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 Dieudonn√©-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)


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)