The Eisenstein descent on \(J_ 0(N)\). (English) Zbl 0594.14027

The paper under review deals with the general question of how to compute the rank of an abelian variety over a number field K. The standard approach uses the injection \(A(K)/n\cdot A(K)\to H^ 1(0_ k,A[n])\) (flat cohomology): One tries to find an upper bound for the righthand side, and a lower bound for the left hand side which coincides with it. Here the authors deal with A \(= Eisenstein\)-quotient of \(J_ 0(N) = Jacobian\) of the modular curve \(X_ 0(N)\), \(N=p\) or \(N=p^ 2\) (p a prime. The case \(K={\mathbb{Q}}\), \(N=p\) has been treated extensively by B. Mazur).
Under certain hypotheses (on Bernoulli-numbers) the authors can show that either the right hand side vanishes, or that the theory of Heegner points suffices to account for all of it. The details of their arguments are a little bit involved, so we prefer not to comment on them.
Reviewer: G.Faltings


14H40 Jacobians, Prym varieties
14K15 Arithmetic ground fields for abelian varieties
11F12 Automorphic forms, one variable
14G25 Global ground fields in algebraic geometry
14K30 Picard schemes, higher Jacobians
Full Text: DOI EuDML


[1] Atkin, A.O.L., Lehner, I.: Hecke operators on? 0(m). Math. Ann.185, 134-160 (1970) · Zbl 0185.15502
[2] Buhler, J.P., Gross, B.H.: Arithmetic on elliptic curves with complex multiplication II. Invent. Math.79, 11-29 (1985) · Zbl 0584.14027
[3] Greenberg, R.: Onp-adicL-functions and cyclotomic fields II. Nagoya Math. J.67, 139-158 (1977) · Zbl 0373.12007
[4] Gross, B.H.: Arithmetic on clliptic curves with complex multiplication. Springer Lect. Notes776, 1-95 (1980)
[5] Gross, B.H.: Heegner points onX 0 (11). Sém. Th. Nomb de Bordeaux, Exposé34, 34.01-34.05 (1981-82)
[6] Gross, B.H.: Heegner points onX 0 (N). In: Rankin, R.A. (ed.): Modular forms. Chichester: Ellis Horwood 1984, pp. 87-106
[7] Gross, B.H., Koblitz, N.: Gauss sums and thep-adic ?-function. Ann. Math.109, 569-581 (1979) · Zbl 0406.12010
[8] Ligozat, G.: Courbes modulaires de niveau 11. Modular functions of one variable V. Springer Lect. Notes609, 149-238 (1977) · Zbl 0357.14006
[9] Lubin, J.: Canonical subgroups of formal groups. Trans. A.M.S.251, 103-127 (1979) · Zbl 0431.14014
[10] Mazur, B.: Modular curves and the Eisenstein ideal. Publ. Math. Inst. Hautes Etud. Sci.47, 33-189 (1977) · Zbl 0394.14008
[11] Mazur, B.: On the arithmetic of special values ofL-functions. Invent. Math.55, 207-240 (1979) · Zbl 0426.14009
[12] Mazur, B., Roberts, L.: Local Euler characteristics. Invent Math.9, 201-234 (1970) · Zbl 0191.19202
[13] Mazur, B., Wiles, A.: Class fields of abelian extension of ?. Invent. Math.76, 179-330 (1983) · Zbl 0545.12005
[14] Ribet, K.: A modular construction of unramifiedp-extensions of ?(? p ). Invent. Math.34, 151-162 (1976) · Zbl 0338.12003
[15] Ribet, K.: Letter to B.H. Gross and J. Lubin, March 3, 1983
[16] Ribet, K., Papier, E.: Eisenstein ideals and ?-adic representations. J. Fac Sci., Tokyo28, 651-665 (1982) · Zbl 0508.12012
[17] Robetts, L.: On the flat cohomology of finite group schemes. Thesis, Harvard University 1968
[18] Serre, J.-P.: Letter to J.-M. Fontaine. May 27, 1979
[19] Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent Math.15, 259-331 (1972) · Zbl 0235.14012
[20] Serre, J.-P., Tate, J.: Good reduction of abelian varieties. Ann. Math.88, 492-517 (1968) · Zbl 0172.46101
[21] 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
[22] Stevens, G.: The cuspidal group and special values ofL-functions. Trans A.M.S. (to appear) · Zbl 0579.10011
[23] Swinnerton-Dyer: Onl-adic representations on congruences for coefficients of modular forms. Modular functions of one variable III. Springer Lect. Notes350, 1-55 (1973)
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