Jeremy Rouse

Galois theory of iterated endomorphisms

Given an abelian algebraic group A over a global field F, � 2 A(F), and a prime `, the set of all preimages ofunder some iterate of (`) generates an extension of F that contains all `-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of A we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes p in the ring of integers of F such that the order of (� mod p) is prime to `. We compute this density in the general case for several classes of A, including elliptic curves and one-dimensional tori. For example, if F is a number field, A=F is an elliptic curve with surjective 2-adic representation and � 2 A(F) with � 㘲 2A(F(A(4))), then the density of p with (� mod p) having odd order is 11=21.

Quadratic Forms Representing All Odd Positive Integers

We consider the problem of classifying all positive-definite integer-valued quadratic forms that represent all positive odd integers. Kaplansky considered this problem for ternary forms, giving a list of 23 candidates, and proving that 19 of those represent all positive odds. (Jagy later dealt with a 20th candidate.) Assuming that the remaining three forms represent all positive odds, we prove that an arbitrary, positive-definite quadratic form represents all positive odds if and only if it represents the odd numbers from 1 up to 451. This result is analogous to Bhargava and Hankes celebrated 290-theorem. In addition, we prove that these three remaining ternaries represent all positive odd integers, assuming the Generalized Riemann Hypothesis. This result is made possible by a new analytic method for bounding the cusp constants of integer-valued quaternary quadratic forms

Zagier Duality for the Exponents of Borcherds Products for Hilbert Modular Forms

Q

Combinatorial Proofs of Fermat's, Lucas's, and Wilson's Theorems

with fundamental discriminant. This method is based on the analytic properties of Rankin-Selberg

Bounds for coefficients of cusp forms and extremal lattices

L

The explicit Sato-Tate Conjecture and densities pertaining to Lehmer-type questions

-functions, and we use it to prove that if

Divisibility properties of the Fibonacci entry point

Q

Traces of singular moduli on Hilbert modular surfaces

is a quaternary form with fundamental discriminant, the largest locally represented integer

Explicit bounds for the number of

n

p

for which