In this talk, we will discuss the application of modularity and the holomorphic projection operator on a question involving moments for class numbers where the discriminant is restricted to a certain arithmetic progression, with an application to moments for elliptic curves over finite fields where the trace of Frobenius is restricted to the same arithmetic progression.

The focus of this talk is on Sato-Tate groups of abelian varieties -- compact groups predicted to determine the limiting distributions of local zeta functions. In recent years, complete classifications of Sato-Tate groups in dimensions 1, 2, and 3 have been given, but there are obstacles to providing classifications in higher dimensions. In this talk, I will describe my recent work on families of higher dimensional Jacobian varieties. This work is partly joint with Melissa Emory.

The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

In 2004, Montgomery and Soundararajan showed (conditionally) that the distribution of the number of primes in appropriately sized intervals is approximately Gaussian and has a somewhat smaller variance than you might expect from modeling the primes as a purely random sequence. Their work depends on evaluating sums of certain arithmetic constants that generalize the twin prime constant, known as singular series. In particular, these sums exhibit square-root cancellation in each term if they have an even number of terms, but if they have an odd number of terms, there should be slightly more than square-root cancellation. I will discuss sums of singular series with an odd number of terms, including tighter bounds for small cases and the function field analog. I will also explain how this problem is connected to a simple problem about adding fractions.

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length  $y$ around $x$, where $y\ll(\log x)^2$. In particular, we conjecture that the maximum grows surprisingly slowly as $y$ranges from $\log x$ to $(\log x)^2$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification. This is joint work with Andrew Granville.