Twisted root numbers of elliptic curves semistable at primes above 2 and 3
Let \(E\) be an elliptic curve defined over a number field \(F\). Fix a prime number \(p\) and consider the field \(F^\infty = F(E[p^\infty])\) generated by the \(p\)-power torsion points. Let \(\tau\) be an irreducible self-dual representation of Gal(\(F^\infty/F\)), let \(L(s,E,\tau)\) be the associated L-function and \(W(E,\tau) = (-1)^w\) the corresponding root number. The main result of this article based on the author's dissertation gives explicit formulas for \(w\) modulo \(2\) which are similar to (but more general than) the formulas obtained by \textit{D. E. Rohrlich} [Math. Res. Lett. 13, No. 2--3, 359--376 (2006; Zbl 1124.11026)]. As an application of his results, the author proves that if \(E\) is an elliptic curve over the rationals, semistable at \(2\) and \(3\), and if \(p \equiv 3 \bmod 4\) is a sufficiently large prime, then the rank of \(E\) is unbounded in its \(p\)-division tower.
scientific article; zbMATH DE number 5704470
scientific article