Arithmetic on Elliptic Threefolds
Rania Wazir
TL;DR
This work extends Nagao-type rank formulas from elliptic surfaces to elliptic threefolds by connecting the Mordell–Weil rank over the base to averages of fibral Frobenius traces, assuming Tate’s conjecture. It establishes a cohomological isomorphism H^1_et(S̄) ≅ H^1_et(Ē) as Galois modules, derives a Shioda–Tate-type decomposition for NS(E) on elliptic n-folds, and analyzes singular fibers to relate Frobenius action to the geometry of fibral components. The main result expresses a_p averages through residues of L-functions and equates a principal residue to rank E(S/k), yielding a Nagao-type formula for elliptic threefolds with Tate’s conjecture as the bridge. Overall, the paper advances understanding of rank in higher-dimensional fibrations by tying arithmetic invariants to the geometry of singular fibers and global cohomology.
Abstract
In a recent paper, Rosen and Silverman showed that Tate's conjecture on the order of vanishing of L(E,s) implies Nagao's formula, which gives the rank of an elliptic surface in terms of a weighted average of fibral Frobenius trace values. The aim of this article is to extend their result to the case of elliptic threefolds, and deduce, from Tate's conjecture, a Nagao-type formula for the rank of an elliptic threefold E. This will require a two-pronged approach: on the one hand, we need some cohomological results in order to derive a Shioda-Tate-like formula for elliptic threefolds; on the other, we compute an "average" number of rational points on the singular fibers and relate this to the action of Galois on those fibers.