Alexander Smith's papers
Here's a list of my papers with arXiv links.
Selmer groups in twist families
I am most well known for my work on the distribution of Selmer groups in twist families and its applications to the Cohen-Lenstra-Gerth heuristics and to Goldfeld's conjecture.
The distribution of \ell^\infty Selmer groups in degree \ell twist families I (2022), submitted.
The distribution of \ell^\infty Selmer groups in degree \ell twist families II (2022), submitted.
These papers subsume the following two papers. The first of these had already subsumed the second.
2^\infty-Selmer groups, 2^\infty-class groups, and Goldfeld's conjecture (2017), unsubmitted.
Governing fields and statistics for 4-Selmer groups and 8-class groups (2016), unsubmitted.
My other work in this area:
The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture (2025), submitted.
Sums of rational cubes and the 3-Selmer group with Peter Koymans (2024), submitted.
The Cassels-Tate pairing for finite Galois modules, with Adam Morgan (2022), submitted.
Field change for the Cassels-Tate pairing and applications to class groups, with Adam Morgan, Research in Number Theory vol 10 article 61 (2024).
The congruent numbers have positive natural density (2016), in preparation.
Trace problem and abelian varieties
Algebraic integers with conjugates in a prescribed distribution. Annals of Mathematics 200.1 (2024).
Abelian varieties of prescribed order over finite fields, with Raymond van Bommel, Edgar Costa, Wanlin Li, and Bjorn Poonen, to appear in Math. Ann.
New Lower Bounds for the Schur-Siegel-Smyth Trace Problem, with Bryce Orloski and Naser Sardari. Mathematics of Computation, 2024.
Other projects
Faithful Artin induction and the Chebotarev density theorem, with Robert J. Lemke Oliver (2024), submitted.
Undergraduate work
Ball Packings with Periodic Constraints, with Robert Connelly and Jeffrey Shen, published in Discrete & Computational Geometry, 52.4 (2014).
Irreducible Canonical Representations in Positive Characteristic, with Benjamin Gunby and Allen Yuan, published in Research in Number Theory, 1.3 (2015).
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