Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In genus one, the Ogg–Saito formula shows that these two invariants are equal, and in genus two, Qing Liu showed that they are related by an inequality. In this talk, we will show that Liu’s inequality extends to hyperelliptic curves of arbitrary genus in the absence of wild ramification phenomena. The key ingredients in this proof are an explicit analysis of regular models arising from Jung’s method of resolving surface singularities, and an understanding of the behaviour of associated metric trees under a natural inductive process. Time permitting, we will also outline joint work with Andrew Obus to extend this to all odd residue characteristics.

As was made famous by Mazur, the mod-5 Galois representation associated to the elliptic curve X₀(11) is reducible. In fact, the mod-25 Galois representation is also reducible. We’ll talk about this kind of extra reducibility phenomenon more generally, for forms of even weight k and prime level. The characters appearing in the reducible representation are related, on one hand, to a ‘tame deriviative’ of an L-function, and, on the other hand, to an algebraic invariant. This type of relation is predicted by the Bloch-Kato conjecture.

Almost 60 years ago Golod and Shafarevich gave examples of number fields with infinite Hilbert class field towers. These fields generally had quite large root discriminants. Forty and twenty years ago respectively, Martinet and Hajir-Maire gave examples of tamely ramified p-towers of number fields with relatively small root discriminant, though these are still far away from the GRH lower bounds. I will explain new recent records and work on better understanding tame Shafarevich groups, kernels of H² localization maps. This is joint with Hajir and Maire.

An isogeny class of elliptic curves over a finite field is determined by a quadratic Weil polynomial. For each rational prime ℓ, one could compute how likely it is that a random ℓ-adic matrix has the specified characteristic polynomial, and compare this to the average among all characteristic polynomials. An irrationally exuberant interpretation of equidistribution might lead one to believe that the product, over all ℓ, of this quantity might somehow compute the size of the isogeny class. Gekeler actually proved that this relation holds.

In this talk, I'll explain a new, transparent proof of this formula, as well as its generalization to principally polarized abelian varieties of arbitrary dimension.

(This is joint work with Ali Altug, Luis Garcia and Julia Gordon.)