## 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

Zbl 0731.11065