A generalization of Greenberg's L -invariant

Abstract

Using the theory of (ϕ, Γ)-modules we generalize Greenberg's construction of the L-invariant to p-adic representations which are semistable at p.

Iwasawa theory for symmetric powers of CM modular forms at nonordinary primes, II

“…There, the index set S = {±} r−1 is used, and s ∈ S corresponds to t ∈ T where t i = s i p (r−i)(k−1) if m is even, and t i = s i αp (r−i)(k−1) if m is odd.) Just as in the case m = 1, these functions can be decomposed in terms of appropriate products of twists of "plus and minus" logarithms and "plus and minus" p-adic L-functions (Corollary 6.9); their trivial zeroes and Linvariants are known (Theorem 6.13), using work of Benois [Ben1,Ben2].…”

On Extra Zeros of p-Adic L-Functions: The Crystalline Case

“…Then D cris (D m ) is the one dimensional Q p -vector space generated by t −m e m . As in [Ben2], we normalise the basis…”

“…The cohomology of such modules was studied in detail in [Ben2], Proposition 1.5.9 and section 1.5.10. Namely, H 0 (W ) = 0, dim Q p H 1 (W ) = 2e and dim Q p (W ) = e. There exists a canonical decomposition …”

“…From Lemma 3.1.4 iii) it follows that ρ D,c is an isomorphism. The following definition generalise (in the crystalline case) the main construction of [Ben2] where we assumed in addition that…”

“…Assume that 0 is a critical point for L (M, s) and that H 0 (M ) = H 0 (M * (1)) = 0. In [Ben2] using the theory of (ϕ, Γ)-modules we associated to each regular D an invariant L (V, D) ∈ Q p generalising both Greenberg's L -invariant [G] and Fontaine-Mazur's L -invariant [M]. This allows to formulate a quite general conjecture about the behavior of p-adic L-functions at extra zeros in the spirit of [G].…”