Local base change via Tate cohomology

Ronchetti, Niccolò

doi:10.1090/ert/486

Local base change via Tate cohomology
HTML articles powered by AMS MathViewer

by Niccolò Ronchetti PDF

Represent. Theory 20 (2016), 263-294 Request permission

Abstract:

We propose a new way to realize cyclic base change (a special case of Langlands functoriality) for prime degree extensions of characteristic zero local fields. Let $F / E$ be a prime degree $l$ extension of local fields of residue characteristic $p \neq l$. Let $\pi$ be an irreducible cuspidal $l$-adic representation of $\mathrm {GL}_n(E)$ and let $\rho$ be an irreducible cuspidal $l$-adic representation of $\mathrm {GL}_n(F)$ which is Galois-invariant. Under some minor technical conditions on $\pi$ and $\rho$ (for instance, we assume that both are level zero) we prove that the $\bmod l$-reductions $r_l(\pi )$ and $r_l(\rho )$ are in base change if and only if the Tate cohomology of $\rho$ with respect to the Galois action is isomorphic, as a modular representation of $\mathrm {GL}_n(E)$, to the Frobenius twist of $r_l(\pi )$. This proves a special case of a conjecture of Treumann and Venkatesh as they investigate the relationship between linkage and Langlands functoriality.

References

James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR 1007299
James Arthur, The principle of functoriality, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 1, 39–53. Mathematical challenges of the 21st century (Los Angeles, CA, 2000). MR 1943132, DOI 10.1090/S0273-0979-02-00963-1
I. N. Bernstein and K. E. Rummelhart, Draft of: Representations of $p$-adic groups, lectures at Harvard University, 1992.
I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR 579172, DOI 10.24033/asens.1333
I. N. Bernšteĭn and A. V. Zelevinskiĭ, Representations of the group $GL(n,F),$ where $F$ is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), no. 3(189), 5–70 (Russian). MR 0425030
Richard E. Borcherds, Modular moonshine. III, Duke Math. J. 93 (1998), no. 1, 129–154. MR 1620091, DOI 10.1215/S0012-7094-98-09305-X
Daniel Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55, Cambridge University Press, Cambridge, 1997. MR 1431508, DOI 10.1017/CBO9780511609572
Colin J. Bushnell and Guy Henniart, Modular local Langlands correspondence for $\textrm {GL}_n$, Int. Math. Res. Not. IMRN 15 (2014), 4124–4145. MR 3244922, DOI 10.1093/imrn/rnt063
Colin J. Bushnell and Guy Henniart, The essentially tame local Langlands correspondence. I, J. Amer. Math. Soc. 18 (2005), no. 3, 685–710. MR 2138141, DOI 10.1090/S0894-0347-05-00487-X
Colin J. Bushnell and Guy Henniart, The essentially tame local Langlands correspondence, III: the general case, Proc. Lond. Math. Soc. (3) 101 (2010), no. 2, 497–553. MR 2679700, DOI 10.1112/plms/pdp053
Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
I. B. Fesenko and S. V. Vostokov, Local fields and their extensions, 2nd ed., Translations of Mathematical Monographs, vol. 121, American Mathematical Society, Providence, RI, 2002. With a foreword by I. R. Shafarevich. MR 1915966, DOI 10.1090/mmono/121
Stephen Gelbart, An elementary introduction to the Langlands program, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219. MR 733692, DOI 10.1090/S0273-0979-1984-15237-6
S. I. Gel′fand, Representations of the full linear group over a finite field, Mat. Sb. (N.S.) 83 (125) (1970), 15–41. MR 0272916
J. A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80 (1955), 402–447. MR 72878, DOI 10.1090/S0002-9947-1955-0072878-2
I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 553598
Gopal Prasad, Galois-fixed points in the Bruhat-Tits building of a reductive group, Bull. Soc. Math. France 129 (2001), no. 2, 169–174 (English, with English and French summaries). MR 1871292, DOI 10.24033/bsmf.2391
David Renard, Représentations des groupes réductifs $p$-adiques, Cours Spécialisés [Specialized Courses], vol. 17, Société Mathématique de France, Paris, 2010 (French). MR 2567785
Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380, DOI 10.1007/978-1-4684-9458-7
Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237, DOI 10.1007/978-1-4757-5673-9
Takuro Shintani, Two remarks on irreducible characters of finite general linear groups, J. Math. Soc. Japan 28 (1976), no. 2, 396–414. MR 414730, DOI 10.2969/jmsj/02820396
T. A. Springer, Cusp forms for finite groups, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 97–120. MR 0263942
T. A. Springer, Characters of special groups, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 121–166. MR 0263943
T. A. Springer and R. Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) Lecture Notes in Mathematics, Vol. 131, Springer, Berlin, 1970, pp. 167–266. MR 0268192
J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607
D. Treumann and A. Venkatesh, Functoriality, Smith theory and the Brauer homomorphism, http://arxiv.org/pdf/1407.2346.pdf.
Marie-France Vignéras, Correspondance de Langlands semi-simple pour $\textrm {GL}(n,F)$ modulo ${\scr l}\not = p$, Invent. Math. 144 (2001), no. 1, 177–223 (French). MR 1821157, DOI 10.1007/s002220100134
Marie-France Vignéras, Représentations $l$-modulaires d’un groupe réductif $p$-adique avec $l\ne p$, Progress in Mathematics, vol. 137, Birkhäuser Boston, Inc., Boston, MA, 1996 (French, with English summary). MR 1395151
David A. Vogan Jr., The local Langlands conjecture, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 305–379. MR 1216197, DOI 10.1090/conm/145/1216197