Tamagawa numbers and positive rank of elliptic curves
Edwina Aylward
TL;DR
The paper shows that predictions of positive rank obtained from the norm-relations Tamagawa-product test are subsumed by parity-based (twist) methods for elliptic curves. It develops a representation-theoretic framework using K-relations and regulator constants to express products of local Tamagawa data modulo quadratic-norms as twisted-root-number data, linking local invariants to global rank via regulator constants. The main technical contribution is extending regulator constants from Brauer relations to K-relations and proving a local-to-global compatibility between Tamagawa terms and root numbers. As a consequence, the norm-relations test cannot predict rank growth beyond what parity twists already predict, and BSD-quotients obey norm-relations compatible with parity, under suitable hypotheses. The work unifies Tamagawa-number-based and parity-based approaches, clarifying the limitations and exact scope of norm-relations predictions in rank behavior.
Abstract
This paper addresses the prediction of positive rank for elliptic curves without the need to find a point of infinite order or compute L-functions. While the most common method relies on parity conjectures, a recent technique introduced by Dokchitser, Wiersema, and Evans predicts positive rank based on the value of a certain product of Tamagawa numbers, raising questions about its relationship to parity. We show that their method is a subset of the parity conjectures approach: whenever their method predicts positive rank, so does the use of parity conjectures. To establish this, we extend previous work on Brauer relations and regulator constants to a broader setting involving combinations of permutation modules known as K-relations. A central ingredient in our argument is demonstrating a compatibility between Tamagawa numbers and local root numbers.