Let x be a curve in positive characteristic. A Hasse–Witt matrix for x is a matrix that
represents the action of the Frobenius operator on the cohomology group H1 (x, OX) with …

Heuristics based on the Sato–Tate conjecture and the Lang–Trotter philosophy suggest that
an abelian surface defined over a number field has infinitely many places of split reduction …

This article is a survey of our work (joint with Davesh Maulik, Arul Shankar, and Salim
Tayou) on arithmetic intersection theory on GSpin Shimura varieties with applications to …

We introduce and study a new way to categorize supersingular abelian varieties defined
over a finite field by classifying them as fully maximal, mixed or fully minimal. The type of A …

The main goal of this thesis is to provide results to explicitly compute Tamagawa numbers
for maximal tori of similitude groups. These numbers correspond to volumes inherently …

We study the extent to which curves over finite fields are characterized by their zeta functions
and the zeta functions of certain of their covers. Suppose $ C $ and $ C'$ are curves over a …

Let $ A $ be an absolutely simple abelian surface defined over a number field $ K $ with a
commutative (geometric) endomorphism ring. Let $\pi_ {A,\text {split}}(x) $ denote the …

Let A be a g-dimensional abelian variety over Q whose adelic Galois representation has
open image in GSp 2 g Z ˆ. We investigate the Frobenius fields Q (π p)= End (A p)⊗ Q of the …

The vertical Sato--Tate conjectures gives expected trace distributions for for families of
curves. We develop exact expression for the distribution associated to degree-$4 …

Let A be a geometrically simple g-dimensional abelian variety over the rationals. This thesis
investigates the behavior of the reductions A p of A modulo its primes p of good reduction …