zbMATH — the first resource for mathematics

\(p\)-adic \(L\)-functions associated with a modular form and an imaginary quadratic field. (Fonctions \(L\) \(p\)-adiques associées à une forme modulaire et à un corps quadratique imaginaire.) (French) Zbl 0656.10019
Let K be an imaginary quadratic field of discriminant -D, and let \(f=\sum a_ nq^ n\) be a weight two newform on \(\Gamma_ 0(N)\) with character \(\psi\). Choose an odd prime p, and fix an embedding \(i_ p\) of \({\bar {\mathbb{Q}}}\) in \({\bar {\mathbb{Q}}}_ p\). Finally, assume that \(i_ p(a_ p)\) is a p-adic unit in \({\mathbb{Q}}_ p\). If \(\Omega\) is a character of K of finite order, let c be the largest power of p that divides \(N\cdot conductor\) of \(\Omega\), and assume that N and Dc are relatively prime. Let \(R_{\infty}\) be the compositum of the maximal p-cyclotomic extension of \({\mathbb{Q}}\), and of the ring class fields of K. For each \(C>1\), prime to NDc, the author shows that there is a measure (associated to f and \(\Omega)\) on \(Gal(R_{\infty}/K)\) which is then used to construct the p-adic L-function attached to f and K. The proof uses an idea of Hida for a p-adic interpretation of the Rankin convolution, and an explicit expression of the measure in terms of Eisenstein measures and theta functions.
Reviewer: S.Kamienny

11F33 Congruences for modular and \(p\)-adic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F11 Holomorphic modular forms of integral weight
Full Text: DOI