ON TRIVIAL p-ADIC ZEROES FOR ELLIPTIC CURVES OVER KUMMER EXTENSIONS

We prove the exceptional zero conjecture is true for semistable elliptic curves E/Q over number fields of the form F ( e2πi/q n ,∆ 1/q 1 , . . . ,∆ 1/q d ) where F is a totally real field, and the split multiplicative prime p 6= 2 is inert in F (e2πi/q n ) ∩ R. In 1986 Mazur, Tate and Teitelbaum [9] attached a p-adic L-function to an elliptic curve E/Q with split multiplicative reduction at p. To their great surprise, the corresponding p-adic object Lp(E, s) vanished at s = 1 irrespective of how the complex L-function L(E, s) behaves there. They conjectured a formula for the derivative Lp(E, 1) = logp(qE) ordp(qE) × L(E, 1) period where E(Qp) ∼= Qp / qZ E , and this was subsequently proven for p ≥ 5 by Greenberg and Stevens [6] seven years later. In recent times there has been considerable progress made on generalising this formula, both for elliptic curves over totally real fields [10, 15], and for their adjoint L-functions [12]. In this note, we outline how the techniques in [3] can be used to establish some new cases of the exceptional zero formula over solvable extensions K/Q that are not totally real. 1. Constructing the p-adic L-function Let E be an elliptic curve defined over Q, and p ≥ 3 a prime of split multiplicative reduction. First we fix a finite normal extension K/Q whose Galois group is a semidirect product Gal(K/Q) = Γ nH where Γ,H are both abelian groups, with H = Gal(K/K ∩ Q) and likewise Γ ∼= Gal(K ∩ Q/Q). Secondly we choose a totally real number field F disjoint from K, and in addition suppose: (H1) the elliptic curve E is semistable over F ; (H2) the prime p is unramified in K; (H3) the prime p is inert in the compositum F · k for all CM fields k ⊂ K ∩Q. 2010 Mathematics Subject Classification 11F33, 11F41, 11F67.