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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).

##### MSC:
 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
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