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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].

11F85 \(p\)-adic theory, local fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11R23 Iwasawa theory