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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)

- Melanie Matchett Wood
- J. London Math. Society
- 2011

- Van H. Vu, Melanie Matchett Wood, Philip Matchett Wood
- J. London Math. Society
- 2011

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)

- Jeffrey D Achter, Daniel Erman, Kiran S Kedlaya, Melanie Matchett Wood, David Zureick-Brown
- Philosophical transactions. Series A…
- 2015

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)

For a number field K and a finite abelian group G, we determine the probabilities of various local completions of a random G-extension of K when extensions are ordered by conductor. In particular, for a fixed prime ℘ of K, we determine the probability that ℘ splits into r primes in a random G-extension of K that is unramified at ℘. We find that these… (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)

This course will cover the questions of counting number fields and determining the distribution of class groups of number fields. The starting point is the very basic question: how many number fields are there? This is commonly interpreted as determining the aymptotics in X for #{K ⊂ ¯ Q | |Disc(K)| ≤ X}.

Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil’s number field/ function field analogy to predict coincidences for limiting homological densities of various sequences Z1m n pXq of spaces of 0-cycles on manifolds X. The main… (More)

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