Melanie Matchett Wood

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Bhargava proved a formula for counting, with certain weights, degree n ´ etale extensions of a local field, or equivalently, local Galois representations to S n. This formula is motivation for his conjectures about the density of discriminants of S n-number fields. We prove there are analogous " mass formulas " that count local Galois representations to any(More)
We show that any finite system S in a characteristic zero integral domain can be mapped to Z/pZ, for infinitely many primes p, preserving all algebraic incidences in S. This can be seen as a generalization of the well-known Freiman isomorphism lemma, which asserts that any finite subset of a torsion-free group can be mapped into Z/pZ, preserving all linear(More)
How many rational points are there on a random algebraic curve of large genus g over a given finite field Fq? We propose a heuristic for this question motivated by a (now proven) conjecture of Mumford on the cohomology of moduli spaces of curves; this heuristic suggests a Poisson distribution with mean q+1+1/(q-1). We prove a weaker version of this(More)
These are expanded lecture notes for a series of five lectures at the Arizona Winter School on " Arithmetic statistics " held March 15-19, 2014 in Tucson, Arizona. They are not intended for publication; in fact, they are largely drawn from articles that have already been published and are referenced below. We start by giving an introduction to some of the(More)
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