Formulas for central values of twisted spin 𝐿-functions attached to paramodular forms

Ryan, Nathan; Tornaría, Gonzalo

doi:10.1090/mcom/2988

Formulas for central values of twisted spin $L$-functions attached to paramodular forms
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by Nathan C. Ryan and Gonzalo Tornaría; with an appendix by Ralf Schmidt PDF

Math. Comp. 85 (2016), 907-929 Request permission

Abstract:

In the 1980s Böcherer formulated a conjecture relating the central values of the imaginary quadratic twists of the spin $L$-function attached to a Siegel modular form $F$ to the Fourier coefficients of $F$. This conjecture has been proved when $F$ is a lift. More recently, we formulated an analogous conjecture for paramodular forms $F$ of prime level, even weight and in the plus-space. In this paper, we examine generalizations of this conjecture. In particular, our new formulations relax the conditions on $F$ and allow for twists by real characters. Moreover, these formulations are more explicit than the earlier ones. We prove the conjecture in the case of lifts and provide numerical evidence in the case of nonlifts.

References

Mahdi Asgari and Ralf Schmidt, Siegel modular forms and representations, Manuscripta Math. 104 (2001), no. 2, 173–200. MR 1821182, DOI 10.1007/PL00005869
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Armand Brumer and Kenneth Kramer, Paramodular abelian varieties of odd conductor, Trans. Amer. Math. Soc. 366 (2014), no. 5, 2463–2516. MR 3165645, DOI 10.1090/S0002-9947-2013-05909-0
S. Böcherer, Bemerkungen über die Dirichletreihen von Koecher und Maaß. (Remarks on the Dirichlet series of Koecher and Maaß). Technical report, Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 68, 36 S., 1986.
A. Borel, Automorphic $L$-functions, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, pages 27–61. Amer. Math. Soc., Providence, R.I., 1979.
Siegfried Böcherer and Rainer Schulze-Pillot, The Dirichlet series of Koecher and Maass and modular forms of weight $\frac 32$, Math. Z. 209 (1992), no. 2, 273–287. MR 1147818, DOI 10.1007/BF02570834
Tim Dokchitser, Computing special values of motivic $L$-functions, Experiment. Math. 13 (2004), no. 2, 137–149. MR 2068888
Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
Masaaki Furusawa and Kimball Martin, On central critical values of the degree four $L$-functions for $\textrm {GSp}(4)$: the fundamental lemma, II, Amer. J. Math. 133 (2011), no. 1, 197–233. MR 2752939, DOI 10.1353/ajm.2011.0007
Masaaki Furusawa and Joseph A. Shalika, The fundamental lemma for the Bessel and Novodvorsky subgroups of $\textrm {GSp}(4)$, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 2, 105–110 (English, with English and French summaries). MR 1669031, DOI 10.1016/S0764-4442(99)80146-3
Masaaki Furusawa and Joseph A. Shalika, The fundamental lemma for the Bessel and Novodvorsky subgroups of $\textrm {GSp}(4)$. II, C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), no. 8, 593–598 (English, with English and French summaries). MR 1799095, DOI 10.1016/S0764-4442(00)01683-9
Masaaki Furusawa and Joseph A. Shalika, On central critical values of the degree four $L$-functions for GSp(4): the fundamental lemma, Mem. Amer. Math. Soc. 164 (2003), no. 782, x+139. MR 1983033, DOI 10.1090/memo/0782
Masaaki Furusawa, On $L$-functions for $\textrm {GSp}(4)\times \textrm {GL}(2)$ and their special values, J. Reine Angew. Math. 438 (1993), 187–218. MR 1215654, DOI 10.1515/crll.1993.438.187
B. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of $L$-series. II, Math. Ann. 278 (1987), no. 1-4, 497–562. MR 909238, DOI 10.1007/BF01458081
Valeri Gritsenko, Arithmetical lifting and its applications, Number theory (Paris, 1992–1993) London Math. Soc. Lecture Note Ser., vol. 215, Cambridge Univ. Press, Cambridge, 1995, pp. 103–126. MR 1345176, DOI 10.1017/CBO9780511661990.008
Wee Teck Gan and Shuichiro Takeda, The local Langlands conjecture for $\textrm {GSp}(4)$, Ann. of Math. (2) 173 (2011), no. 3, 1841–1882. MR 2800725, DOI 10.4007/annals.2011.173.3.12
William B. Hart, Gonzalo Tornaría, and Mark Watkins, Congruent number theta coefficients to $10^{12}$, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 6197, Springer, Berlin, 2010, pp. 186–200. MR 2721421, DOI 10.1007/978-3-642-14518-6_{1}7
Winfried Kohnen and Michael Kuss, Some numerical computations concerning spinor zeta functions in genus 2 at the central point, Math. Comp. 71 (2002), no. 240, 1597–1607. MR 1933046, DOI 10.1090/S0025-5718-01-01399-0
Winfried Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), no. 2, 237–268. MR 783554, DOI 10.1007/BF01455989
Cris Poor and David S. Yuen, Paramodular cusp forms, Math. Comp. 84 (2015), no. 293, 1401–1438. MR 3315514, DOI 10.1090/S0025-5718-2014-02870-6
Cris Poor and David Yuen, The paramodular forms website, http://math.lfc.edu/~yuen/paramodular, 2014.
Martin Raum, Efficiently generated spaces of classical Siegel modular forms and the Böcherer conjecture, J. Aust. Math. Soc. 89 (2010), no. 3, 393–405. MR 2785912, DOI 10.1017/S1446788711001121
Brooks Roberts and Ralf Schmidt, On modular forms for the paramodular groups, Automorphic forms and zeta functions, World Sci. Publ., Hackensack, NJ, 2006, pp. 334–364. MR 2208781, DOI 10.1142/9789812774415_{0}015
Brooks Roberts and Ralf Schmidt, Local newforms for GSp(4), Lecture Notes in Mathematics, vol. 1918, Springer, Berlin, 2007. MR 2344630, DOI 10.1007/978-3-540-73324-9
Nathan C. Ryan and Gonzalo Tornaría, A Böcherer-type conjecture for paramodular forms, Int. J. Number Theory 7 (2011), no. 5, 1395–1411. MR 2825979, DOI 10.1142/S1793042111004629
N. C. Ryan and G. Tornaría, Central values of paramodular form L-functions, http://www.ma.utexas.edu/users/tornaria/cnt/paramodular, 2013.
M. O. Rubinstein, lcalc: The L-function calculator, a C $++$ class library and command line program, http://www.math.uwaterloo.ca/~mrubinst, 2008.
Ralf Schmidt, On classical Saito-Kurokawa liftings, J. Reine Angew. Math. 604 (2007), 211–236. MR 2320318, DOI 10.1515/CRELLE.2007.024
W. Stein, Sage: Open Source Mathematical Software (Version 4.8). The Sage Group, 2011.
M. Stoll, Genus 2 curves with small odd discriminant, http://www.faculty.iu-bremen.de/stoll/data/, 2008.
Nils-Peter Skoruppa and Don Zagier, Jacobi forms and a certain space of modular forms, Invent. Math. 94 (1988), no. 1, 113–146. MR 958592, DOI 10.1007/BF01394347
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
J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, 375–484 (French). MR 646366
Rainer Weissauer, Endoscopy for $\textrm {GSp}(4)$ and the cohomology of Siegel modular threefolds, Lecture Notes in Mathematics, vol. 1968, Springer-Verlag, Berlin, 2009. MR 2498783, DOI 10.1007/978-3-540-89306-6