# ANNALES DE L'INSTITUT FOURIER

Nearly overconvergent Siegel modular forms
Annales de l'Institut Fourier, Volume 69 (2019) no. 6, p. 2439-2506

We introduce a sheaf-theoretic formulation of Shimura’s theory of nearly holomorphic Siegel modular forms and differential operators. We use it to define and study nearly overconvergent Siegel modular forms and their $p$-adic families.

Nous introduisons une formulation faisceau-théorique de la théorie de Shimura des formes modulaires de Siegel quasi holomorphes et des opérateurs différentiels. Nous l’utilisons pour définir et étudier les formes modulaires de Siegel quasi surconvergentes et leurs familles p-adiques

Revised : 2018-10-11
Accepted : 2018-11-06
Published online : 2019-10-29
DOI : https://doi.org/10.5802/aif.3299
Classification:  11F46,  11F33,  11F60,  14J15
Keywords: nearly holomorphic Siegel modular forms, nearly overconvergent Siegel modular forms, differential operators, overconvergent families
@article{AIF_2019__69_6_2439_0,
author = {Liu, Zheng},
title = {Nearly overconvergent Siegel modular forms},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {69},
number = {6},
year = {2019},
pages = {2439-2506},
doi = {10.5802/aif.3299},
language = {en},
url={aif.centre-mersenne.org/item/AIF_2019__69_6_2439_0/}
}

Liu, Zheng. Nearly overconvergent Siegel modular forms. Annales de l'Institut Fourier, Volume 69 (2019) no. 6, pp. 2439-2506. doi : 10.5802/aif.3299. https://aif.centre-mersenne.org/item/AIF_2019__69_6_2439_0/

[1] Andreatta, Fabrizio; Iovita, Adrian Triple product $p$-adic $L$-functions associated to finite slope $p$-adic families of modular forms (2017) (with an appendix by Eric Urban, https://arxiv.org/abs/1708.02785)

[2] Andreatta, Fabrizio; Iovita, Adrian; Pilloni, Vincent $p$-adic families of Siegel modular cuspforms, Ann. Math., Tome 181 (2015) no. 2, pp. 623-697 | Article | MR 3275848 | Zbl 1394.11045

[3] Bernshtein, Iosef N.; Gelʼfand, Israil M.; Gelʼfand, Sergei I. Structure of representations that are generated by vectors of highest weight, Funkts. Anal. Prilozh., Tome 5 (1971) no. 1, pp. 1-9 | Article | MR 291204

[4] Bertolini, Massimo; Darmon, Henri; Prasanna, Kartik Generalized Heegner cycles and $p$-adic Rankin $L$-series, Duke Math. J., Tome 162 (2013) no. 6, pp. 1033-1148 (With an appendix by Brian Conrad) | Article | MR 3053566 | Zbl 1302.11043

[5] Bijakowski, Stéphane; Pilloni, Vincent; Stroh, Benoît Classicité de formes modulaires surconvergentes, Ann. Math., Tome 183 (2016) no. 3, pp. 975-1014 | Article | Zbl 1407.11078

[6] Böcherer, Siegfried Über die Fourier-Jacobi-Entwicklung Siegelscher Eisensteinreihen II, Math. Z., Tome 189 (1985) no. 1, pp. 81-110 | Article | Zbl 0558.10022

[7] Böcherer, Siegfried; Schmidt, Claus-Günther $p$-adic measures attached to Siegel modular forms, Ann. Inst. Fourier, Tome 50 (2000) no. 5, pp. 1375-1443 | Article | MR 1800123 | Zbl 0962.11023

[8] Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold Non-Archimedean analysis. A systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften, Tome 261, Springer, 1984, xii+436 pages | Zbl 0539.14017

[9] Buzzard, Kevin Eigenvarieties, $L$-functions and Galois representations (London Mathematical Society Lecture Note Series) Tome 320, Cambridge University Press, 2007, pp. 59-120 | Article | MR 2392353 | Zbl 1230.11054

[10] Coleman, Robert F. $p$-adic Banach spaces and families of modular forms, Invent. Math., Tome 127 (1997) no. 3, pp. 417-479 | Article | MR 1431135 | Zbl 0918.11026

[11] Courtieu, Michel; Panchishkin, Alexei Non-Archimedean $L$-functions and arithmetical Siegel modular forms, Lecture Notes in Mathematics, Tome 1471, Springer, 2004, viii+196 pages | MR 2034949 | Zbl 1070.11023

[12] Eischen, Ellen $p$-adic differential operators on automorphic forms on unitary groups, Ann. Inst. Fourier, Tome 62 (2012) no. 1, pp. 177-243 | Article | MR 2986270 | Zbl 1257.11054

[13] Eischen, Ellen; Fintzen, Jessica; Mantovan, Elena; Varma, Ila Differential operators and families of automorphic forms on unitary groups of arbitrary signature, Doc. Math., Tome 23 (2018), pp. 445-495 | MR 3846052 | Zbl 1408.14089

[14] Eischen, Ellen; Harris, Michael; Li, Jianshu; Skinner, Christopher $p$-adic $L$-functions for unitary groups, part II: zeta-integral calculations (2016) (https://arxiv.org/abs/1602.01776)

[15] Eischen, Ellen; Wan, Xin $p$-adic Eisenstein series and $L$-functions of certain cusp forms on definite unitary groups, J. Inst. Math. Jussieu, Tome 15 (2016) no. 3, pp. 471-510 | Article | MR 3505656 | Zbl 1404.11051

[16] Faltings, Gerd; Chai, Ching-Li Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Tome 22, Springer, 1990, xii+316 pages (With an appendix by David Mumford) | MR 1083353 | Zbl 0744.14031

[17] Fargues, Laurent La filtration canonique des points de torsion des groupes $p$-divisibles, Ann. Sci. Éc. Norm. Supér., Tome 44 (2011) no. 6, pp. 905-961 (With collaboration of Yichao Tian) | Article | MR 2919687 | Zbl 1331.14044

[18] Fresnel, Jean; van der Put, Marius Rigid analytic geometry and its applications, Progress in Mathematics, Tome 218, Birkhäuser, 2004, xii+296 pages | MR 2014891 | Zbl 1096.14014

[19] Harris, Michael Arithmetic vector bundles and automorphic forms on Shimura varieties. II, Compos. Math., Tome 60 (1986) no. 3, pp. 323-378 | MR 869106 | Zbl 0612.14019

[20] Harris, Michael $L$-functions and periods of polarized regular motives, J. Reine Angew. Math., Tome 483 (1997), pp. 75-161 | MR 1431843 | Zbl 0859.11032

[21] Harris, Michael Cohomological automorphic forms on unitary groups. II. Period relations and values of $L$-functions, Harmonic analysis, group representations, automorphic forms and invariant theory (Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore) Tome 12, World Scientific, 2007, pp. 89-149 | Article | MR 2401812 | Zbl 1390.11088

[22] Harris, Michael A simple proof of rationality of Siegel-Weil Eisenstein series, Eisenstein series and applications (Progress in Mathematics) Tome 258, Birkhäuser, 2008, pp. 149-185 | Article | MR 2402683 | Zbl 1225.11069

[23] Harron, Robert; Xiao, Liang Gauss-Manin connections for $p$-adic families of nearly overconvergent modular forms, Ann. Inst. Fourier, Tome 64 (2014) no. 6, pp. 2449-2464 | Article | MR 3331170 | Zbl 1310.11054

[24] Hida, Haruzo A $p$-adic measure attached to the zeta functions associated with two elliptic modular forms. II, Ann. Inst. Fourier, Tome 38 (1988) no. 3, pp. 1-83 | Article | MR 976685 | Zbl 0645.10028

[25] Hida, Haruzo Elementary theory of $L$-functions and Eisenstein series, London Mathematical Society Student Texts, Tome 26, Cambridge University Press, 1993 | MR 1216135 | Zbl 0942.11024

[26] Hida, Haruzo $p$-adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer, 2004, xii+390 pages | MR 2055355 | Zbl 1055.11032

[27] Ibukiyama, Tomoyoshi On differential operators on automorphic forms and invariant pluri-harmonic polynomials, Comment. Math. Univ. St. Pauli, Tome 48 (1999) no. 1, pp. 103-118 | MR 1684769 | Zbl 1007.11023

[28] Ichikawa, Tomoshi Integrality of nearly (holomorphic) Siegel modular forms (2015) (https://arxiv.org/abs/1508.03138)

[29] Jakobsen, Hans Plesner; Vergne, Michèle Restrictions and expansions of holomorphic representations, J. Funct. Anal., Tome 34 (1979) no. 1, pp. 29-53 | Article | MR 551108 | Zbl 0433.22011

[30] Katz, Nicholas M. $p$-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 Mathematics) Tome 350, Springer, 1973, pp. 69-190 | Article | MR 447119 | Zbl 0271.10033

[31] Katz, Nicholas M. Travaux de Dwork, Séminaire Bourbaki (1971/1972) (Lecture Notes in Mathematics) Tome 317, Springer, 1973, pp. 167-200 | Article | MR 498577 | Zbl 0259.14007

[32] Katz, Nicholas M. $p$-adic $L$-functions for CM fields, Invent. Math., Tome 49 (1978) no. 3, pp. 199-297 | Article | MR 513095 | Zbl 0417.12003

[33] Katz, Nicholas M.; Oda, Tadao On the differentiation of de Rham cohomology classes with respect to parameters, J. Math. Kyoto Univ., Tome 8 (1968), pp. 199-213 | Article | MR 237510 | Zbl 0165.54802

[34] Kisin, Mark; Lai, King Fai Overconvergent Hilbert modular forms, Am. J. Math., Tome 127 (2005) no. 4, pp. 735-783 | Article | MR 2154369 | Zbl 1129.11020

[35] Lan, Kai-Wen Toroidal compactifications of PEL-type Kuga families, Algebra Number Theory, Tome 6 (2012) no. 5, pp. 885-966 | MR 2968629

[36] Lan, Kai-Wen Compactifications of PEL-type Shimura varieties and Kuga families with ordinary loci, World Scientific, 2017 | Zbl 1403.14002

[37] Liu, Zheng $p$-adic $L$-functions for ordinary families of symplectic groups (2016) (To appear in J. Inst. Math. Jussieu)

[38] Matsumura, Hideyuki Commutative algebra, Mathematics Lecture Note Series, Tome 56, Benjamin/Cummings Publishing Co., 1980, xv+313 pages | Zbl 0441.13001

[39] Mumford, David Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, Tome 5, Oxford University Press, 1970, viii+242 pages | MR 282985 | Zbl 0223.14022

[40] Nappari, Mark Alan Holomorphic Forms Canonically Attached to Nearly Holomorphic Automorphic forms (1992) (Ph. D. Thesis) | MR 2687759

[41] Pitale, Ameya; Saha, Abhishek; Schmidt, Ralf Lowest weight modules of ${\mathrm{Sp}}_{4}\left(ℝ\right)$ and nearly holomorphic Siegel modular forms (2015) (https://arxiv.org/abs/1501.00524)

[42] Shimura, Goro The special values of the zeta functions associated with cusp forms, Commun. Pure Appl. Math., Tome 29 (1976) no. 6, pp. 783-804 | Article | MR 434962 | Zbl 0348.10015

[43] Shimura, Goro Invariant differential operators on Hermitian symmetric spaces, Ann. Math., Tome 132 (1990) no. 2, pp. 237-272 | Article | MR 1070598 | Zbl 0718.11020

[44] Shimura, Goro Euler products and Eisenstein series, Regional Conference Series in Mathematics, Tome 93, American Mathematical Society, 1997, xx+259 pages | MR 1450866 | Zbl 0906.11020

[45] Shimura, Goro Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, Tome 82, American Mathematical Society, 2000, x+302 pages | MR 1780262 | Zbl 0967.11001

[46] The Stacks Project Authors Stacks Project, https://stacks.math.columbia.edu, 2015

[47] Tilouine, Jacques Companion forms and classicality in the ${\mathbf{GL}}_{2}\left(ℚ\right)$-case, Number theory (Ramanujan Mathematical Society Lecture Notes Series) Tome 15, Ramanujan Mathematical Society, 2011, pp. 119-141 | MR 2905493 | Zbl 1312.11047

[48] Urban, Eric On the rank of Selmer groups for elliptic curves over $ℚ$, Automorphic representations and $L$-functions (Studies in Mathematics. Tata Institute of Fundamental Research) Tome 22, Tata Institute of Fundamental Research, 2013, pp. 651-680 | MR 3156865 | Zbl 1371.11094

[49] Urban, Eric Nearly overconvergent modular forms, Iwasawa theory 2012 (Contributions in Mathematical and Computational Sciences) Tome 7, Springer, 2014, pp. 401-441 | Article | MR 3586822 | Zbl 1328.11052

[50] Wiles, Andrew J. On ordinary $\lambda$-adic representations associated to modular forms, Invent. Math., Tome 94 (1988) no. 3, pp. 529-573 | Article | MR 969243 | Zbl 0664.10013