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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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