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p-adic measures attached to automorphic representations of GL(3). (English) Zbl 0656.10023
The object of this paper is the L-function that is defined by the symmetric squares of \(\ell\)-adic representations (these form a compatible system of 3-dimensional \(\ell\)-adic representations for the Galois group of \({\bar {\mathbb{Q}}}\) over \({\mathbb{Q}})\). This L-function is interpreted as the twisted L-function of a certain automorphic representation of GL(3).
For an “ordinary” prime p (i.e. the p-th Fourier coefficient of the corresponding primitive cusp form of weight \(\geq 2\) for the congruence subgroup \(\Gamma_ 0(N)\) is a p-adic unit and p does not divide N) the author constructs a p-adic analogue of this twisted L-function and proves its p-adic holomorphy and functional equation.
The essential part of the paper is the p-adic interpolation of all critical values.
Reviewer: N.V.Kuznetsov

11F33 Congruences for modular and \(p\)-adic modular forms
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F80 Galois representations
Full Text: DOI EuDML
[1] Carayol, H.: Courbes de Shimura, formes automorphes et représentations galoisiennes. Thèse, Paris 1984
[2] Coates, J., Schmidt, C.-G.: Iwasawa theory for the symmetric square of an elliptic curve. J. Reine Angew. Math.375/376, 104-156 (1987) · Zbl 0609.14013 · doi:10.1515/crll.1987.375-376.104
[3] Deligne, P.: Les constantes des équations fonctionelles des fonctionsL. (Lect. Notes Math., Vol. 349, pp. 501-595). Berlin-Heidelberg-New York: Springer 1973
[4] Deligne, P.: Valeurs de fonctionsL et périodes d’intégrales. Proc. Symp. Pure Math.33, (Part 2) 313-346 (1979)
[5] Gelbart, S., Jacquet, H.: A relation between automorphic representations ofGL(2) andGL(3). Ann. Sci. Ec. Norm. Super., IV. Ser.11, 471-542 (1978) · Zbl 0406.10022
[6] Henniart, G.: La conjecture de Langlands locale pourGL(3). Mém. Soc. Math. Fr. Nouv. Ser.11-12, 186 (1984)
[7] Hida, H.: Ap-adic measure attached to the zeta functions associated with two elliptic modular forms. Invent. Math.79, 159-185 (1985) · Zbl 0573.10020 · doi:10.1007/BF01388661
[8] Jacquet, H., Langlands, R.P.: Automorphic forms onGL(2). (Lect. Notes Math., Vol. 114). Berlin-Heidelberg-New York: Springer 1970 · Zbl 0236.12010
[9] Schmidt, C.-G.: Thep-adicL-functions attached to Rankin convolutions of modular forms. J. Reine Angew. Math.368, 201-220 (1986) · Zbl 0585.10020 · doi:10.1515/crll.1986.368.201
[10] Shimura, G.: On modular forms of half integral weight. Ann. Math.97, 440-481 (1973) · Zbl 0266.10022 · doi:10.2307/1970831
[11] Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. London Math. Soc.31, 79-98 (1975) · Zbl 0311.10029 · doi:10.1112/plms/s3-31.1.79
[12] Shimura, G.: The special values of the zeta functions associated with cusp forms. Commun. Pure Appl. Math.29, 783-804 (1976) · Zbl 0348.10015 · doi:10.1002/cpa.3160290618
[13] Sturm, J.: Special values of zeta functions and Eisenstein series of half integral weight. Am. J. Math.102, 219-240 (1980) · Zbl 0433.10015 · doi:10.2307/2374237
[14] Sturm, J.: Evaluation of the symmetric square at the near center point. Preprint 1987 · Zbl 0705.11027
[15] Tate, J.: Number theoretic background. Proc. Symp. Pure Math.33, (Part 2) 3-26 (1979) · Zbl 0422.12007
[16] Zagier, D.: Modular forms whose Fourier coefficients involve zeta functions of quadratic fields. (Lect. Notes Math., Vol. 627, pp. 105-169). Berlin-Heidelberg-New York: Springer 1977 · Zbl 0372.10017
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