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- W2952498902 abstract "For a prime number $pge 5$, we consider three classical cusp eigenforms $f_j(z)$ of weights $k_1, k_2, k_3$, of conductors $N_1, N_2, N_3$, and of nebentypus characters $psi_j bmod N_j$. According to H.Hida and R.Coleman, one can include each $f_j$ into a {$p$-adic analytic family} $k_j mapsto {f_{j,k_j}}$ of cusp eigenforms $f_{j,k_j}$ of weights $k_j$ in such a way that $f_{j,k_j}=f_j$, and that all their Fourier coefficients $a_n(f_{j, k_j})$ are given by certain $p$-adic analytic functions $k_j{}mapsto a_{n, j}(k_j{})$. The purpose of this paper is to describe a four variable $p$-adic $L$-function attached to Garrett's triple product of three Coleman's families $k_j mapsto {f_{j,k_j}}$ of cusp eigenforms of three fixed slopes $sigma_j=v_p(alpha_{p, j}^{(1)}(k_j{}))ge 0$ where $alpha_{p,j}^{(1)} = al_{p,j}^{(1)}(k_j{})$ is an eigenvalue (which depends on $k_j{}$) of Atkin's operator $U=U_p$ acting on Fourier expansions by $U(sum_{nge 0}^infty a_{n}q^n) = sum_{n ge 0}^infty a_{np} q^n$. We consider the $p$-adic weight space $X$ containing all $(k{}_j, psi_j)$. Our $p$-adic $L$-functions are Mellin transforms of certain measures with values in $Ar$, where $Ar=Ar({cal B})$ denotes an affinoid algebra associated with an affinoid space ${cal B}$ as in cite{CoPB}, where ${cal B}={cal B}_1times{cal B}_2times{cal B}_3$, is an affinoid neighbourhood around $(k_1, k_2, k_3)in X^3$ (with a given integers $k_j$ and fixed Dirichlet characters $psi_j bmod N$). We construct such a measure from higher twists of classical Siegel-Eisenstein series, which produce distributions with values in certain Banach $Ar$-modules $Mr = Mr(N;Ar)$ of triple modular forms with coefficients in the algebra $Ar$." @default.
- W2952498902 created "2019-06-27" @default.
- W2952498902 creator A5043320556 @default.
- W2952498902 date "2006-07-07" @default.
- W2952498902 modified "2023-09-27" @default.
- W2952498902 title "p-adic Banach modules of arithmetical modular forms and triple products of Coleman's families" @default.
- W2952498902 hasPublicationYear "2006" @default.
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