# Number Theory

## Torsion of Rational Elliptic Curves over the Cyclotomic Extensions of ℚ

Let E be an elliptic curve defined over ℚ. Let p > 3 be a prime such that p - 1 is not divisible by 3, 4, 5, 7, 11. In this article, we classify the groups that can arise as E(ℚ(ζp))tors up to isomorphism. The method illustrates techniques for eliminating possible structures that can appear as a subgroup of E(ℚab)tors.

## Subconvexity for GL(2) L-functions and Shifted MDS

Subconvexity problems have maintained extreme interest in analytic number theory for decades. Critical barriers such as the convexity, Burgess, and Weyl bounds hold particular interest because one usually needs to drastically adjust the analytic techniques involved in order to break through them. It has recently come to light that shifted Dirichlet series can be used to obtain subconvexity results. While these Dirichlet series do not admit Euler products, they are amenable to study via spectral methods. In this talk, we construct a shifted multiple Dirichlet series (MDS) and leverage its analytic continuation via spectral decompositions to obtain the Weyl bound in the conductor-aspect for the L-function of a holomorphic cusp form twisted by an arbitrary Dirichlet character. This improves upon the corresponding bound for quadratic characters obtained by Iwaniec-Conrey in 2000. This work is joint with Jeff Hoffstein, Nikos Diamantis, and Min Lee.

## Orienteering with One Endomorphism

Given two elliptic curves, the path finding problem asks to find an isogeny (i.e. a group homomorphism) between them, subject to certain degree restrictions. Path finding has uses in number theory as well as applications to cryptography. For supersingular curves, this problem is known to be easy when one small endomorphism or the entire endomorphism ring are known. Unfortunately, computing the endomorphism ring, or even just finding one small endomorphism, is hard. How difficult is path finding in the presence of one (not necessarily small) endomorphism? We use the volcano structure of the oriented supersingular isogeny graph to answer this question. We give a classical algorithm for path finding that is subexponential in the degree of the endomorphism and linear in a certain class number, and a quantum algorithm for finding a smooth isogeny (and hence also a path) that is subexponential in the discriminant of the endomorphism. A crucial tool for navigating supersingular oriented isogeny volcanoes is a certain class group action on oriented elliptic curves which generalizes the well-known class group action in the setting of ordinary elliptic curves.

## Parametrization of rings of finite rank - a geometric approach and their use in counting number fields

We describe parametrizations of rings that generalize the notions of monogenic rings and binary rings. We use these parametrizations to give better lower bounds on the number of number fields of degree n and bounded discriminant.

## Moments of real Dirichlet L-functions and multiple Dirichlet series

There are two ways to compute moments in families of L-functions: one uses the approximation by Dirichlet polynomials, and the other is based on multiple Dirichlet series. We will introduce the two methods to study the family of real Dirichlet L-functions, compare them and show that they lead to the same results. We will then focus on obtaining the meromorphic continuation of the associated multiple Dirichlet series.

## A Study of Twisted Sums of Arithmetic Functions

We study sums of the form $\sum_{n\leq x} f(n) n^{-iy}$, where $f$ is an arithmetic function, and we establish an equivalence between the Riemann Hypothesis and estimates on these sums. In this talk, we will explore examples of such sums with specific arithmetic functions, as well as discuss potential implications and future research directions.

## Hybrid Statistics of the Maxima of a Random Model of the Zeta Function over Short Intervals

We will present a matching upper and lower bound for the right tail

probability of the maximum of a random model of the Riemann zeta function over

short intervals. In particular, we show that the right tail interpolates

between that of log-correlated and IID random variables as the interval varies

in length. We will also discuss a new normalization for the moments over short

intervals. This result follows the recent work of Arguin-Dubach-Hartung and is inspired by a conjecture by Fyodorov-Hiary-Keating on the local maximum over

short intervals.

## On the vertical distribution of the zeros of the Riemann zeta-function

In 1973, assuming the Riemann hypothesis (RH), Montgomery studied the vertical distribution of zeta zeros, and conjectured that they behave like the eigenvalues of some random matrices. We will discuss some models for zeta zeros starting from the random matrix model but going beyond it and related questions, conjectures and results on statistical information on the zeros. In particular, assuming RH and a conjecture of Chan for how often gaps between zeros can be close to a fixed non-zero value, we will discuss our proof of a conjecture of Berry (1988) for the number variance of zeta zeros, in a regime where random matrix models alone do not accurately predict the actual behavior (based on joint work with Meghann Moriah Lugar and Micah B. Milinovich).

## Discrete mathematics in continuous quantum walks

Let $G$ be a graph with adjacency matrix $A$. A continuous quantum walk on $G$ is determined by the complex unitary matrix $U(t)=\exp(itA)$, where $i^2=−1 and $t$ is a real number. Here, $G$ represents a quantum spin network, and its vertices and edges represent the particles and their interactions in the network. The propagation of quantum states in the quantum system determined by $G$ is then governed by the matrix $U(t)$. In particular, $|U(t)_{u,v}|^2$ may be interpreted as the probability that the quantum state assigned at vertex $u$ is transmitted to vertex $v$ at time $t$. Quantum walks are of great interest in quantum computing because not only do they produce algorithms that outperform classical counterparts, but they are also promising tools in the construction of operational quantum computers. In this talk, we give an overview of continuous quantum walks, and discuss old and new results in this area with emphasis on the concepts and techniques that fall under the umbrella of discrete mathematics.

## Remarks on a formula of Ramanujan

In this talk, we will discuss a well-known formula of Ramanujan and its relationship with the partial sums of the Möbius function. Under some conjectures, we analyze a finer structure of the involved terms. It is a joint work with Steven M. Gonek (University of Rochester).