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On standard $$p$$-adic $$L$$-functions of families of elliptic cusp forms. (English) Zbl 0841.11028
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, 81-110 (1994).
The purpose of this paper is to give a generalization of Mazur’s construction of the two variable $$p$$-adic $$L$$-function for the $$p$$-adic analytic families of ordinary elliptic cusp forms. A functional equation of the two variable $$p$$-adic $$L$$-function is also obtained as a consequence of the construction.
Let $$K$$ be a finite extension of the field of $$p$$-adic numbers and $${\mathcal O}_K$$ the ring of integers. Let $$\{f_p\}$$ be a $$p$$-adic analytic family of cusp forms. The author constructs the $${\mathcal O}_K [[X]]$$-valued modular symbol which interpolates algebraic parts of the modular symbols $$\xi_{f_p}$$. Then, he obtains the corresponding $${\mathcal O}_K [[X]]$$-valued measure by a method parallel to the case of a single cusp form. Note that R. Greenberg and G. Stevens [Invent. Math. 111, 407-447 (1993; Zbl 0778.11034)] and M. Ohta [On $$p$$-adic theory of modular symbols (preprint 1992)] have given other constructions of the two variable $$p$$-adic $$L$$-functions.
For the entire collection see [Zbl 0794.00016].

##### MSC:
 11F85 $$p$$-adic theory, local fields 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11R23 Iwasawa theory