In this talk, we discuss moments of Rankin-Selberg $L$-functions in the weight aspect. After reviewing some known results for the first and second moments of $L(1/2,f\otimes g)$, I will discuss recent joint work with Naomi Tanabe on a twisted double second moment. Averaging over forms $f\in H_k$ and $g\in H_{k+a}$, with $a$ fixed, and over the weight $k$, we obtain an asymptotic formula with a power-saving error term for twists $\lambda_f(r_1)\lambda_g(r_2)$ in the range $r
Murmurations were first observed as surprising correlations between Frobenius traces and root numbers of elliptic curves. They were then rigorously established for the family of modular forms in the level aspect by Zubrillina, and have since been studied in many families of L-functions in the number field setting. As pointed out by Sarnak, murmurations can also be interpreted as a phase transition in the one-level density of the low-lying zeros of the family. The goal of this talk is to investigate the function field analogue of this phenomenon.
In 1996, Merel showed that for any elliptic curve defined over a number field $F$ of degree $d$, the size of the torsion subgroup is bounded by a constant $B(d)$ that depends only on $d$. It is conjectured that one can choose $B(d)$ to be polynomial in the degree $d$. For the special class of elliptic curves with $j$-invariant in $\mathbb{Q}$, polynomial bounds were first established by Clark and Pollock in 2018. In this talk, I will discuss recent improvements to these bounds that are essentially optimal for the exponent of the torsion subgroup.
Many problems in number theory boil down to bounding the size of a set contained in a certain set of residue classes mod $p$ for various sets of primes $p$; and then sieve methods are the primary tools for doing so.
Chambert-Loir and Ducros have introduced a theory of real-valued smooth differential forms on Berkovich spaces that play the role of smooth forms on complex varieties. We compute the associated Dolbeault cohomology groups of curves by reducing to the case of metric graphs. I'll introduce smooth forms on graphs, and explain how the theory in CLD has to be modified in order to get finite-dimensional cohomology groups.
The Chebotarev density theorem is a powerful tool in number theory, in part because it guarantees the existence of primes whose Frobenius lies in a given conjugacy class in a fixed Galois extension of number fields. However, for some applications, it is necessary to know not just that such primes exist, but to additionally know something about their size, say in terms of the degree and discriminant of the extension. In this talk, I'll discuss recent work with Peter Cho and Asif Zaman on a closely related problem, namely determining the least prime with a given cy
Let $f$ be an endomorphism of projective space defined over a number field. When counting rational points ordered by a certain "canonical" height function attached to $f$, we encounter a mysterious asymptotic constant in the main term. This constant is a product of local factors over the primes of bad reduction of $f$; and these local factors (which take the form of $v$-adic integrals) are rather difficult to calculate explicitly. In this talk I will present my partial progress towards evaluating these integrals. No knowledge of arithmetic dynamics will be assu
I will present some recent work with Debmalya Basak and Alexandru Zaharescu on potential improvements to the Siegel—Walfisz upper bound on the greatest real zero of a Dirichlet $L$-function.
Contact Information
Pages
