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A \(p\)-adic Shimura isomorphism and \(p\)-adic periods of modular forms. (English) Zbl 0838.11033
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, 21-51 (1994).
Let \(f\) be a cuspidal Hecke eigenform of integral weight \(k \geq 2\) on \(\Gamma_1 (Np)\) \((N \in \mathbb{N},p\) a prime not dividing \(N)\). Using the rigid \(p\)-adic analytic uniformization of the abelian variety attached to \(f\), B. Mazur, J. Tate and J. Teitelbaum defined a certain \(p\)-adic invariant \({\mathcal L}_p (f)\) in the case \(k = 2\) which plays an important role in \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer and conjectures about exceptional zeros of the \(p\)-adic \(L\)-functions of \(f\). An analogous quantity \({\mathcal L}_p (f)\) also was conjecturally defined for even \(k > 2\). On the other hand, if \(N\) is squarefree and has an even number of prime factors, for arbitrary \(k \geq 2\) in [J. Teitelbaum, Invent. Math. 101, 395-410 (1990; Zbl 0731.11065)] an invariant \(\widetilde {\mathcal L}_p (f)\) was defined using the \(p\)-adic uniformization of the corresponding Shimura curve. It was shown that \(\widetilde {\mathcal L}_p (f) = {\mathcal L}_p (f)\) if \(k = 2\).
In the present paper, the author defines an invariant \(\widehat {\mathcal L}_p (f)\) for arbitrary \(k\) and \(N\). The definition is formally analogous to that of \(\widetilde {\mathcal L}_p (f)\), however one uses the rigid geometry of the modular curve itself rather than that of the Shimura curve. In weight 2 one has \(\widehat {\mathcal L}_p (f) = \widetilde {\mathcal L}_p (f)\). One conjectures that all three invariants should coincide whenever they are defined.
For the entire collection see [Zbl 0794.00016].

MSC:
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F85 \(p\)-adic theory, local fields
11F11 Holomorphic modular forms of integral weight
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