×

On universal elliptic curves over Igusa curves. (English) Zbl 0705.14024

Let E be the universal elliptic curve over an Igusa curve. Arithmetic properties of E are studied aiming toward the Birch and Swinnerton-Dyer conjecture.
Let K denote the function field of an Igusa curve, and let E be the universal elliptic curve over an Igusa curve. Weierstrass models for E/K, the discriminant and the Hasse invariant of E over \({\mathbb{F}}_ p(j)\) are explicitly determined. Then the Hasse-Weil L-function of E over \({\mathbb{F}}_ p(j)\) is computed in terms of modular forms:
Proposition. \(L(E/{\mathbb{F}}_ p(j),s)=H_ q(S_ 3(\Gamma_ 0(p),(\frac{.}{p})),q^{-s})\) where q is a power of p, \((\frac{.}{p})\) the Legendre symbol, and \(H_ q\) is the Hecke polynomial.
Further the invariants of E over \({\mathbb{F}}_ p(j)\) such as torsion points, local invariants and Tamagawa numbers are explicitly computed. The functional equation for E is proved:
Proposition. Let \(\Lambda(s)=N_ E^{s/2}D^{2s}_{{\mathbb{F}}_ q(j)}L(E/{\mathbb{F}}_ q(j),s)\). Then \(\Lambda(s)= \pm \Lambda(2-s).\)
The Tate-Shafarevich group \(\text{Russian {Sh}}({\mathbb{F}}_ p(j),E)\) is computed for some primes p, and the Birch and Swinnerton-Dyer conjecture is verified numerically for these examples.
Reviewer: N.Yui

MSC:

14G25 Global ground fields in algebraic geometry
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H52 Elliptic curves
14H25 Arithmetic ground fields for curves
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Birch, B.J., Kuyk, W. (Eds.): Modular functions of one variable IV. (Lect. Notes in Math. Vol. 476.) Berlin-Heidelberg-New York: Springer 1973
[2] Cohen, H., Oesterlé, J.: Dimensions des espaces de formes modulaires. In: Serre, J.-P., Zagier, D.B. (Eds.). Modular functions of one variable VI. (Lect. Notes in Math. Vol. 627, pp. 69–73). Berlin-Heidelberg-New York: Springer 1977
[3] Crew, R.:L-functions ofp-adic characters and geometric Iwasawa theory. Invent. Math.88, 395–403 (1987) · Zbl 0615.14013 · doi:10.1007/BF01388914
[4] Deuring, M.: Die Typen der Multiplikatorenringe elliptischer Funktionenkörper. Abh. Math. Semi. Univ. Hamb.14, 197–272 (1941) · Zbl 0025.02003 · doi:10.1007/BF02940746
[5] Eichler, M.: The basis problem for modular forms and the traces of the Hecke operators. In: Kuyk, W. (Eds) Modular functions of one variable I. (Lect. Notes in Math. Vol. 320, pp. 75–151) Berlin-Heidelberg-New York: Springer 1973
[6] Gross, B.: Heegner points and the modular curve of prime level. J. Math. Soc. Japan39, 345–362 (1987) · Zbl 0623.14010 · doi:10.2969/jmsj/03920345
[7] Hasse, H.: Existenz separabler zyklishcer unverzweigter Erweiterungskörper vom Primzahlgradep über elliptischen Funktionenkörpern der Charakteristikp. J. Reine Angew. Math.172, 77–85 (1934) · Zbl 0010.14803
[8] Igusa J.: On the algebraic theory of elliptic modular functions. J. Math. Soc. Japan20, 96–106 (1968) · Zbl 0164.21101 · doi:10.2969/jmsj/02010096
[9] Ihara, Y.: Hecke polynomials as congruence {\(\zeta\)}-functions in elliptic modular case. Ann. Math. (2)85, 267–295 (1967) · Zbl 0181.36501 · doi:10.2307/1970442
[10] Katz, N., Mazur, B.: Arithmetic moduli of elliptic curves. Princeton: Princeton University Press 1985 · Zbl 0576.14026
[11] Manin, J.: The Hasse-Witt matrix of an algebraic curve. Transl. Am. Math. Soc. (2)45, 245–264 (1965) · Zbl 0148.28002
[12] Milne, J.S.: On a conjecture of Artin and Tate. Ann. Math. (2)102, 517–533 (1975) · Zbl 0343.14005 · doi:10.2307/1971042
[13] Mumford, D.: Abelian varieties. Oxford: Oxford University Press 1970 · Zbl 0223.14022
[14] Ogg, A.: On the eigenvalues of Hecke operators. Math. Ann.179, 101–108 (1969) · Zbl 0169.10102 · doi:10.1007/BF01350121
[15] Ross, S.: Hecke operators for Г0(N), their traces, and applications. Ph.D. Thesis, University of Rochester (1985)
[16] Shimura, G.: On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields. Nagoya Math. J.43, 199–208 (1971) · Zbl 0225.14015
[17] Shimura, G.: Introduction to the arithmetic theory of automorphic functions. Princeton: Princeton University Press 1971 · Zbl 0221.10029
[18] Swinnerton-Dyer, H.P.F.: Onl-adic representations and congruences for coefficients of modular forms. In: Kuyk, W., Serre, J.-P. (Eds.) Modular functions of one variable III. (Lect. Notes in Math. Vol. 350, pp. 1–56). Berlin-Heidelberg-New York: Springer 1973
[19] Tate, J.: On the conjecture of Birch and Swinnerton-Dyer and a geometric analog, Seminaire Bourbaki 1965/66, Exposé 306. In: Grothendieck, A. (ed.) Dix exposés sur la cohomologie des schemas, pp. 189–214. Amsterdam: North-Holland 1968
[20] Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. In: Birch, B.J., Kuyk, W. (Eds.) Modular functions of variable IV. (Lect. Notes in Math. Vol. 476, pp. 33–52). Berlin-Heidelberg-New York: Springer 1973 · Zbl 1214.14020
[21] Tate, J., Oort, F.: Group schemes of prime order. Ann. Sci. Ec. Norm. Super., IV. Ser.3, 1–21 (1970) · Zbl 0195.50801
[22] Ulmer, D.L.:L-functions of universal elliptic curves over Igusa curves. Am. J. Math. (to appear) · Zbl 0731.14013
[23] Weil, A.: Adeles and algebraic groups. Boston: Birkhäuser 1982 · Zbl 0493.14028
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.