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 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
