## 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

### MSC:

 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:

### References:

 [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 [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 [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 [10] Shimura, G.: On modular forms of half integral weight. Ann. Math.97, 440-481 (1973) · Zbl 0266.10022 [11] Shimura, G.: On the holomorphy of certain Dirichlet series. Proc. London Math. Soc.31, 79-98 (1975) · Zbl 0311.10029 [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 [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 [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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.