Analytic ranks of elliptic curves over cyclotomic fields (Q2781425)

Let \(E\) be an elliptic curve defined over \(\mathbb Q\) and let \(L(s,E,\chi)\) denote the \(L\)-function of \(E\) twisted by a primitive Dirichlet character \(\chi\). Let \(K_q\) denote the cyclotomic extension of \(\mathbb Q\) obtained by adjoining the \(q\)th roots of unity. The author defines the analytic rank of \(E(K_q)\) to be NEWLINE\[NEWLINE \text{ord}_{s=1/2} \left( \prod_{\chi \pmod{q}} L(s,E,\chi) \right). NEWLINE\]NEWLINE Extending techniques of \textit{D. E. Rohrlich} [Sémin. Théor. Nombres, Univ. Bordeaux I, 1983-1984, Exp. No. 14 (1984; Zbl 0565.14007)] the author proves: NEWLINENEWLINENEWLINETheorem 1. Let \(E\) be an elliptic curve defined over \(\mathbb Q\) of conductor \(N\). For any \(\varepsilon > 0\) and \(q\) a sufficiently large prime (in terms of \(N\) and \(\varepsilon\)), NEWLINE\[NEWLINE\text{analytic rank of }E(K_q) < q^{7/8+\varepsilon}. NEWLINE\]NEWLINE The author also defines \(\eta(p)\) to be the smallest \(k\geq 0\) such that \(L(1/2,E,\chi) \neq 0\) for all primitive Dirichlet characters of conductor \(p^j\) with \(j>k\) and proves NEWLINENEWLINENEWLINETheorem 2. Let \(E\) be an elliptic curve defined over \(\mathbb Q\). Then NEWLINE\[NEWLINE \eta(p) \ll_{E} 1.NEWLINE\]