Note on equality of \(L\)-functions of elliptic curves (Q1383578)

Let \(K,M\) be Galois fields over \(\mathbb{Q}\); assume that \(K,M\) are Galois and quadratic over \(K\cap M\). Let \(A,B\) be elliptic curves over \(\mathbb{Q}\) satisfying \(L(A,K,s)= L(B,M,s)\). The author proves: (i) \(A\) and \(B\) have complex multiplication, (ii) \(A\) and \(B\) are isogenous over \(KM\), (iii) if \(K_0\) is the corresponding field of complex multiplication then \(K_0\subset KM\) and \(K_0\not\subset K\), \(K_0\not\subset M\). Key ingredients in the proof are: Faltings' theorem on isogenies of abelian varieties and Serre's theorem on supersingular reduction.