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 (2022), in preparation.

  • The distribution of fixed point Selmer groups in twist families (2022), in preparation.

    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.

    I am organizing a learning seminar about this work that will start in September 2022. Details are here.

    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 (2022), submitted.

    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 (2021), 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