Talk

Modular forms live on moduli spaces of abelian varieties and curves as sections of vector bundles. We indicate
geometric ways of constructing such forms.

This is joint work with Cléry, Faber and Kouvidakis.

Let k be a natural number and A be a subset of the natural numbers. Can one determine the lower exponential density of the set
$$\{ x^k + a:  a\in A, x\in\mathbb N \}$$
in terms of lower exponential density of A? This problem is trivial for k=1, and has an elementary complete solution when k=2. Eighty years ago Davenport gave lower bounds in all cases of interest. In this talk we describe joint work with Simon Myerson. Davenport's estimates are improved when A is already fairly dense. In some range, our results are best possible.

For $\ell$ an odd prime number and $d$ a squarefree integer, a central question in arithmetic statistics is to give pointwise bounds for the size of the $\ell$-torsion of the class group of $\mathbb{Q}(\sqrt{d})$. This is in general a difficult problem, and unconditional pointwise bounds are only available for $\ell = 3$ due to work of Pierce, Helfgott—Venkatesh and Ellenberg—Venkatesh. The current record is $h_3(d) \ll_\epsilon d^{1/3 + \epsilon}$ due to Ellenberg—Venkatesh. We will discuss how to improve this to $h_3(d) \ll d^{0.32}$. This is joint work with Stephanie Chan.