3 papers
math.NT2026
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…
math.NT2025
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…
math.NT2025
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…