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


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