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).

    		• Markov