Quinary forms and paramodular forms

Dummigan, N.; Pacetti, A.; Rama, G.; Tornaría, G.

doi:10.1090/mcom/3815

Quinary forms and paramodular forms
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by N. Dummigan, A. Pacetti, G. Rama and G. Tornaría;

Math. Comp. 93 (2024), 1805-1858

DOI: https://doi.org/10.1090/mcom/3815

Published electronically: February 16, 2024

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Abstract:

We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev.

References

Hiraku Atobe, Masataka Chida, Tomoyoshi Ibukiyama, Hidenori Katsurada, and Takuya Yamauchi, Harder’s conjecture I, J. Math. Soc. Japan 75 (2023), no. 4, 1339–1408. MR 4659316, DOI 10.2969/jmsj/87988798
Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over $\mathbf Q$: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), no. 4, 843–939. MR 1839918, DOI 10.1090/S0894-0347-01-00370-8
George Boxer, Frank Calegari, Toby Gee, and Vincent Pilloni, Abelian surfaces over totally real fields are potentially modular, Publ. Math. Inst. Hautes Études Sci. 134 (2021), 153–501. MR 4349242, DOI 10.1007/s10240-021-00128-2
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
B. J. Birch, Hecke actions on classes of ternary quadratic forms, Computational number theory (Debrecen, 1989) de Gruyter, Berlin, 1991, pp. 191–212. MR 1151865
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
Armand Brumer and Kenneth Kramer, Corrigendum to “Paramodular abelian varieties of odd conductor", Trans. Amer. Math. Soc. 372 (2019), no. 3, 2251–2254. MR 3976591, DOI 10.1090/tran/7792
Jim Brown and Huixi Li, Congruence primes for Siegel modular forms of paramodular level and applications to the Bloch-Kato conjecture, Glasg. Math. J. 63 (2021), no. 3, 660–681. MR 4301048, DOI 10.1017/S0017089520000439
Nicolas Billerey and Ricardo Menares, On the modularity of reducible $\textrm {mod}\, l$ Galois representations, Math. Res. Lett. 23 (2016), no. 1, 15–41. MR 3512875, DOI 10.4310/MRL.2016.v23.n1.a2
Armand Brumer, Ariel Pacetti, Cris Poor, Gonzalo Tornaría, John Voight, and David S. Yuen, On the paramodularity of typical abelian surfaces, Algebra Number Theory 13 (2019), no. 5, 1145–1195. MR 3981316, DOI 10.2140/ant.2019.13.1145
J. Brzeziński, On orders in quaternion algebras, Comm. Algebra 11 (1983), no. 5, 501–522. MR 693798, DOI 10.1080/00927878308822861
J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
Clifton Cunningham and Lassina Dembélé, Computing genus-2 Hilbert-Siegel modular forms over $\Bbb Q(\sqrt 5)$ via the Jacquet-Langlands correspondence, Experiment. Math. 18 (2009), no. 3, 337–345. MR 2555703, DOI 10.1080/10586458.2009.10129048
Ping-Shun Chan and Wee Teck Gan, The local Langlands conjecture for $\rm GSp(4)$ III: Stability and twisted endoscopy, J. Number Theory 146 (2015), 69–133. MR 3267112, DOI 10.1016/j.jnt.2013.07.009
Gaëtan Chenevier and Jean Lannes, Automorphic forms and even unimodular lattices, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 69, Springer, Cham, 2019. Kneser neighbors of Niemeier lattices; Translated from the French by Reinie Erné. MR 3929692, DOI 10.1007/978-3-319-95891-0
J. E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997. MR 1628193
Lassina Dembélé, On the computation of algebraic modular forms on compact inner forms of $\rm {GSp}_4$, Math. Comp. 83 (2014), no. 288, 1931–1950. MR 3194136, DOI 10.1090/S0025-5718-2014-02374-0
Neil Dummigan and Daniel Fretwell, Ramanujan-style congruences of local origin, J. Number Theory 143 (2014), 248–261. MR 3227346, DOI 10.1016/j.jnt.2014.04.008
Fred Diamond, Congruence primes for cusp forms of weight $k\ge 2$, Astérisque 196-197 (1991), 6, 205–213 (1992) (English, with French summary). Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). MR 1141459
Luis V. Dieulefait, On the images of the Galois representations attached to genus 2 Siegel modular forms, J. Reine Angew. Math. 553 (2002), 183–200. MR 1944811, DOI 10.1515/crll.2002.098
Luis Dieulefait, Remarks on Serre’s modularity conjecture, Manuscripta Math. 139 (2012), no. 1-2, 71–89. MR 2959671, DOI 10.1007/s00229-011-0503-4
Neil Dummigan, Congruences of Saito-Kurokawa lifts and denominators of central spinor $L$-values, Glasg. Math. J. 64 (2022), no. 2, 504–525. MR 4404111, DOI 10.1017/S0017089521000331
Martin Eichler, Quadratische Formen und orthogonale Gruppen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band LXIII, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1952 (German). MR 51875, DOI 10.1007/978-3-662-01212-3
M. Eichler, The basis problem for modular forms and the traces of the Hecke operators, Modular functions of one variable, I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 320, Springer, Berlin-New York, 1973, pp. 75–151. MR 485698
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Jean-Marc Fontaine and Guy Laffaille, Construction de représentations $p$-adiques, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 4, 547–608 (1983) (French). MR 707328, DOI 10.24033/asens.1437
Dan Fretwell, Genus 2 paramodular Eisenstein congruences, Ramanujan J. 46 (2018), no. 2, 447–473. MR 3803970, DOI 10.1007/s11139-017-9884-7
Wee Teck Gan, The Saito-Kurokawa space of $\textrm {PGSp}_4$ and its transfer to inner forms, Eisenstein series and applications, Progr. Math., vol. 258, Birkhäuser Boston, Boston, MA, 2008, pp. 87–123. MR 2402681, DOI 10.1007/978-0-8176-4639-4_{3}
Benedict H. Gross and Mark Reeder, From Laplace to Langlands via representations of orthogonal groups, Bull. Amer. Math. Soc. (N.S.) 43 (2006), no. 2, 163–205. MR 2216109, DOI 10.1090/S0273-0979-06-01100-1
B. Kh. Gross, On the Langlands correspondence for symplectic motives, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 4, 49–64 (Russian, with Russian summary); English transl., Izv. Math. 80 (2016), no. 4, 678–692. MR 3535358, DOI 10.4213/im8431
Matthew Greenberg and John Voight, Lattice methods for algebraic modular forms on classical groups, Computations with modular forms, Contrib. Math. Comput. Sci., vol. 6, Springer, Cham, 2014, pp. 147–179. MR 3381452, DOI 10.1007/978-3-319-03847-6_{6}
Günter Harder, A congruence between a Siegel and an elliptic modular form, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 247–262. MR 2409680, DOI 10.1007/978-3-540-74119-0_{4}
G. Harder, Secondary operations in the cohomology of Harish-Chandra modules, 2013, http://www.math.uni-bonn.de/people/harder/Manuscripts/Eisenstein/SecOPs.pdf.
Jeffery Hein, Orthogonal modular forms: An application to a conjecture of birch, algorithms and computations, ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–Dartmouth College. MR 3553638
Hiroaki Hijikata, Explicit formula of the traces of Hecke operators for $\Gamma _{0}(N)$, J. Math. Soc. Japan 26 (1974), 56–82. MR 337783, DOI 10.2969/jmsj/02610056
J. Hein, G. Tornarìa, and J. Voight. Hilbert modular forms as orthogonal modular forms. Preprint.
Tomoyoshi Ibukiyama, Conjectures of Shimura type and of Harder type revisited, Comment. Math. Univ. St. Pauli 63 (2014), no. 1-2, 79–103. MR 3328425
Tomoyoshi Ibukiyama, Type numbers of quaternion hermitian forms and supersingular abelian varieties, Osaka J. Math. 55 (2018), no. 2, 369–384. MR 3787750
Tomoyoshi Ibukiyama, Quinary lattices and binary quaternion hermitian lattices, Tohoku Math. J. (2) 71 (2019), no. 2, 207–220. MR 3973249, DOI 10.2748/tmj/1561082596
T. Ibukiyama, Dimensions of paramodular forms and compact twist modular forms with involutions, arXiv:2208.13578, 2022.
Tomoyoshi Ibukiyama and Hidetaka Kitayama, Dimension formulas of paramodular forms of squarefree level and comparison with inner twist, J. Math. Soc. Japan 69 (2017), no. 2, 597–671. MR 3638279, DOI 10.2969/jmsj/06920597
Hiroshi Ishimoto, Proofs of Ibukiyama’s conjectures on Siegel modular forms of half-integral weight and of degree 2, Math. Ann. 383 (2022), no. 1-2, 645–698. MR 4444133, DOI 10.1007/s00208-021-02232-4
Jennifer Johnson-Leung and Brooks Roberts, Siegel modular forms of degree two attached to Hilbert modular forms, J. Number Theory 132 (2012), no. 4, 543–564. MR 2887605, DOI 10.1016/j.jnt.2011.08.004
Mark Kisin, Modularity of 2-adic Barsotti-Tate representations, Invent. Math. 178 (2009), no. 3, 587–634. MR 2551765, DOI 10.1007/s00222-009-0207-5
Oliver D. King, Cris Poor, Jerry Shurman, and David S. Yuen, Using Katsurada’s determination of the Eisenstein series to compute Siegel eigenforms, Math. Comp. 87 (2018), no. 310, 879–892. MR 3739221, DOI 10.1090/mcom/3218
Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture. I, Invent. Math. 178 (2009), no. 3, 485–504. MR 2551763, DOI 10.1007/s00222-009-0205-7
Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture. II, Invent. Math. 178 (2009), no. 3, 505–586. MR 2551764, DOI 10.1007/s00222-009-0206-6
G. Lachaussée, Autour de l’énumération des représentations automorphes cuspidales algébriques de $\operatorname {gl}_n$ sur $\mathbb {Q}$ de conducteur $>1$, arXiv:2011.08237, 2020.
W. B. Ladd, Algebraic modular forms on $\mathrm {SO}5(\mathbb {Q})$ and the computation of paramodular forms, Ph.D. Thesis, Berkeley, 2018, https://digitalassets.lib.berkeley.edu/etd/ucb/text/Ladd_berkeley_0028E_17895.pdf.
T. Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005. MR 2104929, DOI 10.1090/gsm/067
S. Lemurell, Quaternion orders and ternary quadratic forms, arXiv:1103.4922, 2011.
The LMFDB Collaboration, The L-functions and modular forms database, 2021, http://www.lmfdb.org. [Online; accessed 10 October 2021].
Chung Pang Mok, Galois representations attached to automorphic forms on $\textrm {GL}_2$ over CM fields, Compos. Math. 150 (2014), no. 4, 523–567. MR 3200667, DOI 10.1112/S0010437X13007665
O. T. O’Meara, Introduction to quadratic forms, Die Grundlehren der mathematischen Wissenschaften, Band 117, Springer-Verlag, New York-Heidelberg, 1971. Second printing, corrected. MR 347768
The PARI Group, Univ. Bordeaux, PARI/GP version 2.11.0, 2018, http://pari.math.u-bordeaux.fr/.
Arnold Pizer, An algorithm for computing modular forms on $\Gamma _{0}(N)$, J. Algebra 64 (1980), no. 2, 340–390. MR 579066, DOI 10.1016/0021-8693(80)90151-9
W. Plesken and B. Souvignier, Computing isometries of lattices, J. Symbolic Comput. 24 (1997), no. 3-4, 327–334. Computational algebra and number theory (London, 1993). MR 1484483, DOI 10.1006/jsco.1996.0130
Cris Poor, Jerry Shurman, and David S. Yuen, Siegel paramodular forms of weight 2 and squarefree level, Int. J. Number Theory 13 (2017), no. 10, 2627–2652. MR 3713095, DOI 10.1142/S1793042117501469
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
G. Rama, Módulo de Brandt generalizado, M.Sc. Thesis, Universidad de la República, 2014, http://cmat.edu.uy/cnt/msc.2014.0001.
G. Rama, Cálculo de formas paramodulares utilizando formas modulares ortogonales de $O(5)$, Ph.D. Thesis, Universidad de la RepÚblica, 2020, http://cmat.edu.uy/cnt/phd.2021.0001.
G. Rama, Quinary orthogonal modular forms code repository, 2020, https://gitlab.fing.edu.uy/grama/quinary.
Kenneth A. Ribet, Congruence relations between modular forms, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 503–514. MR 804706
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
Gustavo Rama and Gonzalo Tornaría, Computation of paramodular forms, ANTS XIV—Proceedings of the Fourteenth Algorithmic Number Theory Symposium, Open Book Ser., vol. 4, Math. Sci. Publ., Berkeley, CA, 2020, pp. 353–370. MR 4235123, DOI 10.2140/obs.2020.4.353
M. Rösner and R. Weissauer, Global liftings between inner forms of GSp(4), arXiv:2103.14715, 2021.
The Sage Developers, SageMath, the Sage Mathematics software system (version 9.4), 2021, https://www.sagemath.org.
Ralf Schmidt, Iwahori-spherical representations of $\textrm {GSp}(4)$ and Siegel modular forms of degree 2 with square-free level, J. Math. Soc. Japan 57 (2005), no. 1, 259–293. MR 2114732
Hiro-aki Narita, Jacquet-Langlands-Shimizu correspondence for theta lifts to $GSp(2)$ and its inner forms I: An explicit functorial correspondence, J. Math. Soc. Japan 69 (2017), no. 4, 1443–1474. With an appendix by Ralf Schmidt. MR 3715811, DOI 10.2969/jmsj/06941443
Ralf Schmidt, Paramodular forms in CAP representations of $\textrm {GSp}(4)$, Acta Arith. 194 (2020), no. 4, 319–340. MR 4103275, DOI 10.4064/aa180606-23-9
R. Schulze-Pillot, Lecture notes on quadratic forms and their arithmetic, arXiv:2008.12847, 2020.
Abhishek Saha and Ralf Schmidt, Yoshida lifts and simultaneous non-vanishing of dihedral twists of modular $L$-functions, J. Lond. Math. Soc. (2) 88 (2013), no. 1, 251–270. MR 3092267, DOI 10.1112/jlms/jdt008
Gonzalo Tornaría, The Brandt module of ternary quadratic lattices, ProQuest LLC, Ann Arbor, MI, 2005. Thesis (Ph.D.)–The University of Texas at Austin. MR 2717378
Pei-Yu Tsai, On Newforms for Split Special Odd Orthogonal Groups, ProQuest LLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–Harvard University. MR 3167284
Gerard van der Geer, Siegel modular forms and their applications, The 1-2-3 of modular forms, Universitext, Springer, Berlin, 2008, pp. 181–245. MR 2409679, DOI 10.1007/978-3-540-74119-0_{3}
Pol van Hoften, A geometric Jacquet-Langlands correspondence for paramodular Siegel threefolds, Math. Z. 299 (2021), no. 3-4, 2029–2061. MR 4329279, DOI 10.1007/s00209-021-02756-0
Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980 (French). MR 580949, DOI 10.1007/BFb0091027
Rainer Weissauer, Four dimensional Galois representations, Astérisque 302 (2005), 67–150 (English, with English and French summaries). Formes automorphes. II. Le cas du groupe $\rm GSp(4)$. MR 2234860
Rainer Weissauer, Existence of Whittaker models related to four dimensional symplectic Galois representations, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, 2008, pp. 285–310. MR 2530981, DOI 10.1017/CBO9780511543371.016
Ariel Weiss, On the images of Galois representations attached to low weight Siegel modular forms, J. Lond. Math. Soc. (2) 106 (2022), no. 1, 358–387. MR 4454492, DOI 10.1112/jlms.12576
Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. MR 1333035, DOI 10.2307/2118559
Hwajong Yoo, The index of an Eisenstein ideal and multiplicity one, Math. Z. 282 (2016), no. 3-4, 1097–1116. MR 3473658, DOI 10.1007/s00209-015-1579-4
Hwajong Yoo, Non-optimal levels of a reducible $\rm {mod}\,\ell$ modular representation, Trans. Amer. Math. Soc. 371 (2019), no. 6, 3805–3830. MR 3917209, DOI 10.1090/tran/7314
Hiroyuki Yoshida, Siegel’s modular forms and the arithmetic of quadratic forms, Invent. Math. 60 (1980), no. 3, 193–248. MR 586427, DOI 10.1007/BF01390016