2.2 DISEÑO ELECTRÓNICO
2.2.1 DISEÑO DE LOS ACTUDORES DEL EXOESQUELETO
By combining the results of the previous two sections, we have
Proposition B.5.1. Let π be a cuspidal automorphic representation of GL(4,AF)
with central character χ2 such that the exterior squareL-functionLS(s, π,∧2⊗χ−1)
has a pole at s= 1. Then the theta liftΘ2(Veπ)of eπ to GSp(2,AF) does not vanish,
and if we letΠ be an irreducible constituent ofΘ2(Veπ), thenΠ is a generic, cuspidal
automorphic representation of GSp(2,AF) with central character χ. Moreover for
each unramified place v, πv is the Langlands functorial lift of Πv via GSp(2,C) ,→
GL(4,C).
At this moment, our lift Π has not been shown to satisfy all the desired proper- ties. Namely, we need to show the functoriality at the archimedean places. However, this easily follows from the following recent result by Asgari and Shahidi [A-S2].
Proposition B.5.2. Let Π be a generic cuspidal automorphic representation of GSp(2,Ak) with central character χ. Then Π has a unique (weak) functorial lift π
to GL(4,AF)with central characterχ2. Moreoverπ is either cuspidal or an isobaric
sum π1π2 of two inequivalent cuspidal automorphic representations of GL(2,AF).
The latter is the case if and only if Π is obtained as a theta lift from the split orthogonal group GSO(4,AF).
Proof. This is (part of) Theorem 2.4. of [A-S2]. Note that theta lift is called Weil lift there. Also although [A-S2] does not explicitly mention this, the orthogonal group GSO(4,Ak) must be split.
Finally, we can prove Theorem B.0.4.
Proof of Theorem B.0.4. First, we can easily see that our lift Π is not a theta lift from the rank 2 orthogonal group GSO(4,AF) by using the latter part of the main
theorem of [Mo]. Therefore Π has a unique cuspidal (weak) functorial lift π0 to GL(4,AF) whose central character isχ2. But we already know thatπv is a functorial
lift of Πvfor almost all placesv, and so by strong multiplicity one theorem for GL(4),
we have π=π0. Thus Π indeed has the desired functorial property. This completes the proof.
Bibliography
[A] Adams, J. : Reductive dual pair correspondence for complex groups, J. Funct. Anal. 132 (1995), 1−42.
[Ad-Pr] Adler, J. and Prasad, D. : On certain multiplicity one theorems, Israel J. Math. 153 (2006), 221−245.
[A-S1] Asgari, M. and Shahidi, F. : Generic transfer for general spin groups, To appear in Duke Math. J.
[A-S2] Asgari, M. and Shahidi, F. : Generic transfer from GSp(4) to GL(4), To appear in Comps. Math.
[BR] Blasius, D. and Ramakrishnan, D. : Maass forms and Galois representations, in “Galois groups overQ”, Math. Sci. Res. Inst. Publ.,16, Springer, New York, (1989), 33−77.
[B-S] B¨ocherer, S. and Schulze-Pillot, R. : Siegel modular forms and theta series attached to quaternion algebras, Nagoya Math. J. 121(1991), 35−96.
[B1] Bump, D. Automorphic Forms and Representations, Cambridge University Press, Cambridge, UK, 1997.
[Cs] Casselman, W. : Canonical extensions of Harish-Chandra modules to repre- sentations of G, Canad. J. Math. 41 (1989), 385−438.
[Co1] Cogdell, J.W. : Dual groups and Langlands functoriality in Bernstein, J. and Gelbart, S. eds., An introduction to the Langlands program, Birkh¨auser Boston, Boston, (2003), 251−268.
[Co2] Cogdell, J. : Lectures onL-functions, converse theorems, and functoriality of
GL(n), in Lectures on AutomorphicL-functions, Fields Institute Monographs, AMS (2004), 5−100.
[F1] Flicker, Y. : Twisted tensors and Euler products, Bull. Soc. math. France.116
[F2] Flicker, Y. : On zeros of the twisted tensorL-function, Math. Ann.297(1993), 199−219.
[F-Z] Flicker, Y. and Zinoviev, D. : On poles of twisted tensor L-functions, ; Pro- ceedings of the Japan Academy 71 (1995), 114−116.
[GRS] Ginzburg, D., Rallis, S. and Soudry, D. : Periods, poles of L-functions and symplectic-orthogonal theta lifts, J. reine angew. Math. 487 (1997), 85−114. [H-K] Harris, M. and Kudla, S. S. : Arithmetic automorphic forms for the nonholo-
morphic discrete series of GSp(2), Duke Math. J. 66 (1992), 59−121.
[HST] Harris, M., Soudry, D. and Taylor, R. : l-adic representations associated to modular forms over imaginary quadratic fields I: lifting to GSp4(Q), Invent.
Math. 112 (1993), 377−411.
[H-PS] Howe, R. and Piatetski-Shapiro, I. : Some examples of automorphic forms on Sp4, Duke Math. J. 50 (1983), 55−106.
[Ik] Ikeda, T. : On the location of the triple L-functions, Compositio Math. 83
(1992), 187−237.
[J1] Jacquet, H. : PrincipalL-functions of the linear group, in Automorphic Forms, Representations, L-functions, Proc. Symposia Pure Math. vol. XXXIII - Part 2, American Mahtematical Society. Providence, (1979) 63−86.
[J2] Jacquet, H. : Private notes communicated to the author.
[J-S1] Jacquet, H. and Shalika, J. : On Euler Products and the Classification of Automorphic Representations I, Am. J. Math. 103 (1981), 499−558.
[J-S2] Jacquet, H. and Shalika, J. : Exterior square L-functions, in Automorphic forms, Shimura varieties, and L-functions, Vol. II (Ann Arbor, MI, 1988), Perspect. Math. 11, Academic Press (1990), 143−226.
[K-S] Kim, H. and Shahidi, F. : Cuspidality of symmetric powers with applications., Duke Math. J. 112 (2002), 177−197
[Kn] Knapp, A. : Representation Theory of Semi-simple groups. An overview based on examples. Princeton University Press, Princeton, NJ, (2001).
[Kd1] Kudla, S. : On the local theta correspondence, Invent. Math. 83 (1986), 229−255
[Kd2] Kudla, S. : Splitting metaplectic covers of dual reductive pairs, Israel J. Math.
[Kd3] Kudla, S. : Notes on the Local Theta Correspondence, unpublished notes, available online.
[Kd-R1] Kudla, S. and Rallis, S. : Poles of Eisenstein series and L-functions, Festschrit in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II, Weizmann, Jerusalem, (1990), 81−110
[Kd-R2] Kudla, S. and Rallis, S. : A regularized Siegel-Weil formula: the first term identity, Ann. Math. 140 (1994), 1−80.
[Lm] Lam, T. Y. : Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics, Vol. 67, A.M.S. (2004).
[L] Lang, S. : Algebra, third edition, Addison Wesley, Reading, Mass. (1993). [Ln] Langlands, R. : Base Change for GL(2), Ann. Math. Studies, Princenton,
(1980).
[M] Marcus, D. A. : Number Fields, Springer, (1977).
[M] Maritn, K. : A symplectic case of Artin’s conjecture, Math. Research Letters,
4 (2003), 483−492
[Mo] Moeglin, C. : Quelques propri´etes de base de series theta, J. of Lie Theory 7
(1997), 231−238.
[P] Paul, A. : On the Howe correspondence for symplectic-orthogonal dual pairs, J. Functional Analysis, 228 (2005), 270−310.
[PS-R] Piatetski-Shapiro, I. I. and Rallis, S. : L-functions for classical groups, in Lecture Notes in Math. 1254, Springer-Verlag, New York, pp. 1-52.
[Pr] Prasad, D. Weil representation, Howe duality, and the theta correspondence, in CRM Proceedings and Lecture Notes, Volume 1 (1993).
[Pr-Rm] Prasad, D. and Ramakrishnan, D. : On the global root numbers of GL(n)×GL(m), in Automorphic Forms, Automorphic Representations, and Arithmetic, Proc. Sympos. Pure Math. 66, Part 2, Amer. Math. Soc., Provi- dence, (1999), 311−330.
[R] Rallis, S. : On the Howe duality conjecture, Compositio Math. 51 (1984), 333−399.
[Ra] Rao, R. R. : On some explicit formulas in the theory of Weil representation, Pacific J. Math. 157, (1993) 335−371.
[Rb1] Roberts, B. : The theta correspondence for similitudes, Israel J. Math. 94
[Rb2] Roberts, B. : Tempered representations and the theta correspondence, Canad. J. Math. 50 (1998), 1105−1108.
[Rb3] Roberts, B. : The nonarchimedean theta correspondence for GSp(2) and GO(4), Trans. AMS 351 (1999), 781−811.
[Rb4] Roberts, B. : Nonvanishing of global theta lifts from orthogonal groups, J. Ramanujan Math. Soc. 14 (1999), 131−194.
[Rb5] Roberts, B. : Global L-packets for GSp(2) and theta lifts, Documenta Math.