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. This work was the subject of my dissertation at Harvard. The following two paper sequence is the complete form of this work.

  • 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.

    With Peter Koymans, I've also calculated the distribution of 3-Selmer groups in cubic twist families of rational elliptic curves of j-invariant 0.

  • Sums of rational cubes and the 3-Selmer group with Peter Koymans (2024), submitted.

    Other recent work

    I wrote two papers on the Cassels-Tate pairing with Adam Morgan. This material is used in my work on the distribution of Selmer groups.

  • 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).

    Thanks to the Honda-Tate theorem, the next two papers are about very similar topics in spite of their dissimilar titles.

  • Abelian varieties of prescribed order over finite fields, with Raymond van Bommel, Edgar Costa, Wanlin Li, and Bjorn Poonen (2021), submitted.

  • Algebraic integers with conjugates in a prescribed distribution, to appear in the Annals of Mathematics.

    I like finite groups and the large sieve.

  • Faithful Artin induction and the Chebotarev density theorem, with Robert J. Lemke Oliver (2024), submitted.

    Less recent work

    The following paper was based on my undergraduate work at Princeton under Shou-Wu Zhang. It pushed me into arithmetic statistics.

  • The congruent numbers have positive natural density (2016), in preparation.

    My undergraduate papers:

  • 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).

  • Markov