Mixed modular symbols and the generalized cuspidal $1$-motive
Author:
Emmanuel Lecouturier
Journal:
Trans. Amer. Math. Soc. 374 (2021), 2823-2872
MSC (2020):
Primary 11Fxx
DOI:
https://doi.org/10.1090/tran/8310
Published electronically:
February 8, 2021
MathSciNet review:
4223035
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We define and study the space of mixed modular symbols for a given finite index subgroup $\Gamma$ of $\operatorname {SL}_2(\mathbf {Z})$. This is an extension of the usual space of modular symbols that, in some cases, carries more information about Eisenstein series. We make use of mixed modular symbols to construct some $1$-motives related to the generalized Jacobian of modular curves. In the case $\Gamma = \Gamma _0(p)$ for some prime $p$, we relate our construction to the work of Ehud de Shalit on $p$-adic periods of $X_0(p)$.
- L. Barbieri-Viale, A. Rosenschon, and M. Saito, Deligne’s conjecture on 1-motives, Ann. of Math. (2) 158 (2003), no. 2, 593–633. MR 2018930, DOI https://doi.org/10.4007/annals.2003.158.593
- Luca Barbieri-Viale and Bruno Kahn, On the derived category of 1-motives, Astérisque 381 (2016), xi+254 (English, with English and French summaries). MR 3545132
- Joël Bellaïche and Samit Dasgupta, The $p$-adic $L$-functions of evil Eisenstein series, Compos. Math. 151 (2015), no. 6, 999–1040. MR 3357177, DOI https://doi.org/10.1112/S0010437X1400788X
- Adel Betina and Emmanuel Lecouturier, On the $p$-adic periods of the modular curve $X(\Gamma _0(p)\cap \Gamma (2))$, Trans. Amer. Math. Soc. 371 (2019), no. 1, 413–429. MR 3885149, DOI https://doi.org/10.1090/tran/7236
- 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 https://doi.org/10.1007/s00222-004-0406-z
- Ehud de Shalit, On certain Galois representations related to the modular curve $X_1(p)$, Compositio Math. 95 (1995), no. 1, 69–100. MR 1314697
- Ehud de Shalit, On the $p$-adic periods of $X_0(p)$, Math. Ann. 303 (1995), no. 3, 457–472. MR 1355000, DOI https://doi.org/10.1007/BF01461000
- Pierre Deligne, Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. 44 (1974), 5–77 (French). MR 498552
- V. G. Drinfel′d, Two theorems on modular curves, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 83–84 (Russian). MR 0318157
- Matthew Emerton, Supersingular elliptic curves, theta series and weight two modular forms, J. Amer. Math. Soc. 15 (2002), no. 3, 671–714. MR 1896237, DOI https://doi.org/10.1090/S0894-0347-02-00390-9
- Klaus Haberland, Perioden von Modulformen einer Variabler and Gruppencohomologie. I, II, III, Math. Nachr. 112 (1983), 245–282, 283–295, 297–315 (German). MR 726861, DOI https://doi.org/10.1002/mana.19831120113
- David Harari and Tamás Szamuely, Arithmetic duality theorems for 1-motives, J. Reine Angew. Math. 578 (2005), 93–128. MR 2113891, DOI https://doi.org/10.1515/crll.2005.2005.578.93
- Ian Kiming, Matthias Schütt, and Helena A. Verrill, Lifts of projective congruence groups, J. Lond. Math. Soc. (2) 83 (2011), no. 1, 96–120. MR 2763946, DOI https://doi.org/10.1112/jlms/jdq062
- Daniel S. Kubert and Serge Lang, Modular units, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 244, Springer-Verlag, New York-Berlin, 1981. MR 648603
- Emmanuel Lecouturier. On the Mazur–Tate conjecture for prime conductor and Mazur’s Eisenstein ideal, Preprint.
- Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 19–66 (Russian). MR 0314846
- Mathoverflow user abx, Extended Abel-Jacobi theorem, https://mathoverflow.net/questions/319650/extended-abel-jacobi-theorem.
- 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
- Barry Mazur, An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 243–311. MR 1638481
- L. Merel, Homologie des courbes modulaires affines et paramétrisations modulaires, Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993) Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 110–130 (French). MR 1363498
- Loïc Merel, Universal Fourier expansions of modular forms, On Artin’s conjecture for odd $2$-dimensional representations, Lecture Notes in Math., vol. 1585, Springer, Berlin, 1994, pp. 59–94. MR 1322319, DOI https://doi.org/10.1007/BFb0074110
- 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 https://doi.org/10.1515/crll.1996.477.71
- Loïc Merel, Symboles de Manin et valeurs de fonctions $L$, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, Progr. Math., vol. 270, Birkhäuser Boston, Boston, MA, 2009, pp. 283–309 (French, with English summary). MR 2641193, DOI https://doi.org/10.1007/978-0-8176-4747-6_9
- Vicenţiu Paşol and Alexandru A. Popa, Modular forms and period polynomials, Proc. Lond. Math. Soc. (3) 107 (2013), no. 4, 713–743. MR 3108829, DOI https://doi.org/10.1112/plms/pdt003
- Vicentiu Pasol and Alexandru A. Popa, On the Petersson scalar product of arbitrary modular forms, Proc. Amer. Math. Soc. 142 (2014), no. 3, 753–760. MR 3148510, DOI https://doi.org/10.1090/S0002-9939-2013-11815-0
- Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564
- Glenn Stevens, Arithmetic on modular curves, Progress in Mathematics, vol. 20, Birkhäuser Boston, Inc., Boston, MA, 1982. MR 670070
- Glenn Stevens, The cuspidal group and special values of $L$-functions, Trans. Amer. Math. Soc. 291 (1985), no. 2, 519–550. MR 800251, DOI https://doi.org/10.1090/S0002-9947-1985-0800251-4
- Glenn Stevens, The Eisenstein measure and real quadratic fields, Théorie des nombres (Quebec, PQ, 1987) de Gruyter, Berlin, 1989, pp. 887–927. MR 1024612
- 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 https://doi.org/10.1215/00127094-2019-0039
- Weijia Wang, Intégrales régularisées et fonctions L de formes modulaires via la méthode de Rogers-Zudilin, PhD thesis, ENS Lyon, 2020.
- Lawrence C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1997. MR 1421575
- Takao Yamazaki and Yifan Yang, Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus, Doc. Math. 21 (2016), 1669–1690. MR 3603932
- Don Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 415–437 (1982). MR 656029
Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 11Fxx
Retrieve articles in all journals with MSC (2020): 11Fxx
Additional Information
Emmanuel Lecouturier
Affiliation:
Yau Mathematical Sciences Center and Department of mathematics, Tsinghua University, Beijing, People’s Republic of China
MR Author ID:
969451
Received by editor(s):
July 20, 2019
Received by editor(s) in revised form:
July 6, 2020, July 28, 2020, and August 11, 2020
Published electronically:
February 8, 2021
Additional Notes:
This research was funded by Tsinghua University and the Yau Mathematical Sciences Center
Article copyright:
© Copyright 2021
American Mathematical Society
