Overconvergent modular forms and their explicit arithmetic
HTML articles powered by AMS MathViewer
- by Jan Vonk;
- Bull. Amer. Math. Soc. 58 (2021), 313-356
- DOI: https://doi.org/10.1090/bull/1700
- Published electronically: August 19, 2020
- HTML | PDF | Request permission
Abstract:
In these notes we aim to give a friendly introduction to the theory of overconvergent modular forms and some examples of recent arithmetic applications. The emphasis is on explicit examples and computations.References
- Fabrizio Andreatta, Adrian Iovita, and Vincent Pilloni, Le halo spectral, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), no. 3, 603–655 (French, with English and French summaries). MR 3831033, DOI 10.24033/asens.2362
- Fabrizio Andreatta, Adrian Iovita, and Glenn Stevens, Overconvergent modular sheaves and modular forms for $\textbf {GL}_{2/F}$, Israel J. Math. 201 (2014), no. 1, 299–359. MR 3265287, DOI 10.1007/s11856-014-1045-8
- Daniel Barsky, Fonctions zeta $p$-adiques d’une classe de rayon des corps de nombres totalement réels, Groupe d’Étude d’Analyse Ultramétrique (5e année: 1977/78), Secrétariat Math., Paris, 1978, pp. Exp. No. 16, 23 (French). MR 525346
- Kevin Buzzard and Frank Calegari, A counterexample to the Gouvêa-Mazur conjecture, C. R. Math. Acad. Sci. Paris 338 (2004), no. 10, 751–753 (English, with English and French summaries). MR 2059481, DOI 10.1016/j.crma.2004.03.016
- Kevin Buzzard and Frank Calegari, Slopes of overconvergent 2-adic modular forms, Compos. Math. 141 (2005), no. 3, 591–604. MR 2135279, DOI 10.1112/S0010437X04001034
- Kevin Buzzard and Frank Calegari, The 2-adic eigencurve is proper, Doc. Math. Extra Vol. (2006), 211–232. MR 2290588
- Massimo Bertolini, Francesc Castella, Henri Darmon, Samit Dasgupta, Kartik Prasanna, and Victor Rotger, $p$-adic $L$-functions and Euler systems: a tale in two trilogies, Automorphic forms and Galois representations. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 414, Cambridge Univ. Press, Cambridge, 2014, pp. 52–101. MR 3444223
- 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
- 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
- Kevin Buzzard and Toby Gee, Slopes of modular forms, Families of automorphic forms and the trace formula, Simons Symp., Springer, [Cham], 2016, pp. 93–109. MR 3675164
- Kevin Buzzard and L. J. P. Kilford, The 2-adic eigencurve at the boundary of weight space, Compos. Math. 141 (2005), no. 3, 605–619. MR 2135280, DOI 10.1112/S0010437X05001314
- John Bergdall and Robert Pollack, Arithmetic properties of Fredholm series for $p$-adic modular forms, Proc. Lond. Math. Soc. (3) 113 (2016), no. 4, 419–444. MR 3556487, DOI 10.1112/plms/pdw031
- Kevin Buzzard, Questions about slopes of modular forms, Astérisque 298 (2005), 1–15 (English, with English and French summaries). Automorphic forms. I. MR 2141701
- Johannes Buchmann and Ulrich Vollmer, Binary quadratic forms, Algorithms and Computation in Mathematics, vol. 20, Springer, Berlin, 2007. An algorithmic approach. MR 2300780
- Frank Calegari, The Coleman-Mazur eigencurve is proper at integral weights, Algebra Number Theory 2 (2008), no. 2, 209–215. MR 2377369, DOI 10.2140/ant.2008.2.209
- F. Calegari, Congruences between modular forms, Arizona Winter School, 2013.
- Pierre Charollois and Samit Dasgupta, Integral Eisenstein cocycles on $\textbf {GL}_n$, I: Sczech’s cocycle and $p$-adic $L$-functions of totally real fields, Camb. J. Math. 2 (2014), no. 1, 49–90. MR 3272012, DOI 10.4310/CJM.2014.v2.n1.a2
- Pierre Charollois, Samit Dasgupta, and Matthew Greenberg, Integral Eisenstein cocycles on $\mathbf {GL}_n$, II: Shintani’s method, Comment. Math. Helv. 90 (2015), no. 2, 435–477. MR 3351752, DOI 10.4171/CMH/360
- Frank Calegari and Matthew Emerton, On the ramification of Hecke algebras at Eisenstein primes, Invent. Math. 160 (2005), no. 1, 97–144. MR 2129709, DOI 10.1007/s00222-004-0406-z
- Robert F. Coleman, Fernando Q. Gouvêa, and Naomi Jochnowitz, $E_2$, $\Theta$, and overconvergence, Internat. Math. Res. Notices 1 (1995), 23–41. MR 1317641, DOI 10.1155/S1073792895000031
- T. Clausen, Theorem, Astronomische Nachrichten, 17 (1840), no. 22, 351–352.
- R. Coleman and B. Mazur, The eigencurve, Galois representations in arithmetic algebraic geometry (Durham, 1996) London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998, pp. 1–113. MR 1696469, DOI 10.1017/CBO9780511662010.003
- Pierrette Cassou-Noguès, Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta $p$-adiques, Invent. Math. 51 (1979), no. 1, 29–59 (French). MR 524276, DOI 10.1007/BF01389911
- Henri Cohen, Variations sur un thème de Seigel et Hecke, Acta Arith. 30 (1976/77), no. 1, 63–93 (French). MR 422215, DOI 10.4064/aa-30-1-63-93
- Robert F. Coleman, Classical and overconvergent modular forms, Invent. Math. 124 (1996), no. 1-3, 215–241. MR 1369416, DOI 10.1007/s002220050051
- Robert F. Coleman, On the coefficients of the characteristic series of the $U$-operator, Proc. Nat. Acad. Sci. U.S.A. 94 (1997), no. 21, 11129–11132. Elliptic curves and modular forms (Washington, DC, 1996). MR 1491972, DOI 10.1073/pnas.94.21.11129
- Robert F. Coleman, $p$-adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), no. 3, 417–479. MR 1431135, DOI 10.1007/s002220050127
- Brian Conrad, Several approaches to non-Archimedean geometry, $p$-adic geometry, Univ. Lecture Ser., vol. 45, Amer. Math. Soc., Providence, RI, 2008, pp. 9–63. MR 2482345, DOI 10.1090/ulect/045/02
- P. Cartier and Y. Roy, Certains calculs numériques relatifs à l’interpolation $p$-adique des séries de Dirichlet, 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. 269–349 (French). MR 330113
- Samit Dasgupta, Computations of elliptic units for real quadratic fields, Canad. J. Math. 59 (2007), no. 3, 553–574. MR 2319158, DOI 10.4153/CJM-2007-023-0
- Samit Dasgupta, Shintani zeta functions and Gross-Stark units for totally real fields, Duke Math. J. 143 (2008), no. 2, 225–279. MR 2420508, DOI 10.1215/00127094-2008-019
- Henri Darmon, Michael Daub, Sam Lichtenstein, and Victor Rotger, Algorithms for Chow-Heegner points via iterated integrals, Math. Comp. 84 (2015), no. 295, 2505–2547. MR 3356037, DOI 10.1090/S0025-5718-2015-02927-5
- Samit Dasgupta, Henri Darmon, and Robert Pollack, Hilbert modular forms and the Gross-Stark conjecture, Ann. of Math. (2) 174 (2011), no. 1, 439–484. MR 2811604, DOI 10.4007/annals.2011.174.1.12
- Pierre Deligne, Formes modulaires et représentations $l$-adiques, Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363, Lecture Notes in Math., vol. 175, Springer, Berlin, 1971, pp. Exp. No. 355, 139–172 (French). MR 3077124
- Evan P. Dummit, Márton Hablicsek, Robert Harron, Lalit Jain, Robert Pollack, and Daniel Ross, Explicit computations of Hida families via overconvergent modular symbols, Res. Number Theory 2 (2016), Paper No. 25, 54. MR 3572014, DOI 10.1007/s40993-016-0052-8
- Hansheng Diao and Ruochuan Liu, The eigencurve is proper, Duke Math. J. 165 (2016), no. 7, 1381–1395. MR 3498869, DOI 10.1215/00127094-3450536
- H. Darmon, A. Pozzi, and J. Vonk, Gross–Stark units, Stark–Heegner points, and derivatives of $p$-adic Eisenstein families, Preprint, 2019.
- H. Darmon, A. Pozzi, and J. Vonk, On the RM values of the Dedekind–Rademacher cocycle, Preprint, 2020.
- Pierre Deligne and Kenneth A. Ribet, Values of abelian $L$-functions at negative integers over totally real fields, Invent. Math. 59 (1980), no. 3, 227–286. MR 579702, DOI 10.1007/BF01453237
- Henri Darmon and Victor Rotger, Diagonal cycles and Euler systems I: A $p$-adic Gross-Zagier formula, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 4, 779–832 (English, with English and French summaries). MR 3250064, DOI 10.24033/asens.2227
- Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids $1$, Ann. Sci. École Norm. Sup. (4) 7 (1974), 507–530 (1975) (French). MR 379379, DOI 10.24033/asens.1277
- Fred Diamond and Jerry Shurman, A first course in modular forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. MR 2112196
- H. Darmon and J. Vonk, Singular moduli for real quadratic fields, Preprint.
- Bernard Dwork, On the zeta function of a hypersurface, Inst. Hautes Études Sci. Publ. Math. 12 (1962), 5–68. MR 159823, DOI 10.1007/BF02684275
- Matthew James Emerton, 2-adic modular forms of minimal slope, ProQuest LLC, Ann Arbor, MI, 1998. Thesis (Ph.D.)–Harvard University. MR 2697462
- Matthew Emerton, $p$-adic families of modular forms (after Hida, Coleman, and Mazur), Astérisque 339 (2011), Exp. No. 1013, vii, 31–61. Séminaire Bourbaki. Vol. 2009/2010. Exposés 1012–1026. MR 2906349
- Carl Friedrich Gauss, Disquisitiones arithmeticae, Yale University Press, New Haven, Conn.-London, 1966. Translated into English by Arthur A. Clarke, S. J. MR 197380
- F. Gouvêa and B. Mazur, Families of modular eigenforms, Math. Comp. 58 (1992), no. 198, 793–805. MR 1122070, DOI 10.1090/S0025-5718-1992-1122070-1
- Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225–320. MR 833192, DOI 10.1007/BF01388809
- E. Hecke, Analytische funktionen und algebraische zahlen, Abh. Math. Sem. Univ. Hamburg 3 (1924), no. 1, 213–236 (German). MR 3069428, DOI 10.1007/BF02954625
- E. Hecke, Zur Theorie der elliptischen Modulfunktionen, Math. Ann. 97 (1927), no. 1, 210–242 (German). MR 1512360, DOI 10.1007/BF01447866
- Haruzo Hida, Galois representations into $\textrm {GL}_2(\textbf {Z}_p[[X]])$ attached to ordinary cusp forms, Invent. Math. 85 (1986), no. 3, 545–613. MR 848685, DOI 10.1007/BF01390329
- Haruzo Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 231–273. MR 868300, DOI 10.24033/asens.1507
- 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
- Tomio Kubota and Heinrich-Wolfgang Leopoldt, Eine $p$-adische Theorie der Zetawerte. I. Einführung der $p$-adischen Dirichletschen $L$-Funktionen, J. Reine Angew. Math. 214(215) (1964), 328–339 (German). MR 163900
- Helmut Klingen, Über die Werte der Dedekindschen Zetafunktion, Math. Ann. 145 (1961/62), 265–272 (German). MR 133304, DOI 10.1007/BF01451369
- Nicholas M. Katz and Barry Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108, Princeton University Press, Princeton, NJ, 1985. MR 772569, DOI 10.1515/9781400881710
- E. E. Kummer, Über eine allgemeine Eigenschaft der rationalen Entwickelungscoefficienten einer bestimmten Gattung analytischer Functionen, J. Reine Angew. Math. 41 (1851), 368–372 (German). MR 1578727, DOI 10.1515/crll.1851.41.368
- Alan G. B. Lauder, Computations with classical and $p$-adic modular forms, LMS J. Comput. Math. 14 (2011), 214–231. MR 2831231, DOI 10.1112/S1461157011000155
- Emmanuel Lecouturier, On the Galois structure of the class group of certain Kummer extensions, J. Lond. Math. Soc. (2) 98 (2018), no. 1, 35–58. MR 3847231, DOI 10.1112/jlms.12123
- Joseph Lehner, Further congruence properties of the Fourier coefficients of the modular invariant $j(\tau )$, Amer. J. Math. 71 (1949), 373–386. MR 27802, DOI 10.2307/2372252
- David Loeffler, Spectral expansions of overconvergent modular functions, Int. Math. Res. Not. IMRN 16 (2007), Art. ID rnm050, 17. MR 2353090, DOI 10.1093/imrn/rnm050
- D. Loeffler, Lecture notes for TCC course “Modular Curves", https://warwick.ac.uk/fac/sci/maths/people/staff/david_loeffler/teaching/modularcurves/, 2014.
- A. Lauder and J. Vonk, Computing $p$-adic L-functions of totally real fields, Preprint, 2019.
- Ruochuan Liu, Daqing Wan, and Liang Xiao, The eigencurve over the boundary of weight space, Duke Math. J. 166 (2017), no. 9, 1739–1787. MR 3662443, DOI 10.1215/00127094-0000012X
- B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33–186 (1978). With an appendix by Mazur and M. Rapoport. MR 488287, DOI 10.1007/BF02684339
- Barry Mazur, How can we construct abelian Galois extensions of basic number fields?, Bull. Amer. Math. Soc. (N.S.) 48 (2011), no. 2, 155–209. MR 2774089, DOI 10.1090/S0273-0979-2011-01326-X
- Loïc Merel, L’accouplement de Weil entre le sous-groupe de Shimura et le sous-groupe cuspidal de $J_0(p)$, J. Reine Angew. Math. 477 (1996), 71–115 (French). MR 1405312, DOI 10.1515/crll.1996.477.71
- Barry Mazur, William Stein, and John Tate, Computation of $p$-adic heights and log convergence, Doc. Math. Extra Vol. (2006), 577–614. MR 2290599
- J. Newton and J. Thorne, Symmetric power functoriality for holomorphic modular forms, ArXiv preprint (2019).
- Vincent Pilloni, Overconvergent modular forms, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 1, 219–239 (French, with English and French summaries). MR 3097946, DOI 10.5802/aif.2759
- S. Ramanujan, On certain arithmetical functions, Trans. Cambridge Phil. Soc., 22 (1916), 159–184.
- Xavier-François Roblot, Computing $p$-adic $L$-functions of totally real number fields, Math. Comp. 84 (2015), no. 292, 831–874. MR 3290966, DOI 10.1090/S0025-5718-2014-02889-5
- David Roe, The 3-adic eigencurve at the boundary of weight space, Int. J. Number Theory 10 (2014), no. 7, 1791–1806. MR 3256852, DOI 10.1142/S1793042114500560
- W. A. Stein et al., Sage Mathematics Software (Version 9.0), The Sage Development Team, 2020. http://www.sagemath.org.
- Jean-Pierre Serre, Endomorphismes complètement continus des espaces de Banach $p$-adiques, Inst. Hautes Études Sci. Publ. Math. 12 (1962), 69–85 (French). MR 144186, DOI 10.1007/BF02684276
- Jean-Pierre Serre, Formes modulaires et fonctions zêta $p$-adiques, 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. 191–268 (French). MR 404145
- Goro Shimura, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 (1966), 209–220. MR 188198, DOI 10.1515/crll.1966.221.209
- Takuro Shintani, On evaluation of zeta functions of totally real algebraic number fields at non-positive integers, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 2, 393–417. MR 427231
- Goro Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), no. 3, 637–679. MR 507462
- Carl Ludwig Siegel, Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1969 (1969), 87–102 (German). MR 252349
- K. Slavov, Gross–stark units for totally real number fields, Master’s thesis, Harvard University, 2007.
- 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
- John Tate, A review of non-Archimedean elliptic functions, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 162–184. MR 1363501
- Richard Taylor and Andrew Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), no. 3, 553–572. MR 1333036, DOI 10.2307/2118560
- Jan Vonk, Computing overconvergent forms for small primes, LMS J. Comput. Math. 18 (2015), no. 1, 250–257. MR 3349318, DOI 10.1112/S1461157015000042
- K. G. C. Staudt, Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffen, J. Reine Angew. Math. 21 (1840), 372–374 (German). MR 1578267, DOI 10.1515/crll.1840.21.372
- Daqing Wan, Dimension variation of classical and $p$-adic modular forms, Invent. Math. 133 (1998), no. 2, 449–463. MR 1632794, DOI 10.1007/s002220050251
- Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575, DOI 10.1007/978-1-4612-1934-7
- J. R. Wilton, Congruence Properties of Ramanujan’s Function tau(n), Proc. London Math. Soc. (2) 31 (1930), no. 1, 1–10. MR 1577449, DOI 10.1112/plms/s2-31.1.1
- 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
- Preston Wake and Carl Wang-Erickson, The rank of Mazur’s Eisenstein ideal, Duke Math. J. 169 (2020), no. 1, 31–115. MR 4047548, DOI 10.1215/00127094-2019-0039
Bibliographic Information
- Jan Vonk
- Affiliation: Institute for Advanced Study, Princeton, New Jersey 08540
- MR Author ID: 858428
- ORCID: 0000-0002-7775-8843
- Email: vonk@ias.edu
- Received by editor(s): March 12, 2020
- Published electronically: August 19, 2020
- Additional Notes: The author was supported by NSF Grant No. DMS-1638352.
- © Copyright 2020 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 58 (2021), 313-356
- MSC (2010): Primary 11F33, 11G18, 11S40
- DOI: https://doi.org/10.1090/bull/1700
- MathSciNet review: 4273104