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The geometric distribution of Selmer groups of elliptic curves over function fields
Tony Feng, Aaron Landesman, Eric M. Rains
Fix a positive integer and a finite field . We study the joint distribution of the rank of , the -Selmer group of , and the -torsion in the Tate-Shafar…
Algebraic maps constant on isomorphism classes of unpolarized abelian varieties are constant
Eric Rains, Karl Rubin, Travis Scholl +2
We show that if a rational map is constant on each isomorphism class of unpolarized abelian varieties of a given dimension, then it is a constant map. Our results are motivated by…
Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves
Manjul Bhargava, Daniel M. Kane, Hendrik W. Lenstra +2
Using maximal isotropic submodules in a quadratic module over Z_p, we prove the existence of a natural discrete probability distribution on the set of isomorphism classes of short…
Self cup products and the theta characteristic torsor
Bjorn Poonen, Eric Rains
We give a general formula relating self cup products in cohomology to connecting maps in nonabelian cohomology, and apply it to obtain a formula for the self cup product associated…
Random maximal isotropic subspaces and Selmer groups
Bjorn Poonen, Eric Rains
Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over F_p. A random subspace chos…
On the Integrality of n-th Roots of Generating Functions
Nadia Heninger, E. M. Rains, N. J. A. Sloane
Motivated by the discovery that the eighth root of the theta series of the E_8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we…