zbMATH — the first resource for mathematics

On certain Galois representations related to the modular curve \(X_ 1(p)\). (English) Zbl 0853.11045
Suppose that \(E\) is a modular elliptic curve of conductor \(pN\), \((N, p) =1\), and having split multiplicative reduction at \(p\). Write \(q(E)= u(E) p^m\), where \(q(E)\) is the Tate \(p\)-adic period of \(E\) and \(u(E)\) is a \(p\)-adic unit. B. Mazur and J. Tate [Duke Math. J. 54, 711-750 (1987; Zbl 0636.14004)] have made a delicate conjecture relating the value of \(u(E) \bmod p\) to a certain expression in modular symbols. Mazur and Tate’s conjecture refines the “Exceptional zero conjecture” [B. Mazur, J. Tate and J. Teitelbaum, Invent. Math. 84, 1-48 (1986; Zbl 0699.14028)], which predicts a value for the \(p\)-adic logarithm of \(u(E)\) in terms of the \(p\)-adic \(L\)-function of \(E\).
In this important paper, the author presents one-half of his proof of the conjecture of Mazur and Tate. The techniques of the entire proof are inspired by the ideas of R. Greenberg and G. Stevens [Invent. Math. 111, 407-447 (1993; Zbl 0778.11034)], who used Hida’s theory of deformations of Galois representations to prove the exceptional zero conjecture. This paper interprets the invariant \(u(E) \bmod p\) in terms of deformations of Galois representations. The second part of the proof, which relates this invariant to modular symbols, is published separately [the author, \(p\)-adic periods and modular symbols of elliptic curves of prime conductor, Invent. Math. 121, No. 2, 225-255 (1995)].
Let \(r\) be the highest power of a prime \(l\neq 2, 3\) which divides \(p-1\). The key insight of this paper is the recognition that a certain subgroup \(V\) of the Pontryagin dual to the \(r\)-torsion of the Jacobian \(J_1 (p)\) of \(X_1 (p)\) can be viewed in a natural way as a deformation of the Galois representation \(V= \operatorname{Hom} (J_0 [r], \mathbb{Q}/ \mathbb{Z})\). Furthermore, by analyzing the geometry of \(X_1 (p)\), the author constructs a 2-step filtration of \(V\) as a representation of \(\text{Gal} (\overline {\mathbb{Q}}_p/ \mathbb{Q}_p)\), and this filtration deforms the filtration on \(V\) (see Theorem 1 of the paper). The construction of this filtration relies on and extends the work of B. Mazur and A. Wiles on Igusa curves [Compos. Math. 59, 231-264 (1986; Zbl 0654.12008)].
Having constructed the deformation \(V\) with its filtration, the author constructs an invariant which classifies these deformations (Theorem 2) and computes this invariant to obtain the value \(u(E) \bmod p\) (Theorem 3).

11F80 Galois representations
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G05 Elliptic curves over global fields
14H52 Elliptic curves
11F85 \(p\)-adic theory, local fields
Full Text: Numdam EuDML
[1] Artin, M. Néron models , in: G. Cornell and J. Silverman (eds.) Arithmetic Geometry , Springer-Verlag, New York (1986). · Zbl 0603.14028
[2] Bosch, S. , Lutkebohmert, W. and Raynaud, M. , Néron Models . Springer-Verlag. · Zbl 0705.14001
[3] De Shalit, E. , Kronecker’s polynomial, supersingular elliptic curves, and p-adic periods of modular curves . Contemp. Math. 165, AMS (1994), 135-148. · Zbl 0863.14015
[4] De Shalit, E. , P-adic periods and modular symbols of elliptic curves of prime conductor . To appear in Invent. Math. · Zbl 1044.11576
[5] De Shalit, E. , On the p-adic periods of X0(p) . To appear in Math. Ann. · Zbl 0864.14014
[6] Greenberg, R. and Stevens, G. , P-adic L functions and p-adic periods of modular forms , Invent. Math. 111 (1993) 407-447. · Zbl 0778.11034
[7] Gross, B. , Heights and special values of L series , Proceedings of the 1985 Montreal conference in Number Theory. CMS conference proceedings, vol. 7. · Zbl 0623.10019
[8] Grothendieck, A. , Modeles de néron et monodromie, in: SGA 71 , LNM 288, Springer-Verlag (1972). · Zbl 0248.14006
[9] Hida, H. , Galois representations into GL2(Zp[[X]]) attached to ordinary cusp forms , Inv. Math. 85 (1986) 543-613. · Zbl 0612.10021
[10] Katz, N. and Mazur, B. , Arithmetic moduli of elliptic curves , Ann. Math. Stud. 108. Princeton (1985). · Zbl 0576.14026
[11] Manin, J. and Drinfeld, V.G. , Periods of p-adic Schottky groups , J.f.d. reine u. angew. Math. 262/3 (1973) 239-247. · Zbl 0275.14017
[12] Mazur, B. , Modular curves and the Eisenstein ideal . Publ. Math. I.H.E.S. 47 (1977) 33-186. · Zbl 0394.14008
[13] Mazur, B. and Tate, J. , Refined conjectures of the ”Birch and Swinnerton-Dyer type” , Duke Math. J. 54 (1987) 711-750. · Zbl 0636.14004
[14] Mazur, B. , Tate, J. and Teitelbaum, J. , On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer , Inv. Math. 84 (1986) 1-48. · Zbl 0699.14028
[15] Mazur, B. and Wiles, A. , Class fields of abelian extensions of Q , Inv. Math. 76 (1984) 179-330. · Zbl 0545.12005
[16] Mazur, B. and Wiles, A. , On p-adic analytic families of Galois representations , Compositio Math. 59 (1986) 231-264. · Zbl 0654.12008
[17] Milne, J. , Etale Cohomology , PUP, Princeton (1980). · Zbl 0433.14012
[18] Serre, J.-P. , Groupes Algébriques et Corps de Classes , Hermann, Paris (1959). · Zbl 0097.35604
[19] Wiles, A. , Modular curves and the class group of Q(\zeta p) , Inv. Math. 58 (1980) 1-35. · Zbl 0436.12004
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.