×

zbMATH — the first resource for mathematics

Refined conjectures of the “Birch and Swinnerton-Dyer type”. (English) Zbl 0636.14004
Let A be an elliptic curve over \({\mathbb{Q}}\) which admits a modular parametrization. In an earlier joint work with J. Teitelbaum [Invent. Math. 84, 1-48 (1986)] the authors presented p-adic analogues of the Birch and Swinnerton-Dyer conjectures for A when p is a prime of ordinary reduction for A.
In this paper the authors offer remarkable refinements of these conjectures which eliminate any reference to the prime p. To be slightly more precise let M be a fixed positive integer. Using modular symbols for A the authors construct an element \(\theta_{A,M}\) in the group ring \({\mathbb{Q}}[({\mathbb{Z}}/M{\mathbb{Z}})\) \(*/(\pm 1)]\). If R is a subring of \({\mathbb{Q}}\) containing the coefficients of \(\theta_{A,M}\) then the analogue of having the L-series L(A,s) vanish to order \(\geq r\) at \(s=1\) is having \(\theta_{A,M}\) lie in the r-th power of the augmentation ideal \(I\subseteq R[({\mathbb{Z}}/M{\mathbb{Z}})\) \(*/(\pm 1)]\). The authors conjecture that \(\theta_{A,M}\) lies in I r where r is the rank of A(\({\mathbb{Q}})\) plus the number of primes dividing M which are also primes of split multiplicative reduction for A. In this case the image of \(\theta_{A,M}\) in I \(r/I^{r+1}\) plays the role of the r-th coefficient of the Taylor expansion of the L-function at \(s=1\). Under certain conditions the authors conjecture a precise formula for this image. They also show that if \(r=0\) then their conjecture is implied by the classical Birch and Swinnerton-Dyer conjecture.
Reviewer: S.Kamienny

MSC:
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H52 Elliptic curves
14H45 Special algebraic curves and curves of low genus
11F11 Holomorphic modular forms of integral weight
14K15 Arithmetic ground fields for abelian varieties
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] G. Shimura, Correction to: “Modular forms of half integral weight” , Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), Springer, Berlin, 1975, p. 145. Lecture Notes in Math., Vol. 476.
[2] A. O. L. Atkin and W. Li, Twists of newforms and pseudo-eigenvalues of \(W\)-operators , Invent. Math. 48 (1978), no. 3, 221-243. · Zbl 0369.10016
[3] A. Hales, Stable augmentation quotients of abelian groups , Pacific J. Math. 118 (1985), no. 2, 401-410. · Zbl 0573.20008
[4] S. Lang, Introduction to modular forms , Springer-Verlag, Berlin, 1976. · Zbl 0344.10011
[5] S. Lang, Fundamentals of Diophantine geometry , Springer-Verlag, New York, 1983. · Zbl 0528.14013
[6] S. Lang, Elliptic curves: Diophantine analysis , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 231, Springer-Verlag, Berlin, 1978. · Zbl 0388.10001
[7] Yu. I. Manin, Values of \(p\)-adic Hecke series at lattice points of the critical strip , Mat. Sb. (N.S.) 93(135) (1974), 621-626, 631. · Zbl 0304.14012
[8] B. Mazur and J. Tate, Canonical height pairings via biextensions , Arithmetic and geometry, Vol. I, Progr. Math., vol. 35, Birkhäuser Boston, Boston, MA, 1983, pp. 195-237. · Zbl 0574.14036
[9] B. Mazur, J. Tate, and J. Teitelbaum, On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer , Invent. Math. 84 (1986), no. 1, 1-48. · Zbl 0699.14028
[10] D. Mumford, Abelian varieties , Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970. · Zbl 0223.14022
[11] D. Mumford and J. Fogarty, Geometric invariant theory , Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. · Zbl 0504.14008
[12] D. G. Northcott, Finite free resolutions , Cambridge University Press, Cambridge, 1976. · Zbl 0328.13010
[13] I. B. S. Passi, Group rings and their augmentation ideals , Lecture Notes in Mathematics, vol. 715, Springer, Berlin, 1979. · Zbl 0405.20007
[14] G. Shimura, Introduction to the arithmetic theory of automorphic functions , Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971. · Zbl 0221.10029
[15] A. Grothendieck, M. Raynaud, and D. S. Rim, Groupes de monodromie en géométrie algébrique. I , Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969, Lecture Notes in Mathematics, vol. 288, Springer-Verlag, Berlin, 1972, viii+523. · Zbl 0237.00013
[16] J. T. Tate, The arithmetic of elliptic curves , Invent. Math. 23 (1974), 179-206. · Zbl 0237.00013
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.