Root numbers of semistable elliptic curves in division towers
Let \(F\) be a number field. Let \(E/F\) an elliptic curve. Denote \(F^\infty=F(E[p^\infty])\) the field of all \(p\)-division points on \(E\). The paper under review discusses the root number \(W(E,\tau)\) of the \(L\)-function \(L(s,E,\tau)\) where \(\tau\) is an irreducible self-dual representation of \(F^\infty/F\). Under some assumptions \(E/F\) is semistable, \(p\) is odd, the natural map \(\text{Gal}(F^\infty/F) \hookrightarrow \text{GL}(T_p)\cong \text{GL}_2(\mathbb Z_p)\) is an isomorphism, and for a bad place \(v\) we have that \(\text{ord}_v j(E) \not \equiv 0 \bmod p\)) the root number \(W(E,\tau)\) is computed in terms of invariants of \(E\) and \(F\). The proof consists of a detailed study of linear representations of \(\text{GL}(2,\mathbb Z_p)\).
scientific article; zbMATH DE number 5127173