Rank determines semi-stable conductor. (Q1427984)

The following question posed by Yu. Zarkhin motivates the study in this paper: Let \(X_1\) and \(X_2\) be Abelian varieties defined over \(k\). If \(\text{rank}_{\mathbb{Z}}(X_1(K))= \text{rank}_{\mathbb{Z}}(X_2(K))\) for every finite extension \(K/k\), then are \(X_1\) and \(X_2\) isogenous over \(k\)? The author proves the following: Let \(E_1\), \(E_2\) be elliptic curves defined over \(\mathbb{Q}\), with conductors \(N(E_1)\) and \(N(E_2)\). Suppose that for every number field \(K\) with \([K:\mathbb{Q}]\leq 2\), \(\text{ord}_{s=1}L(E_1/K,s)\equiv \text{ord}_{s=1} L(E_2/K,s)\pmod 2\) holds. Then \(N(E_1)\) and \(N(E_2)\) are equal up to square factors. Under the additional assumption that \(E_i\) are semi-stable and the 2-primary parts of the Tate-Shafarevich groups of \(E_i\) are finite for all such \(K\), he shows that if \(\text{rank}_{\mathbb{Z}}(E_1(K))\equiv \text{rank}_{\mathbb{Z}}(E_2(K))\pmod 2\), then \(N(E_1)= N(E_2)\). In particular, there are only a finite number of isogeny classes of such elliptic curves.