# zbMATH — the first resource for mathematics

On monodromy invariants occurring in global arithmetic, and Fontaine’s theory. (English) Zbl 0846.11039
Mazur, Barry (ed.) et al., $$p$$-adic monodromy and the Birch and Swinnerton-Dyer conjecture. A workshop held August 12-16, 1991 in Boston, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 165, 1-20 (1994).
The paper under review presents a new definition of the “$$L$$-invariant” of a modular form. The $$L$$-invariant is supposed to relate the special value of the classical complex $$L$$-function and the derivative of the $$p$$-adic $$L$$-function. The proposed new definition uses Fontaine’s $$N$$-operator on the Dieudonné-module associated to the $$p$$-adic Galois representation. It is conjectured that this gives the desired relation between $$L$$-functions. The bulk of the paper is an elucidation of Fontaine’s constructions in the special case of modular forms.
For the entire collection see [Zbl 0794.00016].
Reviewer: G.Faltings (Bonn)

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14F30 $$p$$-adic cohomology, crystalline cohomology 14G20 Local ground fields in algebraic geometry 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
##### Keywords:
$$L$$-invariant; $$p$$-adic $$L$$-function; modular form