Skip to main content

The spin $L$ -function on $\text{GSp}_{6}$ for Siegel modular forms

  • Aaron Pollack (a1)

We give a Rankin–Selberg integral representation for the Spin (degree eight) $L$ -function on $\operatorname{PGSp}_{6}$ that applies to the cuspidal automorphic representations associated to Siegel modular forms. If $\unicode[STIX]{x1D70B}$ corresponds to a level-one Siegel modular form $f$ of even weight, and if $f$ has a nonvanishing maximal Fourier coefficient (defined below), then we deduce the functional equation and finiteness of poles of the completed Spin $L$ -function $\unicode[STIX]{x1D6EC}(\unicode[STIX]{x1D70B},\text{Spin},s)$ of  $\unicode[STIX]{x1D70B}$ .

Hide All
[And71] Andrianov, A. N., Dirichlet series with Euler product in the theory of Siegel modular forms of genus two , Trudy Mat. Inst. Steklov. 112 (1971), 7394, 386.
[And74] Andrianov, A. N., Euler products that correspond to Siegel’s modular forms of genus 2 , Uspekhi Mat. Nauk 29 (1974), 43110.
[AS01] Asgari, M. and Schmidt, R., Siegel modular forms and representations , Manuscripta Math. 104 (2001), 173200.
[BH16] Bhargava, M. and Ho, W., Coregular spaces and genus one curves , Camb. J. Math. 4 (2016), 1119.
[Bra28] Brandt, H., Idealtheorie in Quaternionenalgebren , Math. Ann. 99 (1928), 129.
[Bro69] Brown, R. B., Groups of type E 7 , J. Reine Angew. Math. 236 (1969), 79102.
[Brz82] Brzeziński, J., A characterization of Gorenstein orders in quaternion algebras , Math. Scand. 50 (1982), 1924.
[Brz83] Brzeziński, J., On orders in quaternion algebras , Comm. Algebra 11 (1983), 501522.
[Bum05] Bump, D., The Rankin–Selberg method: an introduction and survey , in Automorphic representations, L-functions and applications: progress and prospects, Ohio State University Mathematical Research Institute Publications, vol. 11 (de Gruyter, Berlin, 2005), 4173.
[BG92] Bump, D. and Ginzburg, D., Spin L-functions on symplectic groups , Int. Math. Res. Not. IMRN 1992 (1992), 153160.
[Del96] Deligne, P., La série exceptionnelle de groupes de Lie , C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 321326.
[DG02] Deligne, P. and Gross, B. H., On the exceptional series, and its descendants , C. R. Math. Acad. Sci. Paris 335 (2002), 877881.
[DdM96] Deligne, P. and de Man, R., La série exceptionnelle de groupes de Lie. II , C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 577582.
[Eic52] Eichler, M., Quadratische Formen und orthogonale Gruppen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. 63 (Springer, Berlin, 1952).
[Evd84] Evdokimov, S. A., Dirichlet series, multiple Andrianov zeta-functions in the theory of Euler modular forms of genus 3 , Dokl. Akad. Nauk SSSR 277 (1984), 2529.
[FK94] Faraut, J. and Korányi, A., Analysis on symmetric cones, Oxford Mathematical Monographs (The Clarendon Press, Oxford University Press, New York, 1994).
[Fre54] Freudenthal, H., Beziehungen der E 7 und E 8 zur Oktavenebene. I , Indag. Math. (N.S.) 16 (1954), 218230; Nederl. Akad. Wetensch. Proc. Ser. A, vol. 57.
[Fur93] Furusawa, M., On L-functions for GSp(4) × GL(2) and their special values , J. Reine Angew. Math. 438 (1993), 187218.
[GS05] Gan, W. T. and Savin, G., On minimal representations definitions and properties , Represent. Theory 9 (2005), 4693 (electronic).
[Gar87] Garrett, P. B., Decomposition of Eisenstein series: Rankin triple products , Ann. of Math. (2) 125 (1987), 209235.
[GPSR87] Gelbart, S., Piatetski-Shapiro, I. and Rallis, S., Explicit constructions of automorphic L-functions, Lecture Notes in Mathematics, vol. 1254 (Springer, Berlin, 1987).
[GR94] Ginzburg, D. and Rallis, S., A tower of Rankin–Selberg integrals , Int. Math. Res. Not. IMRN 1994 (1994), 201208 (electronic).
[GL09] Gross, B. H. and Lucianovic, M. W., On cubic rings and quaternion rings , J. Number Theory 129 (2009), 14681478.
[Lan71] Langlands, R. P., Euler products, James K. Whittemore lectures in mathematics given at Yale University, Yale Mathematical Monographs, vol. 1 (Yale University Press, New Haven, CT, 1971).
[Lan76] Langlands, R. P., On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, vol. 544 (Springer, Berlin, 1976).
[Luc03] Lucianovic, M. W., Quaternion rings, ternary quadratic forms, and Fourier coefficients of modular forms on PGSp(6), PhD Thesis, Harvard University (2003).
[Maa71] Maass, H., Siegel’s modular forms and Dirichlet series, Lecture Notes in Mathematics, vol. 216 (Springer, Berlin, 1971).
[PSR87] Piatetski-Shapiro, I. and Rallis, S., Rankin triple L functions , Compos. Math. 64 (1987), 31115.
[PS15] Pollack, A. and Shah, S., The spin -function on via a non-unique model, Preprint (2015), arXiv:1503.08197.
[Rum97] Rumelhart, K. E., Minimal representations of exceptional p-adic groups , Represent. Theory 1 (1997), 133181 (electronic).
[Sah13] Saha, A., Siegel cusp forms of degree 2 are determined by their fundamental Fourier coefficients , Math. Ann. 355 (2013), 363380.
[Sha88] Shahidi, F., On the Ramanujan conjecture and finiteness of poles for certain L-functions , Ann. of Math. (2) 127 (1988), 547584.
[Shi81] Shimura, G., Arithmetic of differential operators on symmetric domains , Duke Math. J. 48 (1981), 813843.
[Spr06] Springer, T. A., Some groups of type E 7 , Nagoya Math. J. 182 (2006), 259284.
[Voi] Voight, J., Quaternion algebras,∼jvoight/quat-book.pdf.
[Voi11] Voight, J., Characterizing quaternion rings over an arbitrary base , J. Reine Angew. Math. 657 (2011), 113134.
[Yam09] Yamana, S., Determination of holomorphic modular forms by primitive Fourier coefficients , Math. Ann. 344 (2009), 853862.
[Yam13] Yamana, S., On the Siegel–Weil formula for quaternionic unitary groups , Amer. J. Math. 135 (2013), 13831432.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification