Computing groups of Hecke characters. (English) Zbl 1498.11243
Summary: We describe algorithms to represent and compute groups of Hecke characters. We make use of an idèlic point of view and obtain the whole family of such characters, including transcendental ones. We also show how to isolate the algebraic characters, which are of particular interest in number theory. This work has been implemented in Pari/GP, and we illustrate our work with a variety of explicit examples using our implementation.
MSC:
| 11Y40 | Algebraic number theory computations |
| 11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |
| 11G10 | Abelian varieties of dimension \(> 1\) |
| 11G15 | Complex multiplication and moduli of abelian varieties |
References:
| [1] | Arvind, V., Kurur, Piyush P.: Upper bounds on the complexity of some Galois theory problems (extended abstract). In: Algorithms and Computation. Lecture Notes in Comput. Sci., vol. 2906, pp. 716-725. Springer, Berlin (2003). doi:10.1007/978-3-540-24587-2_73 · Zbl 1205.11137 |
| [2] | Booker, AR; Then, H., Rapid computation of L-functions attached to Maass forms, Int. J. Number Theory, 14, 5, 1459-1485 (2018) · Zbl 1446.11093 · doi:10.1142/S1793042118500896 |
| [3] | Bouw, I.I., Koutsianas, A., Sijsling, J., Wewers, S.: Conductor and discriminant of Picard curves. J. Lond. Math. Soc. (2) 102(1), 368-404 (2020). doi:10.1112/jlms.12323 · Zbl 1457.14061 |
| [4] | Buchmann, J.: A subexponential algorithm for the determination of class groups and regulators of algebraic number fields. Séminaire de Théorie des Nombres, Paris 1988-1989. Progr. Math., vol. 91, pp. 27-41. Birkhäuser, Boston (1990) · Zbl 0727.11059 |
| [5] | Chevalley, C., Deux théorèmes d’arithmétique, J. Math. Soc. Jpn., 3, 36-44 (1951) · Zbl 0044.03001 · doi:10.2969/jmsj/00310036 |
| [6] | Cohen, H., Strömberg, F.: Modular Forms. Graduate Studies in Mathematics. A Classical Approach, vol. 179, pp. xii+700. American Mathematical Society, Providence (2017). doi:10.1090/gsm/179 · Zbl 1464.11001 |
| [7] | Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics, vol. 138, pp. xii+534. Springer, Berlin (1993). doi:10.1007/978-3-662-02945-9 · Zbl 0786.11071 |
| [8] | Cohen, H.: Advanced Topics in Computational Number Theory. Graduate Texts in Mathematics, vol. 193, pp. xvi+578. Springer, New York (2000). doi:10.1007/978-1-4419-8489-0 · Zbl 0977.11056 |
| [9] | Costa, E.; Mascot, N.; Sijsling, J.; Voight, J., Rigorous computation of the endomorphism ring of a Jacobian, Math. Comp., 88, 317, 1303-1339 (2019) · Zbl 1484.11135 · doi:10.1090/mcom/3373 |
| [10] | Cremona, J., Page, A., Sutherland, A.V.: Sorting and labelling integral ideals in a number field. (2020). arXiv:2005.09491 [math.NT] |
| [11] | Deligne, P.: Valeurs de fonctions L et périodes d’intégrales. French. Automorphic forms, representations and L-functions. In: Proc. Symp. Pure Math. Am. Math. Soc., Corvallis/Oregon 1977, Proc. Symp. Pure Math., vol. 33(2), pp. 313-346 (1979) · Zbl 0449.10022 |
| [12] | Elsenhans, A-S; Klüners, J., Computing subfields of number fields and applications to Galois group computations, J. Symbolic Comput., 93, 1-20 (2019) · Zbl 1419.11142 · doi:10.1016/j.jsc.2018.04.013 |
| [13] | Hecke, E.: Mathematische Werke. Herausgegeben im Auftrage der Akademie der Wissenschaften zu Göttingen. Vandenhoeck & Ruprecht, Göttingen (1 plate), (1959) · Zbl 0092.00102 |
| [14] | Johansson, F., Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput., 66, 8, 1281-1292 (2017) · Zbl 1388.65037 · doi:10.1109/TC.2017.2690633 |
| [15] | Joux, A., Odlyzko, A., Pierrot, C.: The past, evolving present, and future of the discrete logarithm. English. In: Open Problems in Mathematics and Computational Science. Based on the Presentations at the Conference, Istanbul, Turkey, September 18-20, 2013, pp. 5-36. Springer, Cham (2014). doi:10.1007/978-3-319-10683-0_2 · Zbl 1314.94006 |
| [16] | Landau, S.; Miller, GL, Solvability by radicals is in polynomial time, J. Comput. Syst. Sci., 30, 2, 179-208 (1985) · Zbl 0586.12002 · doi:10.1016/0022-0000(85)90013-3 |
| [17] | Lang, S.: Algebraic Number Theory. Second. Graduate Texts in Mathematics, vol. 110, pp. xiv+357. Springer, New York (1994). doi:10.1007/978-1-4612-0853-2 · Zbl 0811.11001 |
| [18] | Lang, S.: Complex Multiplication. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 255, pp. viii+184. Springer, New York (1983). doi:10.1007/978-1-4612-5485-0 · Zbl 0536.14029 |
| [19] | Lenstra, A.K.: Factoring polynomials over algebraic number fields. In: Computer Algebra (London, 1983). Lecture Notes in Comput. Sci., vol. 162, pp. 245-254. Springer, Berlin (1983). doi:10.1007/3-540-12868-9_108 · Zbl 0539.68030 |
| [20] | Lombardo, D., Computing the geometric endomorphism ring of a genus-2 Jacobian, Math. Comp., 88, 316, 889-929 (2019) · Zbl 1410.11043 · doi:10.1090/mcom/3358 |
| [21] | Milne, J.S.: Complex multiplication (2020). https://www.jmilne.org/math/CourseNotes/cm.html. Accessed 21 Feb 2022 |
| [22] | Milne, JS, On the arithmetic of abelian varieties, Invent. Math., 17, 177-190 (1972) · Zbl 0249.14012 · doi:10.1007/BF01425446 |
| [23] | Molin, P., Page, A.: Hecke Grossencharacters support. Version 2.15.0. (2022). https://pari.math.u-bordeaux.fr/dochtml/html/General_number_fields.html#gcharinit |
| [24] | Morris, S.A.: Duality and structure of locally compact abelian groups for the layman. Math. Chronicle 8, 39-56 (1979) · Zbl 0452.22004 |
| [25] | Morris, S.A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups. London Mathematical Society Lecture Note Series, vol. 29, pp. viii+128. Cambridge University Press, Cambridge (1977) · Zbl 0446.22006 |
| [26] | Moy, RA; Specter, J., There exist non-CM Hilbert modular forms of partial weight 1, Int. Math. Res. Not. IMRN, 24, 13047-13061 (2015) · Zbl 1388.11020 · doi:10.1093/imrn/rnv089 |
| [27] | PARI/GP version 2.15.0. (2022). http://pari.math.u-bordeaux.fr/.ThePARIGroup.Univ.Bordeaux |
| [28] | Patrikis, S.: Variations on a theorem of Tate. Mem. Am. Math. Soc. 258(1238), viii+156 (2019). doi:10.1090/memo/1238 |
| [29] | Ribet, K.A.: Galois representations attached to eigenforms with Nebentypus. In: Modular Functions of One Variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976). Lecture Notes in Math., vol. 601, pp. 17-51 (1977) · Zbl 0363.10015 |
| [30] | Schappacher, N.: Periods of Hecke Characters. Lecture Notes in Mathematics, vol. 1301, pp. xvi+160. Springer, Berlin (1988). doi:10.1007/BFb0082094 · Zbl 0659.14001 |
| [31] | Shimura, G., On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields, Nagoya Math. J., 43, 199-208 (1971) · Zbl 0225.14015 · doi:10.1017/S0027763000014471 |
| [32] | Shimura, G., On the zeta-function of an abelian variety with complex multiplication, Ann. Math., 2, 94, 504-533 (1971) · Zbl 0242.14009 · doi:10.2307/1970768 |
| [33] | Shimura, G., Class fields over real quadratic fields and Hecke operators, Ann. Math., 2, 95, 130-190 (1972) · Zbl 0255.10032 · doi:10.2307/1970859 |
| [34] | Taniyama, Y., L-functions of number fields and zeta functions of abelian varieties, J. Math. Soc. Jpn., 9, 330-366 (1957) · Zbl 0213.22803 · doi:10.2969/jmsj/00930330 |
| [35] | Tate, J.T.: Fourier analysis in number fields, and Hecke’s zeta-functions. In: Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), pp. 305-347. Thompson, Washington, DC (1967) · Zbl 1492.11154 |
| [36] | The LMFDB Collaboration. The L-Functions and Modular Forms Database, Home Page of Classical Modular Forms (2022). https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/. Accessed 21 Feb 2022 |
| [37] | The LMFDB Collaboration. The L-Functions and Modular Forms Database, Home Page of the Genus 2 Curve 28561.a.371293.1 (2022). https://www.lmfdb.org/Genus2Curve/Q/28561/a/371293/1. Accessed 21 Feb 2022 |
| [38] | The LMFDB Collaboration. The L-Functions and Modular Forms Database, Home Page of the Genus 3 curve 3.9-1.0.3-9-9.6 (2022). https://www.lmfdb.org/HigherGenus/C/Aut/3.9-1.0.3-9-9.6. Accessed 21 Feb 2022 |
| [39] | van Hoeij, M.; Klüners, J.; Novocin, A., Generating subfields, J. Symbolic Comput., 52, 17-34 (2013) · Zbl 1278.11111 · doi:10.1016/j.jsc.2012.05.010 |
| [40] | Visser, R.: L-values of Elliptic curves twisted by Hecke Grössencharacters. Phd project. (2021). https://warwick.ac.uk/fac/sci/maths/people/staff/visser/firstyearphd_project2_rvisser.pdf |
| [41] | Waldschmidt, M.: Sur certains caractères du groupe des classes d’idèles d’un corps de nombres. Seminar on Number Theory, Paris 1980-81 (Paris, 1980/1981). Progr. Math., vol. 22, pp. 323-335. Birkhäuser, Boston (1982) · Zbl 0494.12007 |
| [42] | Watkins, M.: Computing with Hecke Grössencharacters. Actes de la Conférence “Théorie des Nombres et Applications”. Vol.: Publ. Math. Besançon Algèbre Théorie Nr. Presses Univ. Franche- Comté, Besançon, vol. 2011, pp. 119-135 (2011) · Zbl 1280.11071 |
| [43] | Weil, A.: On a certain type of characters of the idèle-class group of an algebraic number-field. In: Proceedings of the International Symposium on Algebraic Number Theory, Tokyo & Nikko, 1955. Science Council of Japan, Tokyo, pp. 1-7 (1956) · Zbl 0073.26303 |
| [44] | Weil, A., Sur la théorie du corps de classes, J. Math. Soc. Jpn., 3, 1-35 (1951) · Zbl 0044.02901 · doi:10.2969/jmsj/00310001 |
| [45] | Weil, A., Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann., 168, 149-156 (1967) · Zbl 0158.08601 · doi:10.1007/BF01361551 |
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