28 citations · 54 across the 16 of their papers we have counts for
16 papers
Tamagawa ratios and unbounded Selmer moments
Peter Koymans, Alexander Smith
We develop a framework to predict whether a family of Selmer groups has average size that is bounded or unbounded. Applying this framework to certain geometric families of abelian…
Lattice points in thickened parabolas and rational points near hypersurfaces
Alexander Smith
Among the nondegenerate C^4 hypersurfaces M in R^n, we characterize the rational quadrics as the hypersurfaces that are the least well approximated by rational points. Given M othe…
The Birch and Swinnerton-Dyer conjecture implies Goldfeld's conjecture
Alexander Smith
Given an elliptic curve E/Q, we show that 50% of the quadratic twists of E have -Selmer corank 0 and 50% have -Selmer corank 1. As one consequence, we prove…
Sums of rational cubes and the -Selmer group
Peter Koymans, Alexander Smith
Recently, Alpöge-Bhargava-Shnidman determined the average size of the -Selmer group in the cubic twist family of any elliptic curve over with -invariant . We…
New Lower Bounds for the Schur-Siegel-Smyth Trace Problem
Bryce Joseph Orloski, Naser Talebizadeh Sardari, Alexander Smith
We derive and implement a new way to find lower bounds on the smallest limiting trace-to-degree ratio of totally positive algebraic integers and improve the previously best known b…
The distribution of -Selmer groups in degree twist families I
Alexander Smith
In this paper and its sequel, we develop a technique for finding the distribution of -Selmer groups in degree twist families of Galois modules over number fie…
The distribution of -Selmer groups in degree twist families II
Alexander Smith
We continue the investigation of the distribution of -Selmer groups in degree twist families of Galois modules over number fields begun in the previous paper.…
Field change for the Cassels-Tate pairing and applications to class groups
Adam Morgan, Alexander Smith
In previous work, the authors defined a category of finite Galois modules decorated with local conditions for each global field . In this paper, given an extension $K/F…
Algebraic integers with conjugates in a prescribed distribution
Alexander Smith
Given a compact subset of the real numbers obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in . The distribution of con…
Abelian varieties of prescribed order over finite fields
Raymond van Bommel, Edgar Costa, Wanlin Li +2
Given a prime power and , we prove that every integer in a large subinterval of the Hasse--Weil interval is $#A(\mathbb{F}_q)$…
The Cassels-Tate pairing for finite Galois modules
Adam Morgan, Alexander Smith
Given a global field with absolute Galois group , we define a category whose objects are finite -modules decorated with local conditions. We define this cate…
-Selmer groups, -class groups, and Goldfeld's conjecture
Alexander Smith
We prove that the -class groups of the imaginary quadratic fields have the distribution predicted by the Cohen-Lenstra heuristic. Given an elliptic curve E/Q with full ra…
Governing fields and statistics for 4-Selmer groups and 8-class groups
Alexander Smith
Taking A to be an abelian variety with full 2-torsion over a number field k, we investigate how the 4-Selmer rank of the quadratic twist A^d changes with d. We show that this rank…
The congruent numbers have positive natural density
Alexander Smith
We prove that the rational elliptic curve y^2 = x^3 - n^2x satisfies the full Birch and Swinnerton-Dyer conjecture for at least 41.9% of positive squarefree integers n equal to 1,…
Irreducible Canonical Representations in Positive Characteristic
Benjamin Gunby, Alexander Smith, Allen Yuan
For a curve over a field of positive characteristic, we investigate when the canonical representation of on is irreducible. Any curve with an irre…
Ball Packings with Periodic Constraints
Robert Connelly, Jeffrey D. Shen, Alexander D. Smith
We call a periodic ball packing in d-dimensional Euclidean space periodically (strictly) jammed with respect to a period lattice if there are no nontrivial motions of the balls tha…