Non-solvable base change for Hilbert modular representations and zeta functions of twisted quaternionic Shimura varieties

Virdol, Cristian

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 831-848.

Dans cet article, nous montrons un changement de base non-résoluble pour certaines représentations modulaires de Hilbert et nous utilisons ce résultat pour établir le prolongement méromorphe à tout le plan complexe des fonctions zêta de certaines variétés de Shimura quaternioniques tordues. Les fonctions zêta des variétés de Shimura quaternioniques tordues sont calculées à toutes les places.

In this paper we prove some non-solvable base change for Hilbert modular representations, and we use this result to show the meromorphic continuation to the entire complex plane of the zeta functions of some twisted quaternionic Shimura varieties. The zeta functions of the twisted quaternionic Shimura varieties are computed at all places.

[AC] Arthur (J.), Clozel (L.).— Simple algebras, base change and the advanced theory of the trace formula, Ann. of Math. Studies, Princeton University Press, (1989). | MR | Zbl

[B] Blasius (D.).— Hilbert modular forms and the Ramanujan conjecture, Noncommutative geometry and number theory, p. 35-56, Aspects Math., E37, Vieweg, Wiesbaden, (2006). | MR | Zbl

[BL] Brylinski (J.L.), Labesse (J.P.).— Cohomologie d’intersection et fonctions L de certaines varietes de Shimura, Annales Scientifiques de l’Ecole Normale Superieure, 17, p. 361-412, (1984). | Numdam | MR | Zbl

[BR] Blasius (D.), Rogawski (J.D.).— Zeta functions of Shimura varieties, Motives, AMS Proc. Symp. Pure Math. 55, Part 2. | MR | Zbl

[C] Carayol (H.).— Sur la mauvaise r$\acute{e}$duction des courbes de Shimura, Compositio Math., 59, nr.2, p. 151-230, (1986). | Numdam | MR | Zbl

[D] Dimitrov (M.).— Galois representations mod $p$ and cohomology of Hilbert modular varieties, Ann. Sci. de l’Ecole Norm. Sup. 38, p. 505-551, (2005). | Numdam | MR | Zbl

[DE] Deligne (P.).— Travaux de Shimura, S$\acute{\mathrm{e}}$m. Bourbaki F$\acute{\mathrm{e}}$b. 71, Expos$\acute{\mathrm{e}}$ 389, Lectures Notes in Math. vol. 244. Berlin-heidelberg-New York; Springer (1971). | Numdam | MR | Zbl

[G] Gelbart (S.S.).— Automorphic forms on adeles groups, Ann. of Math. Studies, Princeton University Press, (1975). | MR | Zbl

[HSBT] Harris (M.), Shepherd-Barron (N.), Taylor (R.).— A family of Calabi-Yau varieties and potential automorphy, to appear in Ann. of Math. | MR

[L] Langlands (R.P.).— Base change for GL${}_2$, Ann. of Math. Studies 96, Princeton University Press, (1980). | MR | Zbl

[R] Reimann (H.).— The semi-simple zeta function of quaternionic Shimura varieties, Lecture Notes in Mathematics 1657, Springer, (1997). | MR | Zbl

[RA1] Ramakrishnan (D.).— Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math., 152, p. 45-111, (2000). | MR | Zbl

[RA2] Ramakrishnan (D.).— Modularity of solvable Artin representations of GO(4)-type, IMRN, No. 1, p. 1-54, (2002). | MR | Zbl

[RT] Rogawski (J.D.), Tunnell (J.B.).— On Artin L-functions associated to Hilbert modular forms of weight one, Inv. Math., 74, p. 1-43, (1983). | MR | Zbl

[SE] Serre (J.-P.).— Linear representations of finite groups, Springer (1977). | MR | Zbl

[T1] Taylor (R.).— On Galois representations associated to Hilbert modular forms, Inv. Math., 98, p. 265-280, (1989). | MR | Zbl

[T2] Taylor (R.).— On the meromorphic continuation of degree two L-functions, Documenta Mathematica, Extra Volume: John Coates’ Sixtieth Birthday, p. 729-779, (2006). | MR | Zbl

[V1] Virdol (C.).— Zeta functions of twisted modular curves, J. Aust. Math. Soc. 80, p. 89-103, (2006). | MR | Zbl

[V2] Virdol (C.).— Tate classes and poles of $L$-functions of twisted quaternionic Shimura surfaces, J. of Number Theory 123, Nr. 2, p. 315-328, (2007). | MR | Zbl

[W] Wiles (A.).— Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 141, p. 443-551, (1995). | MR | Zbl