Root numbers of semistable elliptic curves in division towers

Abstract

The growth of the Mordell-Weil rank of an elliptic curve in a tower of number fields can be discussed at many levels. On the one hand, the issues raised can be embedded in a broader Iwasawa theory of elliptic curves (Mazur [10]); on the other hand, they can be crystallized in a single easily stated question, namely whether the rank of the elliptic curve over subextensions of finite degree is bounded in the tower. But whatever one’s point of view, the case of abelian towers has seen important advances in recent years, including notably the results of Kato on cyclotomic towers, of Cornut [2] and Vatsal [18] on anticyclotomic towers, and of Skinner and Urban on Mazur’s conjecture. In the case of nonabelian towers, by contrast, virtually nothing is known, and even a conjectural framework has begun to emerge only recently [1]. However Greenberg [7] observed more than twenty years ago that some insight could already be gained from the classical conjectures about L-functions: They imply that for certain pairs (E1, E2) of elliptic curves of relatively prime conductor there exist primes p such that the rank of E1 is unbounded in the p-division tower of E2 (which can be chosen not to have complex multiplication, so that the division tower is nonabelian). Greenberg’s idea was elaborated further by L. Howe [8], who gave a conjectural lower bound for the growth of the rank of E1 in the division tower of E2. The aim of the present note is to show that the standard conjectures also have some bearing on the growth of the rank of an elliptic curve in its own division tower. For example, we shall see that if E is a semistable elliptic curve over Q and p is a sufficiently large prime congruent to 3 mod 4 then the rank of E should be unbounded in its p-division tower. As in the papers of Greenberg and Howe, the mechanism for arriving at such conclusions is a root number calculation. Fix a number field F , an elliptic curve E over F , and a prime p, and put F∞ = F (E[p∞]), where E[p∞] denotes the group of points on E of p-power order. The root numbers at issue here are associated to the L-functions L(s,E, τ), where τ runs over irreducible self-dual representations of Gal(F∞/F ). For a precise definition of L(s,E, τ) we refer the reader to [12], pp. 151 and 156, or to [1], §5, but we should at least spell out our group-theoretic conventions: First of all, a representation of a topological group is understood to be continuous, finite-dimensional, and defined over the complex numbers. In particular, in the case of a profinite group like Gal(F∞/F ) a representation is trivial on an open subgroup and hence factors through a finite quotient. It also follows that when the term character is used in the sense of “one-dimensional representation” we are talking about a continuous homomorphism to C×. On the other hand, if we think of a character as the trace of a representation of arbitrary dimension then in the case of a compact group like Gal(F∞/F ) characters provide a criterion