Graded Lie algebras give a uniform approach to many questions in arithmetic statistics. I'll give some background about graded Lie algebras and show how they arise in proofs about families of algebraic curves. For a certain class of graded Lie algebras, Thorne showed that we can construct families of curves as fibers of a certain categorical quotient map. I'll talk about ongoing work with Jef Laga to generalize this construction, including new examples.[-]

Let $L/K$ be an extension of number fields. The norm map $N_{L/K} :L^{*}\to K^{*}$ extends to a norm map from the ideles of L to those of $K$. The Hasse norm principle is said to hold for $L/K$ if, for elements of $K^{*}$, being in the image of the idelic norm map is equivalent to being the norm of an element of L^{*}. The frequency of failure of the Hasse norm principle in families of abelian extensions is fairly well understood, thanks to previous work of Christopher Frei, Daniel Loughran and myself, as well as recent work of Peter Koymans and Nick Rome. In this talk, I will focus on the non-abelian setting and discuss joint work with Ila Varma on the statistics of the Hasse norm principle in field extensions with normal closure having Galois group $S_{4}$ or $S_{5}$.[-]

L-functions of abelian varieties are objects of great interest. In particular, they are believed (and known in some cases) to carry key arithmetic information of the variety via the Birch and Swinnerton-Dyer conjecture. As such, it is useful to be able to compute them in practice. In this talk, we will address the case of a genus 2 curve C/Q with bad reduction at an odd prime p where Jac(C) has good reduction. Our approach relies on counting points on the special fibre of the minimal regular model of the curve, which we extract using the theory of cluster pictures of hyperelliptic curves. Our method yields a fast algorithm in the sense that all computations occur in at most quadratic extensions of Q or finite fields. This is joint work with Andrew Sutherland.[-]

Let $A$ be an abelian variety over a number field. The connected monodromy field of $A$ is the minimal field over which the image of the $l$-adic torsion representations have connected Zariski closure. We show that for all even $g \geq 4$, there exist infinitely many geometrically nonisogenous abelian varieties $A$ over $\mathbb{Q}$ of dimension $g$ where the connected monodromy field is strictly larger than the field of definition of the endomorphisms of $A$. Our construction arises from explicit families of hyperelliptic Jacobians with definite quaternionic multiplication. This is joint work with Victoria Cantoral-Farfan and Davide Lombardo.[-]