Abstract
In this article, we study the action of the \(U_p\) Hecke operator on the normalized spherical vector \(\phi \) in the representation of \({{\mathrm{GSp}}}_4(\mathbf {Q}_p)\) induced from a character on the Borel subgroup. We compute the Petersson norm of \(U_p \phi \) in terms of certain local L-values associated with \(\phi \).
References
Agarwal, M., Klosin, K.: Yoshida lifts and the Bloch–Kato conjecture for the convolution \(L\)-function. J. Number Theory 133(8), 2496–2537 (2013)
Ahlgren, S., Boylan, M.: Arithmetic properties of the partition function. Invent. Math. 153, 487–502 (2003)
Ahlgren, S., Ono, K.: Congruence properties for the partition function. Proc. Natl. Acad. Sci. USA 98(23), 12882–12884 (2001)
Ahlgren, S., Ono, K.: Arithmetic of singular moduli and class polynomials. Compos. Math. 141(2), 293–312 (2005)
Andrianov, A.N.: Action of Hecke operator \(T(p)\) on theta series. Math. Ann. 247(3), 245–254 (1980)
Böcherer, S., Funke, J., Schulze-Pillot, R.: Trace operator and theta series. J. Number Theory 78(1), 119–139 (1999)
Böcherer, S., Dummigan, N., Schulze-Pillot, R.: Yoshida lifts and Selmer groups. J. Math. Soc. Jpn. 64(4), 1353–1405 (2012)
Brown, J.: Saito-Kurokawa lifts and applications to the Bloch–Kato conjecture. Comput. Math. 143(2), 290–322 (2007)
Brown, J., Klosin, K.: On the norm of \(p\)-stabilized elliptic newforms, p. 1–18 (2013) (preprint (With an appendix by K. Conrad)
Buzzard, K.: On \(p\)-adic families of automorphic forms. In: Cremona, J., Lario, J.C., Quer, J., Ribet, K. (eds.) Modular Curves and Abelian Varieties. Progress in Mathematics, vol. 224. Birkhäuser, Basel (2004)
Chenevier, G.: Familles \(p\)-adiques de formes automorphes pour \({\rm GL}_n\). J. Reine Angew. Math. 570, 143–217 (2004)
Chenevier, G.: On the infinite fern of Galois representations of unitary type. Ann. Sci. Éc. Norm. Supér. (4) 44(6), 963–1019 (2011)
Coleman, R., Mazur, B.: The eigencurve. In: Galois Representations in Arithmetic Algebraic Geometry (Durham, 1996). London Mathematical Society Lecture Note Series, vol. 254, pp. 1–113. Cambridge University Press, Cambridge (1998)
Folsom, A., Kent, Z., Ono, K.: \(\ell \)-adic properties of the partition function. Adv. Math. 229, 1586–1609 (2012)
Hida, H.: Galois representations into \({\rm GL}_2({ Z}_p[[X]])\) attached to ordinary cusp forms. Invent. Math. 85(3), 545–613 (1986)
Klosin, K.: Congruences among modular forms on \(U(2,2)\) and the Bloch–Kato conjecture. Ann. Inst. Fourier 59(1), 81–166 (2009)
Kume, T.: Calculation of traces of theta series by means of the Weil representation. J. Math. Kyoto Univ. 38(3), 453–473 (1998)
Maass, H.: Die Primzahlen in der Theorie der Siegelschen Modulfunktionen. Math. Ann. 124, 87–122 (1951)
Ono, K.: Unearthing the visions of a master: harmonic Maass forms and number theory. In: Current Developments in Mathematics, 2008, pp. 347–454. International Press, Somerville (2009)
Pitale, A., Schmidt, R.: Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2. Proc. Am. Math. Soc. 136, 3831–3838 (2008)
Pitale, A., Schmidt, R.: Ramanujan type results for Siegel cusp forms of degree 2. J. Ramanujan Math. Soc. 24(1), 87–111 (2009)
Roberts, B., Schmidt, R.: Local Newforms for GSp(4). Springer Lecture Notes in Mathematics, vol. 1918. Springer, Berlin (2007)
Solomon, L.: A fixed-point formula for the classical groups over a finite field. Trans. Am. Math. Soc. 117, 423–440 (1965)
Taylor, R.: On congruences between modular forms. PhD thesis, Princeton University, Princeton (1988)
Urban, E.: Eigenvarieties for reductive groups. Ann. Math. (2) 174(3), 1685–1784 (2011)
Weissauer, R.: Endoscopy for \({\rm GSp}(4)\) and the Cohomology of Siegel Modular Threefolds. Lecture Notes in Mathematics, vol. 1968. Springer, Berlin (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was partially supported by the National Security Agency under Grant Number H98230-11-1-0137. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein. The work of the second author was supported by a grant from the Simons Foundation (#354890, Krzysztof Klosin). In addition, the second author was partially supported by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.
Rights and permissions
About this article
Cite this article
Brown, J., Klosin, K. On the action of the \(U_p\) operator on the local (at p) representation attached to congruence level Siegel modular forms. Ramanujan J 44, 597–615 (2017). https://doi.org/10.1007/s11139-016-9833-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-016-9833-x