On the computation of general vector-valued modular forms

Magnusson, Tobias; Raum, Martin

doi:10.1090/mcom/3847

On the computation of general vector-valued modular forms
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by Tobias Magnusson and Martin Raum

Math. Comp. 92 (2023), 2861-2891

DOI: https://doi.org/10.1090/mcom/3847

Published electronically: May 23, 2023

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Abstract:

We present and discuss an algorithm and its implementation that is capable of directly determining Fourier expansions of any vector-valued modular form of weight at least $2$ associated with representations whose kernel is a congruence subgroup. It complements two available algorithms that are limited to inductions of Dirichlet characters and to Weil representations, thus covering further applications like Moonshine or Jacobi forms for congruence subgroups. We examine the calculation of invariants in specific representations via techniques from permutation groups, which greatly aids runtime performance. We explain how a generalization of cusp expansions of classical modular forms enters our implementation. After a heuristic consideration of time complexity, we relate the formulation of our algorithm to the two available ones, to highlight the compromises between level of generality and performance that each them makes.

References

Josh Alman and Virginia Vassilevska Williams, A refined laser method and faster matrix multiplication, Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), [Society for Industrial and Applied Mathematics (SIAM)], Philadelphia, PA, 2021, pp. 522–539. MR 4262465, DOI 10.1137/1.9781611976465.32
Eran Assaf, Computing classical modular forms for arbitrary congruence subgroups, Arithmetic geometry, number theory, and computation, Simons Symp., Springer, Cham, [2021] ©2021, pp. 43–104. MR 4427960, DOI 10.1007/978-3-030-80914-0_{2}
Alex J. Best, Jonathan Bober, Andrew R. Booker, Edgar Costa, John E. Cremona, Maarten Derickx, Min Lee, David Lowry-Duda, David Roe, Andrew V. Sutherland, and John Voight, Computing classical modular forms, Arithmetic geometry, number theory, and computation, Simons Symp., Springer, Cham, [2021] ©2021, pp. 131–213. MR 4427962, DOI 10.1007/978-3-030-80914-0_{4}
Karim Belabas and Henri Cohen, Modular forms in Pari/GP, Res. Math. Sci. 5 (2018), no. 3, Paper No. 37, 19. MR 3846373, DOI 10.1007/s40687-018-0155-z
Lev A. Borisov and Paul E. Gunnells, Toric modular forms and nonvanishing of $L$-functions, J. Reine Angew. Math. 539 (2001), 149–165. MR 1863857, DOI 10.1515/crll.2001.071
Lev A. Borisov and Paul E. Gunnells, Toric varieties and modular forms, Invent. Math. 144 (2001), no. 2, 297–325. MR 1826373, DOI 10.1007/PL00005802
Lev A. Borisov and Paul E. Gunnells, Toric modular forms of higher weight, J. Reine Angew. Math. 560 (2003), 43–64. MR 1992801, DOI 10.1515/crll.2003.060
Jan Hendrik Bruinier and Michael Kuss, Eisenstein series attached to lattices and modular forms on orthogonal groups, Manuscripta Math. 106 (2001), no. 4, 443–459. MR 1875342, DOI 10.1007/s229-001-8027-1
Richard E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562. MR 1625724, DOI 10.1007/s002220050232
Daniel Bump, Lie groups, 2nd ed., Graduate Texts in Mathematics, vol. 225, Springer, New York, 2013. MR 3136522, DOI 10.1007/978-1-4614-8024-2
Jan Hendrik Bruinier and Martin Westerholt-Raum, Kudla’s modularity conjecture and formal Fourier-Jacobi series, Forum Math. Pi 3 (2015), e7, 30. MR 3406827, DOI 10.1017/fmp.2015.6
Miranda C. N. Cheng, John F. R. Duncan, and Jeffrey A. Harvey, Umbral moonshine and the Niemeier lattices, Res. Math. Sci. 1 (2014), Art. 3, 81. MR 3449012, DOI 10.1186/2197-9847-1-3
Henri Cohen, Expansions at cusps and Petersson products in Pari/GP, Elliptic integrals, elliptic functions and modular forms in quantum field theory, Texts Monogr. Symbol. Comput., Springer, Cham, 2019, pp. 161–181. MR 3889557
James Demmel, Ioana Dumitriu, and Olga Holtz, Fast linear algebra is stable, Numer. Math. 108 (2007), no. 1, 59–91. MR 2350185, DOI 10.1007/s00211-007-0114-x
David S. Dummit and Richard M. Foote, Abstract algebra, 3rd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2004. MR 2286236
P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 349, Springer, Berlin-New York, 1973, pp. 143–316 (French). MR 337993
Fred Diamond and Jerry Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR 2112196
Moritz Dittmann, Riccardo Salvati Manni, and Nils R. Scheithauer, Harmonic theta series and the Kodaira dimension of $\mathcal A_6$, Algebra Number Theory 15 (2021), no. 1, 271–285. MR 4226989, DOI 10.2140/ant.2021.15.271
Bas Edixhoven, Modular curves, modular forms, lattices, Galois representations, Computational aspects of modular forms and Galois representations, Ann. of Math. Stud., vol. 176, Princeton Univ. Press, Princeton, NJ, 2011, pp. 29–68. MR 2857087, DOI 10.1515/9781400839001
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
Claus Fieker, William Hart, Tommy Hofmann, and Fredrik Johansson, Nemo/Hecke: computer algebra and number theory packages for the Julia programming language, ISSAC’17—Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2017, pp. 157–164. MR 3703682, DOI 10.1145/3087604.3087611
The GAP Group, GAP – groups, algorithms, and programming, Version 4.11.1, 2021.
Matthias R. Gaberdiel, Daniel Persson, Henrik Ronellenfitsch, and Roberto Volpato, Generalized Mathieu Moonshine, Commun. Number Theory Phys. 7 (2013), no. 1, 145–223. MR 3108775, DOI 10.4310/CNTP.2013.v7.n1.a5
Valery Gritsenko, Cris Poor, and David S. Yuen, Borcherds products everywhere, J. Number Theory 148 (2015), 164–195. MR 3283174, DOI 10.1016/j.jnt.2014.07.028
V. Gritsenko, N.-P. Skoruppa, and D. Zagier, Theta blocks, arXiv:1907.00188, 2019.
Derek F. Holt, Bettina Eick, and Eamonn A. O’Brien, Handbook of computational group theory, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2129747, DOI 10.1201/9781420035216
Fredrik Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput. 66 (2017), no. 8, 1281–1292. MR 3681746, DOI 10.1109/TC.2017.2690633
Nicholas M. Katz, $p$-adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972) Lecture Notes in Math., Vol. 350, Springer, Berlin-New York, 1973, pp. 69–190. MR 447119
Chris A. Kurth and Ling Long, Computations with finite index subgroups of $\textrm {PSL}_2(\Bbb Z)$ using Farey symbols, Advances in algebra and combinatorics, World Sci. Publ., Hackensack, NJ, 2008, pp. 225–242. MR 2440118, DOI 10.1142/9789812790019_{0}015
Winfried Kohnen and Yves Martin, Products of two Eisenstein series and spaces of cusp forms of prime level, J. Ramanujan Math. Soc. 23 (2008), no. 4, 337–356. MR 2492572
Kamal Khuri-Makdisi, Moduli interpretation of Eisenstein series, Int. J. Number Theory 8 (2012), no. 3, 715–748. MR 2904927, DOI 10.1142/S1793042112500418
W. Kohnen and D. Zagier, Modular forms with rational periods, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 197–249. MR 803368
B. Mazur, Rational points on modular curves, Modular functions of one variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Lecture Notes in Math., Vol. 601, Springer, Berlin-New York, 1977, pp. 107–148. MR 450283
Toshitsune Miyake, Modular forms, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. MR 1021004, DOI 10.1007/3-540-29593-3
H. Monien, The Sporadic Group $j2$, Hauptmodul and Belyi Map, arXiv:1703.05200, 2017.
Pietro Mercuri and René Schoof, Modular forms invariant under non-split Cartan subgroups, Math. Comp. 89 (2020), no. 324, 1969–1991. MR 4081925, DOI 10.1090/mcom/3503
Cris Poor, Jerry Shurman, and David S. Yuen, Theta block Fourier expansions, Borcherds products and a sequence of Newman and Shanks, Bull. Aust. Math. Soc. 98 (2018), no. 1, 48–59. MR 3819971, DOI 10.1017/S000497271800031X
R. A. Rankin, The scalar product of modular forms, Proc. London Math. Soc. (3) 2 (1952), 198–217. MR 49231, DOI 10.1112/plms/s3-2.1.198
Martin Raum and Jiacheng Xia, All modular forms of weight 2 can be expressed by Eisenstein series, Res. Number Theory 6 (2020), no. 3, Paper No. 32, 16. MR 4134182, DOI 10.1007/s40993-020-00207-z
Nils R. Scheithauer, On the classification of automorphic products and generalized Kac-Moody algebras, Invent. Math. 164 (2006), no. 3, 641–678. MR 2221135, DOI 10.1007/s00222-006-0500-5
Nils R. Scheithauer, The Weil representation of $\textrm {SL}_2(\Bbb Z)$ and some applications, Int. Math. Res. Not. IMRN 8 (2009), 1488–1545. MR 2496771, DOI 10.1093/imrn/rnn166
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 450380, DOI 10.1007/978-1-4684-9458-7
Nils-Peter Skoruppa, Jacobi forms of critical weight and Weil representations, Modular forms on Schiermonnikoog, Cambridge Univ. Press, Cambridge, 2008, pp. 239–266. MR 2512363, DOI 10.1017/CBO9780511543371.013
William Stein, Modular forms, a computational approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, RI, 2007. With an appendix by Paul E. Gunnells. MR 2289048, DOI 10.1090/gsm/079
Volker Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969), 354–356. MR 248973, DOI 10.1007/BF02165411
Jacob Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985) Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 275–280. MR 894516, DOI 10.1007/BFb0072985
The Sage Developers, SageMath, the Sage Mathematics software system (Version 9.2), 2020.
Brandon Williams, Poincaré square series for the Weil representation, Ramanujan J. 47 (2018), no. 3, 605–650. MR 3874810, DOI 10.1007/s11139-017-9986-2
Brandon Williams, Vector-valued Hirzebruch-Zagier series and class number sums, Res. Math. Sci. 5 (2018), no. 2, Paper No. 25, 13. MR 3799151, DOI 10.1007/s40687-018-0142-4
Brandon Williams, Overpartition $M2$-rank differences, class number relations, and vector-valued mock Eisenstein series, Acta Arith. 189 (2019), no. 4, 347–365. MR 3962833, DOI 10.4064/aa170810-21-10
Martin Westerholt-Raum, Products of vector valued Eisenstein series, Forum Math. 29 (2017), no. 1, 157–186. MR 3592597, DOI 10.1515/forum-2014-0198