Analytic ranks of elliptic curves over cyclotomic fields. (English) Zbl 1028.11040
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
\[
\text{ord}_{s=1/2} \left( \prod_{\chi \pmod{q}} L(s,E,\chi) \right).
\]
Extending techniques of D. E. Rohrlich [Sémin. Théor. Nombres, Univ. Bordeaux I, 1983-1984, Exp. No. 14 (1984; Zbl 0565.14007)] the author proves:
Theorem 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 \(\varepsilon\)), \[ \text{analytic rank of }E(K_q) < q^{7/8+\varepsilon}. \] 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
Theorem 2. Let \(E\) be an elliptic curve defined over \(\mathbb Q\). Then \[ \eta(p) \ll_{E} 1. \]
Theorem 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 \(\varepsilon\)), \[ \text{analytic rank of }E(K_q) < q^{7/8+\varepsilon}. \] 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
Theorem 2. Let \(E\) be an elliptic curve defined over \(\mathbb Q\). Then \[ \eta(p) \ll_{E} 1. \]
Reviewer: Kevin L.James (Clemson)
MSC:
11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
11G05 | Elliptic curves over global fields |
Citations:
Zbl 0565.14007References:
[1] | M. Ram Murty, On simple zeros of certain L-series, in: Number theory, de Gruyter, Berlin (1990), 427-439. |
[2] | Rohrlich David E., Exp. No. 14 pp 10– |
[3] | David, Invent. Math. 75 (3) pp 409– (1984) |
[4] | Shimura Goro, Comm. Pure Appl. Math. 29 (6) pp 783– (1976) |
[5] | Shimura Goro, Math. Ann. 229 (3) pp 211– (1977) |
[6] | Stefanicki Tomasz, Math. 474 pp 1– (1996) |
[7] | Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. (2) 141 (3) (1995), 553-572. · Zbl 0823.11030 |
[8] | Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. Math. (2) 141 (3) (1995), 443-551. Department of Mathematics.Box1917, Brown University, Providence, RI 02912 · Zbl 0823.11029 |
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