An analogue of Kida’s formula for the \(p\)-adic \(L\)-functions of modular elliptic curves. (English) Zbl 0970.11021
Let \(p\) be an odd prime and let \(E\) be an elliptic curve defined over \(\mathbb Q\) having good ordinary reduction or multiplicative reduction at \(p\). In these cases, B. Mazur and P. Swinnerton-Dyer [Invent. Math. 25, 1-61 (1974; Zbl 0281.14016)] and B. Mazur, J. Tate, and J. Teitelbaum [Invent. Math. 84, 1-48 (1986; Zbl 0699.14028)] have attached \(p\)-adic \(L\)-functions to \(E\).
Let \(L/K\) be a \(p\)-extension of abelian number fields. Assume that \(E\) has additive reduction at all primes of \(K\) lying above rational primes at which \(E\) has additive reduction. In addition, assume that the \(p\)-adic \(L\)-function of \(E\) satisfies a certain integrality condition (that holds for almost all primes of good ordinary reduction). Then there are \(p\)-adic \(L\)-functions attached to \(E\) over \(L\) and \(K\), and there are associated Iwasawa \(\lambda\) and \(\mu\) invariants.
The author shows that if \(\mu_E(K)=0\), then \(\mu_E(L)=0\) and there is a formula, similar to the Riemann-Hurwitz genus formula, relating \(\lambda_E(L)\) and \(\lambda_E(K)\).
A similar result for Iwasawa invariants of class groups of numbers fields was given by G. Gras [Publ. Math. Fac. Sci. Besançon, Théor. Nombres, Année 1978-1979, Exp. No. 5, 37 p. (1979; Zbl 0472.12009)], L. Kuz’min [Izv. Akad. Nauk. SSSR, Ser. Mat. 43, 483-546 (1979; Zbl 0434.12006)], and Y. Kida [J. Number Theory 12, 519-528 (1980; Zbl 0455.12007)].
The method of the present paper is based on the paper of W. M. Sinnott [Compos. Math. 53, 3-17 (1984; Zbl 0545.12011)]. A result for Selmer groups, similar to the one in the present paper, has been given by Y. Hachimori and the author [J. Algebr. Geom. 8, 581-601 (1999; Zbl 1081.11508)].
Let \(L/K\) be a \(p\)-extension of abelian number fields. Assume that \(E\) has additive reduction at all primes of \(K\) lying above rational primes at which \(E\) has additive reduction. In addition, assume that the \(p\)-adic \(L\)-function of \(E\) satisfies a certain integrality condition (that holds for almost all primes of good ordinary reduction). Then there are \(p\)-adic \(L\)-functions attached to \(E\) over \(L\) and \(K\), and there are associated Iwasawa \(\lambda\) and \(\mu\) invariants.
The author shows that if \(\mu_E(K)=0\), then \(\mu_E(L)=0\) and there is a formula, similar to the Riemann-Hurwitz genus formula, relating \(\lambda_E(L)\) and \(\lambda_E(K)\).
A similar result for Iwasawa invariants of class groups of numbers fields was given by G. Gras [Publ. Math. Fac. Sci. Besançon, Théor. Nombres, Année 1978-1979, Exp. No. 5, 37 p. (1979; Zbl 0472.12009)], L. Kuz’min [Izv. Akad. Nauk. SSSR, Ser. Mat. 43, 483-546 (1979; Zbl 0434.12006)], and Y. Kida [J. Number Theory 12, 519-528 (1980; Zbl 0455.12007)].
The method of the present paper is based on the paper of W. M. Sinnott [Compos. Math. 53, 3-17 (1984; Zbl 0545.12011)]. A result for Selmer groups, similar to the one in the present paper, has been given by Y. Hachimori and the author [J. Algebr. Geom. 8, 581-601 (1999; Zbl 1081.11508)].
Reviewer: Lawrence Washington (College Park)
MSC:
| 11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |
| 11G05 | Elliptic curves over global fields |
| 11S40 | Zeta functions and \(L\)-functions |
Citations:
Zbl 0281.14016; Zbl 0699.14028; Zbl 0472.12009; Zbl 0434.12006; Zbl 0455.12007; Zbl 0545.12011; Zbl 1081.11508References:
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| [2] | Y. Hachimori, Computation of the invariants of \(pL\); Y. Hachimori, Computation of the invariants of \(pL\) |
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| [4] | Kida, Y., \(l\)-extensions of CM-fields and cyclotomic invariants, J. Number Theory, 12, 519-528 (1980) · Zbl 0455.12007 |
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| [11] | Sinnott, W. M., On \(p\)-adic \(L\)-functions and the Riemann-Hurwitz genus formula, Composito Math., 53, 3-17 (1984) · Zbl 0545.12011 |
| [12] | Stevens, G., Stickelberger elements and modular parametrizations of elliptic curves, Invent. Math., 98, 75-106 (1989) · Zbl 0697.14023 |
| [13] | Tang, S.-L, Congruences between modular forms, cyclic isogenies of modular elliptic curves, and integrality of \(p\)-adic \(L\)-functions, Trans. Amer. Math. Soc., 349, 837-856 (1997) · Zbl 1042.11519 |
| [14] | Washington, L. C., Introduction to Cyclotomic Fields. Introduction to Cyclotomic Fields, Graduate Texts in Math., 83 (1997), Springer-Verlag: Springer-Verlag Berlin/New York · Zbl 0966.11047 |
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