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Two variable \(p\)-adic \(L\) functions attached to eigenfamilies of positive slope. (English) Zbl 1065.11025
Let \(p\geq 5\) be a prime number, and \(N\) a positive integer prime to \(p\). Any classical cusp eigenform \(f\) of weight \(k\geq 2\) for \(\Gamma_0(N)\), with a Dirichlet character \(\psi\bmod N\) can be induced (under suitable assumptions on \(p\) and \(f\)) into a family of cusp eigenforms \(f_{(k')}\) of weight \(k'\) in such a way that \(f_{(k)}=f\), and Fourier coefficients \(a_n(f_{(k')})\) are given by certain \(p\)-adic analytic functions \(k'\mapsto a_n(f_{(k')})\) (Hida, Coleman).
The author constructs a two variable \(p\)-adic \(L\)-function attached to a family \(\{f_{(k')}\}\) of cusp eigenforms of a fixed positive slope \(\sigma=v_p(\alpha_p)>0\), where \(\alpha_p=\alpha_p(k)\) is an eigenvalue of the Atkin operator \(U_p\) (Theorem 0.5). Such a \(p\)-adic \(L\)-function interpolates the special values \(L_{f(k')}(s_s\chi)\) at points \((s_sk')\) where \(s=1,\dots,k'-1\) and \(\chi\) is an arbitrary Dirichlet character. The resulting \(p\)-adic \(L\)-functions come from certain Eisenstein distributions with values in Banach modules of families of overconvergent forms (Theorem 0.3).

MSC:
11F33 Congruences for modular and \(p\)-adic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F85 \(p\)-adic theory, local fields
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