\(p\)-adic interpolation of convolutions of Hilbert modular forms. (English) Zbl 0882.11025
Summary: The author constructs \(p\)-adic measures related to the values of convolutions of Hilbert modular forms of integral and half-integral weight at the negative critical points under the assumption that the underlying totally real number field \(F\) has class number \(h_F=1\). This extends the result of A. A. Panchishkin [Non-archimedean \(L\)-functions of Siegel and Hilbert modular forms, Lect. Notes Math. 1471 (1991; Zbl 0732.11026)] who treated the case that both modular forms are of integral weight. In order to define the measures, we need to introduce the twist operator and a certain inverter \(j\) on the space of Hilbert modular forms of half-integral weight. The proof then makes use of the Rankin-Selberg integral representation of the convolution and of explicit formulas for the Fourier coefficients of certain Eisenstein series of half-integral weight derived by G. Shimura [Duke Math. J. 52, 281-314 (1985; Zbl 0577.10025)].
MSC:
| 11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
| 11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |
| 11F85 | \(p\)-adic theory, local fields |
References:
| [1] | [D] , Valeurs de fonctions L et périodes d’intégrales, in: Proc. of Symp. in Pure Math., 33 (1979), Part 2, 313-346. · Zbl 0449.10022 |
| [2] | [DR] , , Values of abelian L-functions at negative integers over totally real fields, Inventiones Math., 59 (1980), 227-286. · Zbl 0434.12009 |
| [3] | [G] , Motives, in: Proc. of Symp. in Pure Math., 55 (1994), Part 2, 193-223. · Zbl 0819.11046 |
| [4] | [H] , Vorlesungen über die Theorie der algebraischen Zahlen, Chelsea, 1948. · Zbl 0041.01102 |
| [5] | [Hi] , On Λ-adic forms of half integral weight for SL2/Q, in: Number Theory Paris 1992-1993, editor S. David, LMSLN 215, 139-166 (1995), Cambridge Univ. Press. · Zbl 0838.11031 |
| [6] | [I] , Special values of Dirichlet series attached to Hilbert modular forms, Am. J. Math., 113 (1991), 975-1017. · Zbl 0756.11014 |
| [7] | [K] , p-adic L-functions for CM fields, Inventiones Math., 49 (1978), 199-297. · Zbl 0417.12003 |
| [8] | [Kob] , Introduction to Elliptic Curves and Modular Forms, Second Edition, GTM 97, Springer Verlag, 1993. · Zbl 0804.11039 |
| [9] | [Koh] , Newforms of half-integral weight, J. Reine Angew. Math., 333 (1982), 32-72. · Zbl 0475.10025 |
| [10] | [M] , Modular forms, Springer Verlag, 1989. · Zbl 0701.11014 |
| [11] | [MRV] , , , On the theory of new-forms of half-integral weight, J. Number Th., 34 (1990), 210-224. · Zbl 0704.11013 |
| [12] | [N] , Algebraische Zahlentheorie, Springer Verlag, 1992. · Zbl 0747.11001 |
| [13] | [P] , Non-archimedean L-functions of Siegel and Hilbert modular forms, Lecture Notes in Mathematics 1471, Springer Verlag, 1991. · Zbl 0732.11026 |
| [14] | [S1] , On modular forms of half-integral weight, Ann. of Math., 97 (1973), 440-481. · Zbl 0266.10022 |
| [15] | [S2] , On the holomorphy of certain Dirichlet series, Proc. London Math. Soc., (3) 31 (1975), 79-98. · Zbl 0311.10029 |
| [16] | [S3] , The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math., 29 (1976), 783-804. · Zbl 0348.10015 |
| [17] | [S4] , The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J., 45 (1978), 637-679. · Zbl 0394.10015 |
| [18] | [S5] , Confluent hypergeometric functions on tube domains, Math. Ann., 260 (1982), 269-302. · Zbl 0502.10013 |
| [19] | [S6] , On Einsenstein Series of half-integral weight, Duke Math. J., 52 (1985), 281-314. · Zbl 0577.10025 |
| [20] | [S7] , On Hilbert modular forms of half-integral weight, Duke Math. J., 55 (1987), 765-838. · Zbl 0636.10024 |
| [21] | [S8] , On the Fourier coefficients of Hilbert modular forms of half-integral weight, Duke Math. J., 71 (1993), 501-557. · Zbl 0802.11017 |
| [22] | [U] , On twisting operators and newforms of half-integral weight, Nagoya Math. J., 131 (1993), 135-205. · Zbl 0778.11027 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
