Let E be a modular elliptic curve over a totally real number field F . We prove the weak exceptional zero conjecture which links a (higher) derivative of the p-adic L-function attached to E to certain padic periods attached to the corresponding Hilbert modular form at the places above p where E has split multiplicative reduction. Under some mild restrictions on p and the conductor of E we deduce the exceptional zero conjecture in the strong form (i.e. where the automorphic p-adic periods are replaced by the L-invariants of E defined in terms of Tate periods) from a special case proved earlier by Mok. Crucial for our method is a new construction of the p-adic L-function of E in terms of local data.
Let χ be a Hecke character of finite order of a totally real number field F . By using Hill's Shintani cocyle we provide a cohomological construction of the p-adic L-series Lp(χ, s) associated to χ. This is used to show that Lp(χ, s) has a trivial zero at s = 0 of order at least equal to the number of places of F above p where the local component of χ is trivial. L(χ, s) at integers n ≤ 0 lie in the algebraic closure Q ⊆ C of Q. In [14] Shintani gave another proof by constructing a nice fundamental domain (i.e. a finite disjoint union of rational cones; a so-called Shintani decomposition) for the canonical action of the positive global units E + of F on R d + . Deligne and Ribet [8] and independently Barsky and Cassou-Noguès [1,3] have shown that there exists a p-adic analytic analogue L p (χ, s) of the Hecke L-series L(χ, s) which is characterized by 1 L p (χ, 1 − n) = L Sp (χω 1−n , 1 − n)
We give an explicit description in terms of logarithmic differential forms of the isomorphism of P. Schneider and U. Stuhler relating de Rham cohomology of p-adic symmetric spaces to boundary distributions. As an application we prove a Hodgetype decomposition for the de Rham cohomology of varieties over p-adic fields which admit a uniformization by a p-adic symmetric space.with the obvious face and degeneracy maps (where H i denotes a linear form defining H i ). It is shown in [SS] that there is a natural isomorphism between H k (X ) and
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