We prove formulas for power moments for point counts of elliptic curves over a finite field k
such that the groups of k‐points of the curves contain a chosen subgroup. These formulas …

The cyclicity and Koblitz conjectures ask about the distribution of primes of cyclic and prime-
order reduction, respectively, for elliptic curves over $\mathbb {Q} $. In 1976, Serre gave a …

The main topic of this thesis is Galois representations of abelian varieties. Following the
introduction, there are four largely independent chapters that each make progress around …

For an elliptic curve $ E $ defined over $\Q $ without complex multiplication and for a
rational prime $ p $ of good reduction, one can consider the subring of the $\F_p …

Given an elliptic curve E without complex multiplication defined over a number field K,
consider the image of the Galois representation defined by letting Galois act on the torsion of …

We consider elliptic curves $ E/\mathbb {Q} $ for which the image of the adelic Galois
representation $\rho_E $ is as large as possible given a constraint on the image modulo 2 …

We apply the character sums method of Lenstra, Moree, and Stevenhagen to explicitly
compute the constants in the Titchmarsh divisor problem for Kummer fields and division …

Background Classical weight 2 newforms are complex functions with certain kinds of
arithmetic symmetries. They are among the most important objects in number theory, and in …

Abstract In 1927, Emil Art in conjectured a product expression for the density of primes p for
which a given non-zero integer a is a primitive root modulo p. The conjectured density was …

Given an elliptic curve defined over the field of rational numbers, what is the frequency with
which its reduction modulo a prime gives rise to a cyclic group? Guided by this question, we …