Abelian surfaces over totally real fields are potentially modular

George Boxer; Frank Calegari; Toby Gee; Vincent Pilloni

doi:10.1007/s10240-021-00128-2

Article

Abelian surfaces over totally real fields are potentially modular

George Boxer ¹; Frank Calegari ¹; Toby Gee ¹; Vincent Pilloni ¹

¹

Publications Mathématiques de l'IHÉS, Volume 134 (2021), pp. 153-501

Abstract

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces $A$ over $𝐐$ with ${End}_{𝐂} A = 𝐙$ . We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

Received: 2018-12-25
Accepted: 2021-10-27
Online First: 2021-11-29
Published online: 2021-12-07

MR Zbl

DOI: 10.1007/s10240-021-00128-2

Author's affiliations:

George Boxer ¹; Frank Calegari ¹; Toby Gee ¹; Vincent Pilloni ¹

¹

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     author = {George Boxer and Frank Calegari and Toby Gee and Vincent Pilloni},
     title = {Abelian surfaces over totally real fields are potentially modular},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {153--501},
     year = {2021},
     publisher = {Springer International Publishing},
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George Boxer; Frank Calegari; Toby Gee; Vincent Pilloni. Abelian surfaces over totally real fields are potentially modular. Publications Mathématiques de l'IHÉS, Volume 134 (2021), pp. 153-501. doi: 10.1007/s10240-021-00128-2

References
Cited by

[AC89.] J. Arthur; L. Clozel Simple algebras, base change, and the advanced theory of the trace formula, 120, Princeton University Press, Princeton, 1989 | Zbl | MR

[ACC+18.] P. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, B. Le Hung, J. Newton, P. Scholze, R. Taylor and J. Thorne, Potential automorphy over CM fields, preprint, 2018. | MR

[AGM20.] A. Ash; P. E. Gunnells; M. McConnell Cohomology with twisted one-dimensional coefficients for congruence subgroups of ${SL}_{4} (ℤ)$ and Galois representations, J. Algebra, Volume 553 (2020), pp. 211-247 | MR | Zbl | DOI

[AIP15.] F. Andreatta; A. Iovita; V. Pilloni $p$ -adic families of Siegel modular cuspforms, Ann. Math. (2), Volume 181 (2015), pp. 623-697 | MR | Zbl | DOI

[All16.] P. B. Allen Deformations of polarized automorphic Galois representations and adjoint Selmer groups, Duke Math. J., Volume 165 (2016), pp. 2407-2460 | MR | Zbl | DOI

[AMRT10.] A. Ash; D. Mumford; M. Rapoport; Y.-S. Tai Smooth compactifications of locally symmetric varieties, Cambridge University Press, Cambridge, 2010 (With the collaboration of Peter Scholze) | Zbl | MR | DOI

[Art04.] J. Arthur Automorphic representations of $GSp (4)$ , Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, 2004, pp. 65-81 | MR | Zbl

[Art13.] J. Arthur The endoscopic classification of representations, 61, Am. Math. Soc., Providence, 2013 (Orthogonal and symplectic groups) | Zbl | MR

[AS08.] A. Ash and G. Stevens, $p$ -adic deformations of arithmetic cohomology, preprint, 2008.

[AT09.] E. Artin; J. Tate Class field theory, AMS Chelsea Publishing, Providence, 2009 (Reprinted with corrections from the 1967 original) | Zbl | MR

[B0̈0.] G. Böckle Demuškin groups with group actions and applications to deformations of Galois representations, Compos. Math., Volume 121 (2000), pp. 109-154 | Zbl | DOI | MR

[Bak46.] H. F. Baker A Locus with 25920 Linear Self-Transformations, 39, Cambridge University Press/The Macmillan Company, Cambridge/New York, 1946 | Zbl | MR

[Bal12.] S. Balaji, G-valued potentially semi-stable deformation rings, ProQuest LLC, Ann Arbor, MI, 2012, Thesis (Ph.D.)–The University of Chicago. | MR

[BC11.] J. Bellaïche; G. Chenevier The sign of Galois representations attached to automorphic forms for unitary groups, Compos. Math., Volume 147 (2011), pp. 1337-1352 | MR | Zbl | DOI

[BCDT01.] C. Breuil; B. Conrad; F. Diamond; R. Taylor On the modularity of elliptic curves over $𝐐$ : wild $3$ -adic exercises, J. Am. Math. Soc., Volume 14 (2001), pp. 843-939 (electronic) | MR | Zbl | DOI

[BCGP21.] G. Boxer, F. Calegari, T. Gee and V. Pilloni, Auxiliary magma files, 2021, https://github.com/fcale75/abeliansurfacesmagmafiles.

[BCP97.] W. Bosma; J. Cannon; C. Playoust The Magma algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997), pp. 235-265 Computational algebra and number theory (London, 1993) | MR | Zbl | DOI

[BDPcS15.] T. Berger; L. Dembélé; A. Pacetti; M. H. Şengün Theta lifts of Bianchi modular forms and applications to paramodularity, J. Lond. Math. Soc. (2), Volume 92 (2015), pp. 353-370 | MR | Zbl | DOI

[BG08.] S. Baba; H. Granath Genus 2 curves with quaternionic multiplication, Can. J. Math., Volume 60 (2008), pp. 734-757 | MR | Zbl | DOI

[BG14.] K. Buzzard; T. Gee The conjectural connections between automorphic representations and Galois representations, Automorphic forms and Galois representations. Vol. 1, 414, Cambridge University Press, Cambridge, 2014, pp. 135-187 | Zbl | MR | DOI

[BG19.] R. Bellovin; T. Gee $G$ -valued local deformation rings and global lifts, Algebra Number Theory, Volume 13 (2019), pp. 333-378 | MR | Zbl | DOI

[BHR94.] D. Blasius; M. Harris; D. Ramakrishnan Coherent cohomology, limits of discrete series, and Galois conjugation, Duke Math. J., Volume 73 (1994), pp. 647-685 | MR | Zbl | DOI

[BJ15.] G. Böckle; A.-K. Juschka Irreducibility of versal deformation rings in the $(p, p)$ -case for 2-dimensional representations, J. Algebra, Volume 444 (2015), pp. 81-123 | MR | Zbl | DOI

[BK14.] A. Brumer; K. Kramer Paramodular abelian varieties of odd conductor, Trans. Am. Math. Soc., Volume 366 (2014), pp. 2463-2516 | MR | Zbl | DOI

[BK19.] A. Brumer; K. Kramer Corrigendum to “Paramodular abelian varieties of odd conductor”, Trans. Am. Math. Soc., Volume 372 (2019), pp. 2251-2254 | MR | Zbl | DOI

[BK20.] T. Berger; K. Klosin Deformations of Saito-Kurokawa type and the paramodular conjecture, Am. J. Math., Volume 142 (2020), pp. 1821-1875 (With and appendix by Chris Poor, Jerry Shurman, and David S. Yuen) | MR | Zbl | DOI

[BL04.] C. Birkenhake; H. Lange Complex abelian varieties, 302, Springer, Berlin, 2004 | Zbl | MR | DOI

[Bla06.] D. Blasius Hilbert modular forms and the Ramanujan conjecture, Noncommutative geometry and number theory, E37, Vieweg, Wiesbaden, 2006, pp. 35-56 | Zbl | DOI | MR

[BLGG13.] T. Barnet-Lamb; T. Gee; D. Geraghty Congruences between Hilbert modular forms: constructing ordinary lifts, II, Math. Res. Lett., Volume 20 (2013), pp. 67-72 | MR | Zbl | DOI

[BLGGT14a.] T. Barnet-Lamb; T. Gee; D. Geraghty; R. Taylor Local-global compatibility for $l = p$ , II, Ann. Sci. Éc. Norm. Supér. (4), Volume 47 (2014), pp. 165-179 | MR | Zbl | DOI | Numdam

[BLGGT14b.] T. Barnet-Lamb; T. Gee; D. Geraghty; R. Taylor Potential automorphy and change of weight, Ann. Math. (2), Volume 179 (2014), pp. 501-609 | MR | Zbl | DOI

[BLGHT11.] T. Barnet-Lamb; D. Geraghty; M. Harris; R. Taylor A family of Calabi-Yau varieties and potential automorphy II, Publ. Res. Inst. Math. Sci., Volume 47 (2011), pp. 29-98 | MR | Zbl | DOI

[BN18.] N. Bruin; B. Nasserden Arithmetic aspects of the Burkhardt quartic threefold, J. Lond. Math. Soc. (2), Volume 98 (2018), pp. 536-556 | MR | Zbl | DOI

[Bol07.] M. Bolognesi On Weddle surfaces and their moduli, Adv. Geom., Volume 7 (2007), pp. 113-144 | MR | Zbl | DOI

[Box15.] G. Boxer, Torsion in the coherent cohomology of Shimura varieties and Galois representations, preprint, 2015. | MR

[BPP+19.] A. Brumer; A. Pacetti; C. Poor; G. Tornaría; J. Voight; D. S. Yuen On the paramodularity of typical abelian surfaces, Algebra Number Theory, Volume 13 (2019), pp. 1145-1195 | MR | Zbl | DOI

[BPS16.] S. Bijakowski; V. Pilloni; B. Stroh Classicité de formes modulaires surconvergentes, Ann. Math. (2), Volume 183 (2016), pp. 975-1014 | MR | Zbl | DOI

[BPY16.] J. Breeding; C. Poor; D. S. Yuen Computations of spaces of paramodular forms of general level, J. Korean Math. Soc., Volume 53 (2016), pp. 645-689 | MR | Zbl | DOI

[BR16.] R. Brasca and G. Rosso, Eigenvarieties for non-cuspidal modular forms over certain PEL Shimura varieties, 2016.

[Bra47.] R. Brauer On Artin’s $L$ -series with general group characters, Ann. Math. (2), Volume 48 (1947), pp. 502-514 | MR | Zbl | DOI

[BT99.] K. Buzzard; R. Taylor Companion forms and weight one forms, Ann. Math. (2), Volume 149 (1999), pp. 905-919 | MR | Zbl | DOI

[Bur91.] H. Burkhardt Untersuchungen aus dem Gebiete der hyperelliptischen Modulfunctionen, Math. Ann., Volume 38 (1891), pp. 161-224 | MR | Zbl | DOI

[Buz03.] K. Buzzard Analytic continuation of overconvergent eigenforms, J. Am. Math. Soc., Volume 16 (2003), pp. 29-55 | MR | Zbl | DOI

[Cai14.] B. Cais On the $U_{p}$ operator in characteristic $p$ , Math. Res. Lett., Volume 21 (2014), pp. 255-260 | MR | Zbl | DOI

[Cal12.] F. Calegari Even Galois Representations and the Fontaine-Mazur conjecture II, J. Am. Math. Soc., Volume 25 (2012), pp. 533-554 | MR | Zbl | DOI

[Cal18.] F. Calegari Non-minimal modularity lifting in weight one, J. Reine Angew. Math., Volume 740 (2018), pp. 41-62 | MR | Zbl | DOI

[Car12.] A. Caraiani Local-global compatibility and the action of monodromy on nearby cycles, Duke Math. J., Volume 161 (2012), pp. 2311-2413 | MR | Zbl | DOI

[Car14.] A. Caraiani Monodromy and local-global compatibility for $l = p$ , Algebra Number Theory, Volume 8 (2014), pp. 1597-1646 | MR | Zbl | DOI

[Cas.] W. Casselman, Introduction to admissible representations of $p$ -adic groups, Available at https://www.math.ubc.ca/~cass/research/publications.html.

[CC20.] F. Calegari and S. Chidambaram, Rationality of twists of the Siegel modular variety of genus 2 and level 3, 2020, | arXiv | MR

[CCG20.] F. Calegari; S. Chidambaram; A. Ghitza Some modular abelian surfaces, Math. Comput., Volume 89 (2020), pp. 387-394 | MR | Zbl | DOI

[CCN+85.] J. H. Conway; R. T. Curtis; S. P. Norton; R. A. Parker; R. A. Wilson Atlas of finite groups, Oxford University Press, Eynsham, 1985 (Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray) | Zbl | MR

[CDT99.] B. Conrad; F. Diamond; R. Taylor Modularity of certain potentially Barsotti-Tate Galois representations, J. Am. Math. Soc., Volume 12 (1999), pp. 521-567 | MR | Zbl | DOI

[CG13.] F. Calegari; T. Gee Irreducibility of automorphic Galois representations of $G L (n)$ , $n$ at most 5, Ann. Inst. Fourier (Grenoble), Volume 63 (2013), pp. 1881-1912 | MR | Zbl | DOI | Numdam

[CG18.] F. Calegari; D. Geraghty Modularity lifting beyond the Taylor–Wiles method, Invent. Math., Volume 211 (2018), pp. 297-433 | MR | Zbl | DOI

[CG20.] F. Calegari; D. Geraghty Minimal modularity lifting for nonregular symplectic representations, Duke Math. J., Volume 169 (2020), pp. 801-896 (With an appendix by Calegari, Geraghty and Michael Harris) | MR | Zbl | DOI

[Che04.] G. Chenevier Familles $p$ -adiques de formes automorphes pour ${GL}_{n}$ , J. Reine Angew. Math., Volume 570 (2004), pp. 143-217 | MR | Zbl

[Chi90.] W. C. Chi On the $l$ -adic representations attached to some absolutely simple abelian varieties of type $II$ , J. Fac. Sci., Univ. Tokyo, Sect. 1A, Math., Volume 37 (1990), pp. 467-484 | MR | Zbl

[Chi92.] W. C. Chi $l$ -adic and $λ$ -adic representations associated to abelian varieties defined over number fields, Am. J. Math., Volume 114 (1992), pp. 315-353 | MR | Zbl | DOI

[CHT08.] L. Clozel; M. Harris; R. Taylor Automorphy for some $l$ -adic lifts of automorphic mod $l$ Galois representations, Publ. Math. IHES, Volume 108 (2008), pp. 1-181 | MR | Zbl | DOI | Numdam

[CL10.] P.-H. Chaudouard; G. Laumon Le lemme fondamental pondéré. I. Constructions géométriques, Compos. Math., Volume 146 (2010), pp. 1416-1506 | MR | Zbl | DOI

[Clo90.] L. Clozel Motifs et formes automorphes: applications du principe de fonctorialité, Automorphic forms, Shimura varieties and $L$ -functions, Vol. I, Volume 10 (1990), pp. 77-159 | MR

[Cob06.] A. B. Coble An invariant condition for certain automorphic algebraic forms, Am. J. Math., Volume 28 (1906), pp. 333-366 | MR | Zbl | DOI

[Col97.] R. F. Coleman P-adic Banach spaces and families of modular forms, Invent. Math., Volume 127 (1997), pp. 417-479 | MR | Zbl | DOI

[Cre84.] J. E. Cremona Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, Compos. Math., Volume 51 (1984), pp. 275-324 | MR | Zbl | Numdam

[Cre92.] J. E. Cremona Abelian varieties with extra twist, cusp forms, and elliptic curves over imaginary quadratic fields, J. Lond. Math. Soc. (2), Volume 45 (1992), pp. 404-416 | MR | Zbl | DOI

[DDT97.] H. Darmon; F. Diamond; R. Taylor Fermat’s last theorem, Elliptic curves, modular forms & Fermat’s last theorem (1997), pp. 2-140 | MR

[Del73.] P. Deligne Les constantes des équations fonctionnelles des fonctions $L$ , Modular functions of one variable, II, Volume 349 (1973), pp. 501-597 | DOI | MR

[Del79.] P. Deligne Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, Automorphic forms, representations and $L$ -functions, Volume XXXIII (1979), pp. 247-289 | DOI | MR

[Dia97.] F. Diamond The Taylor–Wiles construction and multiplicity one, Invent. Math., Volume 128 (1997), pp. 379-391 | MR | Zbl | DOI

[dJ93.] A. J. de Jong The moduli spaces of principally polarized abelian varieties with $Γ_{0} (p)$ -level structure, J. Algebraic Geom., Volume 2 (1993), pp. 667-688 | MR | Zbl

[DK00.] W. Duke; E. Kowalski A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math., Volume 139 (2000), pp. 1-39 (With an appendix by Dinakar Ramakrishnan) | MR | Zbl | DOI

[DK16.] L. Dembélé; A. Kumar Examples of abelian surfaces with everywhere good reduction, Math. Ann., Volume 364 (2016), pp. 1365-1392 | MR | Zbl | DOI

[DK17.] F. Diamond; P. L. Kassaei Minimal weights of Hilbert modular forms in characteristic $p$ , Compos. Math., Volume 153 (2017), pp. 1769-1778 | MR | Zbl | DOI

[DL08.] I. Dolgachev; D. Lehavi On isogenous principally polarized abelian surfaces, Curves and abelian varieties, 465, Am. Math. Soc., Providence, 2008, pp. 51-69 | Zbl | MR | DOI

[DR73.] P. Deligne; M. Rapoport Les schémas de modules de courbes elliptiques, Modular functions of one variable, II, Volume 349 (1973), pp. 143-316 | MR | Zbl | DOI

[Edi92.] B. Edixhoven The weight in Serre’s conjectures on modular forms, Invent. Math., Volume 109 (1992), pp. 563-594 | MR | Zbl | DOI

[EG14.] M. Emerton; T. Gee A geometric perspective on the Breuil-Mézard conjecture, J. Inst. Math. Jussieu, Volume 13 (2014), pp. 183-223 | MR | Zbl | DOI

[Ell05.] J. S. Ellenberg Serre’s conjecture over $𝔽_{9}$ , Ann. Math. (2), Volume 161 (2005), pp. 1111-1142 | MR | Zbl | DOI

[ERX17.] M. Emerton; D. Reduzzi; L. Xiao Unramifiedness of Galois representations arising from Hilbert modular surfaces, Forum Math. Sigma, Volume 5 (2017) | MR | Zbl | DOI

[Fal83.] G. Faltings Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math., Volume 73 (1983), pp. 349-366 | MR | Zbl | DOI

[Far08.] L. Fargues L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld et applications cohomologiques, L’isomorphisme entre les tours de Lubin-Tate et de Drinfeld, 262, Birkhäuser, Basel, 2008, pp. 1-325 | Zbl | DOI | MR

[Far10.] L. Fargues La filtration de Harder-Narasimhan des schémas en groupes finis et plats, J. Reine Angew. Math., Volume 645 (2010), pp. 1-39 | MR | Zbl | DOI

[Far11.] L. Fargues La filtration canonique des points de torsion des groupes $p$ -divisibles, Ann. Sci. Éc. Norm. Supér. (4), Volume 44 (2011), pp. 905-961 (With collaboration of Yichao Tian) | MR | Zbl | Numdam | DOI

[FC90.] G. Faltings; C.-L. Chai Degeneration of abelian varieties, 22, Springer, Berlin, 1990 (With an appendix by David Mumford) | Zbl | MR | DOI

[FKRS12.] F. Fité; K. S. Kedlaya; V. Rotger; A. V. Sutherland Sato-Tate distributions and Galois endomorphism modules in genus 2, Compos. Math., Volume 148 (2012), pp. 1390-1442 | MR | Zbl | DOI

[Fon94.] J.-M. Fontaine Représentations $l$ -adiques potentiellement semi-stables, Astérisque, Volume 223 (1994), pp. 321-347 Périodes $p$ -adiques (Bures-sur-Yvette, 1988) | Zbl | MR | Numdam

[FP21.] N. Fakhruddin and V. Pilloni, Hecke operators and the coherent cohomology of Shimura varieties, J. Inst. Math. Jussieu (2021), 1–69. | MR

[Fre83.] E. Freitag Siegelsche Modulfunktionen, 254, Springer, Berlin, 1983 | Zbl | MR | DOI

[Ger19.] D. Geraghty Modularity lifting theorems for ordinary Galois representations, Math. Ann., Volume 373 (2019), pp. 1341-1427 | MR | Zbl | DOI

[GG12.] T. Gee; D. Geraghty Companion forms for unitary and symplectic groups, Duke Math. J., Volume 161 (2012), pp. 247-303 | MR | Zbl | DOI

[GHLS17.] T. Gee; F. Herzig; T. Liu; D. Savitt Potentially crystalline lifts of certain prescribed types, Doc. Math., Volume 22 (2017), pp. 397-422 | MR | Zbl | DOI

[GJ72.] R. Godement; H. Jacquet Zeta functions of simple algebras, 260, Springer, Berlin, 1972 | Zbl | MR | DOI

[GK19.] W. Goldring; J.-S. Koskivirta Strata Hasse invariants, Hecke algebras and Galois representations, Invent. Math., Volume 217 (2019), pp. 887-984 | MR | Zbl | DOI

[GN20.] T. Gee and J. Newton, Patching and the completed homology of locally symmetric spaces, J. Inst. Math. Jussieu (2020), 1–64. | MR

[GPSR87.] S. Gelbart; I. Piatetski-Shapiro; S. Rallis Explicit constructions of automorphic $L$ -functions, 1254, Springer, Berlin, 1987 | Zbl | MR | DOI

[Gro90.] B. H. Gross A tameness criterion for Galois representations associated to modular forms (mod $p$ ), Duke Math. J., Volume 61 (1990), pp. 445-517 | MR | Zbl | DOI

[Gro16.] B. H. Gross On the Langlands correspondence for symplectic motives, Izv. Ross. Akad. Nauk Ser. Mat., Volume 80 (2016), pp. 49-64 | MR | Zbl

[GRR72.] A. Grothendieck; M. Raynaud; D. S. Rim Groupes de monodromie en géométrie algébrique. I, 288, Springer, Berlin, 1972 Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I), Dirigé par A. Grothendieck. Avec la collaboration de Michèle Raynaud et Dock S. Rim | Zbl

[GT05.] A. Genestier; J. Tilouine Systèmes de Taylor-Wiles pour ${GSp}_{4}$ , Astérisque, Volume 302 (2005), pp. 177-290 (Formes automorphes. II. Le cas du groupe ${GSp}_{4}$ ) | Zbl | MR | Numdam

[GT11a.] W. T. Gan; S. Takeda The local Langlands conjecture for $G S p (4)$ , Ann. Math. (2), Volume 173 (2011), pp. 1841-1882 | MR | Zbl | DOI

[GT11b.] W. T. Gan; S. Takeda Theta correspondences for $GSp (4)$ , Represent. Theory, Volume 15 (2011), pp. 670-718 | MR | Zbl | DOI

[GT19.] T. Gee; O. Taïbi Arthur’s multiplicity formula for ${𝐆𝐒𝐩}_{4}$ and restriction to ${𝐒𝐩}_{4}$ , J. Éc. Polytech. Math., Volume 6 (2019), pp. 469-535 | MR | Zbl | DOI

[Har66.] R. Hartshorne Residues and duality, Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64, 20, Springer, Berlin, 1966 (With an appendix by Pierre Deligne) | Zbl | MR

[Har90a.] M. Harris Automorphic forms and the cohomology of vector bundles on Shimura varieties, Automorphic forms, Shimura varieties, and $L$ -functions, Vol. II, Volume 11 (1990), pp. 41-91 | MR | Zbl

[Har90b.] M. Harris Automorphic forms of $\bar{\partial}$ -cohomology type as coherent cohomology classes, J. Differ. Geom., Volume 32 (1990), pp. 1-63 | MR | Zbl

[Hec20.] E. Hecke Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Math. Z., Volume 6 (1920), pp. 11-51 | MR | Zbl | DOI

[Hen09.] G. Henniart Sur la fonctorialité, pour $GL (4)$ , donnée par le carré extérieur, Mosc. Math. J., Volume 9 (2009), pp. 33-45 (back matter) | MR | Zbl | DOI

[Hid04.] H. Hida $p$ -adic automorphic forms on Shimura varieties, Springer, New York, 2004 | Zbl | MR | DOI

[Hid09.] H. Hida Irreducibility of the Igusa tower, Acta Math. Sin. Engl. Ser., Volume 25 (2009), pp. 1-20 | MR | Zbl | DOI

[HKP10.] T. J. Haines; R. E. Kottwitz; A. Prasad Iwahori-Hecke algebras, J. Ramanujan Math. Soc., Volume 25 (2010), pp. 113-145 | MR | Zbl

[HLTT16.] M. Harris; K.-W. Lan; R. Taylor; J. Thorne On the rigid cohomology of certain Shimura varieties, Res. Math. Sci., Volume 3 (2016), p. 37 | MR | Zbl | DOI

[HS02.] K. Hulek; G. K. Sankaran The geometry of Siegel modular varieties, Higher dimensional birational geometry, Volume 35 (2002), pp. 89-156 | Zbl | DOI | MR

[HT01.] M. Harris; R. Taylor The geometry and cohomology of some simple Shimura varieties, 151, Princeton University Press, Princeton, 2001 (With an appendix by Vladimir G. Berkovich) | Zbl | MR

[Hub96.] R. Huber Étale cohomology of rigid analytic varieties and adic spaces, E30, Friedr. Vieweg & Sohn, Braunschweig, 1996 | Zbl | MR | DOI

[Hun96.] B. Hunt The geometry of some special arithmetic quotients, 1637, Springer, Berlin, 1996 | Zbl | DOI | MR

[HZ01.] M. Harris and S. Zucker, Boundary cohomology of Shimura varieties. III. Coherent cohomology on higher-rank boundary strata and applications to Hodge theory, Mém. Soc. Math. Fr. (N.S.) (2001) vi+116. | MR | Numdam

[Igu64.] J. Igusa On Siegel modular forms genus two. II, Am. J. Math., Volume 86 (1964), pp. 392-412 | MR | Zbl | DOI

[JLR12.] J. Johnson-Leung; B. Roberts Siegel modular forms of degree two attached to Hilbert modular forms, J. Number Theory, Volume 132 (2012), pp. 543-564 | MR | Zbl | DOI

[Joc82.] N. Jochnowitz Congruences between systems of eigenvalues of modular forms, Trans. Am. Math. Soc., Volume 270 (1982), pp. 269-285 | MR | Zbl | DOI

[Joh17.] C. Johansson On the Sato-Tate conjecture for non-generic abelian surfaces, Trans. Am. Math. Soc., Volume 369 (2017), pp. 6303-6325 (With an appendix by Francesc Fité) | MR | Zbl | DOI

[Jon11.] O. T. R. Jones An analogue of the BGG resolution for locally analytic principal series, J. Number Theory, Volume 131 (2011), pp. 1616-1640 | MR | Zbl | DOI

[JS81.] H. Jacquet; J. A. Shalika On Euler products and the classification of automorphic forms. II, Am. J. Math., Volume 103 (1981), pp. 777-815 | MR | Zbl | DOI

[Kas06.] P. L. Kassaei A gluing lemma and overconvergent modular forms, Duke Math. J., Volume 132 (2006), pp. 509-529 | MR | Zbl | DOI

[Kas16.] P. L. Kassaei Analytic continuation of overconvergent Hilbert modular forms, Astérisque, Volume 382 (2016), pp. 1-48 | MR | Zbl

[Kem78.] G. Kempf The Grothendieck–Cousin complex of an induced representation, Adv. Math., Volume 29 (1978), pp. 310-396 | MR | Zbl | DOI

[Kim03.] H. H. Kim Functoriality for the exterior square of ${GL}_{4}$ and the symmetric fourth of ${GL}_{2}$ , J. Am. Math. Soc., Volume 16 (2003), pp. 139-183 (With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak) | DOI

[Kis08.] M. Kisin Potentially semi-stable deformation rings, J. Am. Math. Soc., Volume 21 (2008), pp. 513-546 | MR | Zbl | DOI

[Kis09.] M. Kisin Moduli of finite flat group schemes, and modularity, Ann. Math. (2), Volume 170 (2009), pp. 1085-1180 | MR | Zbl | DOI

[Kle94.] S. L. Kleiman The standard conjectures, Motives, Volume 55 (1994), pp. 3-20 | DOI

[Kot92.] R. E. Kottwitz Points on some Shimura varieties over finite fields, J. Am. Math. Soc., Volume 5 (1992), pp. 373-444 | MR | Zbl | DOI

[KR99.] S. S. Kudla; M. Rapoport Arithmetic Hirzebruch-Zagier cycles, J. Reine Angew. Math., Volume 515 (1999), pp. 155-244 | MR | Zbl | DOI

[KST14.] P. L. Kassaei; S. Sasaki; Y. Tian Modularity lifting results in parallel weight one and applications to the Artin conjecture: the tamely ramified case, Forum Math. Sigma, Volume 2 (2014) | MR | Zbl | DOI

[KT17.] C. B. Khare; J. A. Thorne Potential automorphy and the Leopoldt conjecture, Am. J. Math., Volume 139 (2017), pp. 1205-1273 | MR | Zbl | DOI

[Lan80.] R. P. Langlands Base change for $GL (2)$ , 96, Princeton University Press/University of Tokyo Press, Princeton/Tokyo, 1980

[Lan13.] K.-W. Lan Arithmetic compactifications of PEL-type Shimura varieties, 36, Princeton University Press, Princeton, 2013 | Zbl | DOI

[Lan16.] K.-W. Lan Compactifications of PEL-type Shimura varieties in ramified characteristics, Forum Math. Sigma, Volume 4 (2016) | MR | Zbl | DOI

[Lan17.] K.-W. Lan Integral models of toroidal compactifications with projective cone decompositions, Int. Math. Res. Not., Volume 11 (2017), pp. 3237-3280 | MR | Zbl

[LPS13.] G. Lombardo; C. Peters; M. Schütt Abelian fourfolds of Weil type and certain K3 double planes, Rend. Semin. Mat. Univ. Politec. Torino, Volume 71 (2013), pp. 339-383 | MR | Zbl

[Maz89.] B. Mazur Deforming Galois representations, Galois groups over $𝐐$ , Volume 16 (1989), pp. 385-437 | DOI

[MB89.] L. Moret-Bailly Groupes de Picard et problèmes de Skolem II, Ann. Sci. Éc. Norm. Supér., Volume 22 (1989), pp. 181-194 | MR | Zbl | DOI | Numdam

[MB90.] L. Moret-Bailly Extensions de corps globaux à ramification et groupe de Galois donnés, C. R. Acad. Sci., Sér. 1 Math., Volume 311 (1990), pp. 273-276 | MR | Zbl

[Mes72.] W. Messing The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, 264, Springer, Berlin, 1972 | Zbl | DOI

[Mil79.] J. S. Milne Points on Shimura varieties mod $p$ , Automorphic forms, representations and $L$ -functions, Volume XXXIII (1979), pp. 165-184 | DOI

[Mok14.] C. P. Mok Galois representations attached to automorphic forms on ${GL}_{2}$ over CM fields, Compos. Math., Volume 150 (2014), pp. 523-567 | MR | Zbl | DOI

[Mor84.] D. R. Morrison On $K 3$ surfaces with large Picard number, Invent. Math., Volume 75 (1984), pp. 105-121 | MR | Zbl | DOI

[Mor85.] D. R. Morrison The Kuga-Satake variety of an abelian surface, J. Algebra, Volume 92 (1985), pp. 454-476 | MR | Zbl | DOI

[MT02.] A. Mokrane; J. Tilouine Cohomology of Siegel varieties with $p$ -adic integral coefficients and applications, Astérisque, Volume 280 (2002), pp. 1-95 (Cohomology of Siegel varieties) | MR | Zbl | Numdam

[MT15.] C. P. Mok; F. Tan Overconvergent families of Siegel-Hilbert modular forms, Can. J. Math., Volume 67 (2015), pp. 893-922 | MR | Zbl | DOI

[Mum08.] D. Mumford Abelian varieties, 5, Hindustan Book Agency, New Delhi, 2008 With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition | Zbl

[MW16a.] C. Mœglin; J.-L. Waldspurger Stabilisation de la formule des traces tordue, 1, Springer, Berlin, 2016 | Zbl | DOI

[MW16b.] C. Mœglin; J.-L. Waldspurger Stabilisation de la formule des traces tordue, 2, Springer, Berlin, 2016 | Zbl | DOI

[MZ95.] B. J. J. Moonen; Y. G. Zarhin Hodge classes and Tate classes on simple abelian fourfolds, Duke Math. J., Volume 77 (1995), pp. 553-581 | MR | Zbl | DOI

[Nak84.] S. Nakajima On Galois module structure of the cohomology groups of an algebraic variety, Invent. Math., Volume 75 (1984), pp. 1-8 | MR | Zbl | DOI

[NG02.] B. C. Ngô; A. Genestier Alcôves et $p$ -rang des variétés abéliennes, Ann. Inst. Fourier (Grenoble), Volume 52 (2002), pp. 1665-1680 | MR | Zbl | DOI | Numdam

[Noo13.] R. Noot The system of representations of the Weil-Deligne group associated to an abelian variety, Algebra Number Theory, Volume 7 (2013), pp. 243-281 | MR | Zbl | DOI

[Noo17.] R. Noot The $p$ -adic representation of the Weil-Deligne group associated to an abelian variety, J. Number Theory, Volume 172 (2017), pp. 301-320 | MR | Zbl | DOI

[Pat19.] S. Patrikis Variations on a theorem of Tate, Mem. Am. Math. Soc., Volume 258 (2019), p. viii+156 | MR | Zbl

[Pil11.] V. Pilloni Prolongement analytique sur les variétés de Siegel, Duke Math. J., Volume 157 (2011), pp. 167-222 | MR | Zbl | DOI

[Pil12.] V. Pilloni Modularité, formes de Siegel et surfaces abéliennes, J. Reine Angew. Math., Volume 666 (2012), pp. 35-82 | MR | Zbl

[Pil17.] V. Pilloni Formes modulaires $p$ -adiques de Hilbert de poids 1, Invent. Math., Volume 208 (2017), pp. 633-676 | MR | Zbl | DOI

[Pil20.] V. Pilloni Higher coherent cohomology and $p$ -adic modular forms of singular weights, Duke Math. J., Volume 169 (2020), pp. 1647-1807 | MR | Zbl | DOI

[Pin90.] R. Pink Arithmetical compactification of mixed Shimura varieties, 209, Universität Bonn, Mathematisches Institut, Bonn, 1990 (Dissertation, Rheinische Friedrich-Wilhelms-Universität, Bonn, Bonn, 1989) | Zbl

[Poi08.] J. Poineau Un résultat de connexité pour les variétés analytiques $p$ -adiques: privilège et noethérianité, Compos. Math., Volume 144 (2008), pp. 107-133 | MR | Zbl | DOI

[PS09.] A. Pitale; R. Schmidt Bessel models for lowest weight representations of $GSp (4, 𝐑)$ , Int. Math. Res. Not., Volume 7 (2009), pp. 1159-1212 | MR | Zbl

[PS16a.] V. Pilloni; B. Stroh Cohomologie cohérente et représentations Galoisiennes, Ann. Math. Qué., Volume 40 (2016), pp. 167-202 | MR | Zbl | DOI

[PS16b.] V. Pilloni; B. Stroh Surconvergence, ramification et modularité, Astérisque, Volume 382 (2016), pp. 195-266 | Zbl

[PSY17.] C. Poor; J. Shurman; D. S. Yuen Siegel paramodular forms of weight 2 and squarefree level, Int. J. Number Theory, Volume 13 (2017), pp. 2627-2652 | MR | Zbl | DOI

[PT15.] S. Patrikis; R. Taylor Automorphy and irreducibility of some $l$ -adic representations, Compos. Math., Volume 151 (2015), pp. 207-229 | MR | Zbl | DOI

[PVZ16.] S. Patrikis; J. F. Voloch; Y. G. Zarhin Anabelian geometry and descent obstructions on moduli spaces, Algebra Number Theory, Volume 10 (2016), pp. 1191-1219 | MR | Zbl | DOI

[PY15.] C. Poor; D. S. Yuen Paramodular cusp forms, Math. Comput., Volume 84 (2015), pp. 1401-1438 | MR | Zbl | DOI

[Ram00.] D. Ramakrishnan Modularity of the Rankin-Selberg $L$ -series, and multiplicity one for $SL (2)$ , Ann. Math. (2), Volume 152 (2000), pp. 45-111 | MR | Zbl | DOI

[Ray94.] M. Raynaud 1-motifs et monodromie géométrique, Astérisque, Volume 223 (1994), pp. 295-319 Périodes $p$ -adiques (Bures-sur-Yvette, 1988) | Zbl | Numdam

[Rib04.] K. A. Ribet Abelian varieties over $𝐐$ and modular forms, Modular curves and abelian varieties, 224, Birkhäuser, Basel, 2004, pp. 241-261 | Zbl | DOI

[Rie59.] B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie (1859), 671–680.

[Rob01.] B. Roberts Global $L$ -packets for $GSp (2)$ and theta lifts, Doc. Math., Volume 6 (2001), pp. 247-314 (electronic) | MR | Zbl | DOI

[RS07a.] D. Ramakrishnan; F. Shahidi Siegel modular forms of genus 2 attached to elliptic curves, Math. Res. Lett., Volume 14 (2007), pp. 315-332 | MR | Zbl | DOI

[RS07b.] B. Roberts; R. Schmidt Local newforms for GSp(4), 1918, Springer, Berlin, 2007 | Zbl | DOI

[RS07c.] B. Roberts and R. Schmidt, Tables for representations of GSp(4), 2007, Available at http://www2.math.ou.edu/~rschmidt/papers/tables.pdf.

[RZ96.] M. Rapoport; T. Zink Period spaces for $p$ -divisible groups, 141, Princeton University Press, Princeton, 1996 | Zbl | DOI

[Sai97.] T. Saito Modular forms and $p$ -adic Hodge theory, Invent. Math., Volume 129 (1997), pp. 607-620 | MR | Zbl | DOI

[Sas13.] S. Sasaki On Artin representations and nearly ordinary Hecke algebras over totally real fields, Doc. Math., Volume 18 (2013), pp. 997-1038 | MR | Zbl | DOI

[Saw16.] W. F. Sawin Ordinary primes for Abelian surfaces, C. R. Math. Acad. Sci. Paris, Volume 354 (2016), pp. 566-568 | MR | Zbl | DOI

[SBT97.] N. I. Shepherd-Barron; R. Taylor $\mod 2$ and $\mod 5$ icosahedral representations, J. Am. Math. Soc., Volume 10 (1997), pp. 283-298 | Zbl | DOI

[Sch15.] P. Scholze On torsion in the cohomology of locally symmetric varieties, Ann. Math. (2), Volume 182 (2015), pp. 945-1066 | MR | Zbl | DOI

[Sch17.] R. Schmidt Archimedean aspects of Siegel modular forms of degree 2, Rocky Mt. J. Math., Volume 47 (2017), pp. 2381-2422 | MR | Zbl | DOI

[Sch18.] R. Schmidt Packet structure and paramodular forms, Trans. Am. Math. Soc., Volume 370 (2018), pp. 3085-3112 | MR | Zbl | DOI

[Sch19.] C. Schembri Examples of genuine QM abelian surfaces which are modular, Res. Number Theory, Volume 5 (2019), p. 12 | MR | Zbl | DOI

[Sch20.] R. Schmidt Paramodular forms in CAP representations of $GSp (4)$ , Acta Arith., Volume 194 (2020), pp. 319-340 | MR | Zbl | DOI

[SD73.] H. P. F. Swinnerton-Dyer On $l$ -adic representations and congruences for coefficients of modular forms, Modular functions of one variable, III, Volume 350 (1973), pp. 1-55 | DOI | Zbl

[Ser68.] J.-P. Serre Abelian $l$ -adic representations and elliptic curves, W. A. Benjamin, Inc., New York-Amsterdam, 1968 | Zbl

[Ser70.] J.-P. Serre, Facteurs locaux des fonctions zêta des variétés algébriques (définitions et conjectures), Séminaire Delange-Pisot-Poitou. Théorie des nombres, 11 (1969–1970), 1–15 (fre).

[Ser73.] J.-P. Serre Formes modulaires et fonctions zêta $p$ -adiques, Modular functions of one variable, III, Volume 350 (1973), pp. 191-268 | DOI | Zbl

[Ser77.] J.-P. Serre Modular forms of weight one and Galois representations, Algebraic number fields: $L$ -functions and Galois properties (1977), pp. 193-268 | Zbl

[Ser89.] J.-P. Serre Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15, Friedr. Vieweg & Sohn, Braunschweig, 1989 (Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt) | Zbl

[Ser00.] J.-P. Serre Lettre à Marie-France Vignéras du 10/02/1987. Collected papers IV, Springer, Berlin, 2000, pp. 38-55

[Sha97.] F. Shahidi On non-vanishing of twisted symmetric and exterior square $L$ -functions for $GL (n)$ , Pac. J. Math., Volume 181 (1997), pp. 311-322 (Olga Taussky-Todd: in memoriam) | MR | DOI | Zbl

[Sho18.] J. Shotton The Breuil-Mézard conjecture when $l \neq p$ , Duke Math. J., Volume 167 (2018), pp. 603-678 | MR | Zbl | DOI

[Ski09.] C. Skinner A note on the $p$ -adic Galois representations attached to Hilbert modular forms, Doc. Math., Volume 14 (2009), pp. 241-258 | MR | Zbl | DOI

[Sno09.] A. Snowden, On two dimensional weight two odd representations of totally real fields, 2009, | arXiv

[Sor10.] C. M. Sorensen Galois representations attached to Hilbert-Siegel modular forms, Doc. Math., Volume 15 (2010), pp. 623-670 | MR | Zbl | DOI

[ST93.] P. J. Sally; M. Tadić Induced representations and classifications for $GSp (2, F)$ and $Sp (2, F)$ , Mém. Soc. Math. Fr. (N.S.), Volume 52 (1993), pp. 75-133 | MR | Zbl | Numdam

[Sta13.] The Stacks Project Authors, Stacks Project, 2013, http://stacks.math.columbia.edu.

[Ste04.] W. A. Stein Shafarevich–Tate groups of nonsquare order, Modular curves and abelian varieties, 224, Birkhäuser, Basel, 2004, pp. 277-289 | Zbl | DOI

[Str15.] B. Stroh Mauvaise réduction au bord, Astérisque, Volume 370 (2015), pp. 269-304 | Zbl

[Tat79.] J. Tate Number theoretic background, Automorphic forms, representations and $L$ -functions, Volume XXXIII (1979), pp. 3-26 | DOI

[Tay02.] R. Taylor Remarks on a conjecture of Fontaine and Mazur, J. Inst. Math. Jussieu, Volume 1 (2002), pp. 125-143 | MR | Zbl | DOI

[Tay03.] R. Taylor On icosahedral Artin representations. II, Am. J. Math., Volume 125 (2003), pp. 549-566 | MR | Zbl | DOI

[Tay08.] R. Taylor Automorphy for some $l$ -adic lifts of automorphic mod $l$ Galois representations. II, Publ. Math. IHES, Volume 108 (2008), pp. 183-239 | MR | Zbl | DOI | Numdam

[Tho12.] J. A. Thorne On the automorphy of $l$ -adic Galois representations with small residual image, J. Inst. Math. Jussieu, Volume 11 (2012), pp. 855-920 (With an appendix by Robert Guralnick, Florian Herzig, Richard Taylor and Thorne) | MR | Zbl | DOI

[Tho15.] J. A. Thorne Automorphy lifting for residually reducible $l$ -adic Galois representations, J. Am. Math. Soc., Volume 28 (2015), pp. 785-870 | MR | Zbl | DOI

[Til96.] J. Tilouine, Deformations of Galois representations and Hecke algebras, Published for The Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad (1996).

[Til98.] J. Tilouine Deformations of Siegel-Hilbert Hecke eigensystems and their Galois representations, Number theory, Volume 210 (1998), pp. 195-225 | DOI | Zbl

[Til06a.] J. Tilouine Nearly ordinary rank four Galois representations and $p$ -adic Siegel modular forms, Compos. Math., Volume 142 (2006), pp. 1122-1156 (With an appendix by Don Blasius) | MR | Zbl | DOI

[Til06b.] J. Tilouine, Siegel varieties and $p$ -adic Siegel modular forms, Doc. Math. (2006), 781–817.

[Til09.] J. Tilouine Cohomologie des variétés de Siegel et représentations galoisiennes associées aux représentations cuspidales cohomologiques de ${GSp}_{4} (𝐐)$ , Algèbre et théorie des nombres. Années 2007–2009, Lab. Math. Besançon, Besançon, 2009, pp. 99-114 | Zbl

[Til12.] J. Tilouine Formes compagnons et complexe BGG dual pour $G S p_{4}$ , Ann. Inst. Fourier (Grenoble), Volume 62 (2012), pp. 1383-1436 | MR | Zbl | DOI | Numdam

[TU95.] J. Tilouine; É. Urban Familles $p$ -adiques à trois variables de formes de Siegel et de représentations galoisiennes, C. R. Acad. Sci. Paris, Sér. 1 Math., Volume 321 (1995), pp. 5-10 | Zbl

[TU99.] J. Tilouine; É. Urban Several-variable $p$ -adic families of Siegel-Hilbert cusp eigensystems and their Galois representations, Ann. Sci. Éc. Norm. Supér. (4), Volume 32 (1999), pp. 499-574 | MR | Zbl | Numdam | DOI

[Tun81.] J. Tunnell Artin’s conjecture for representations of octahedral type, Bull. Am. Math. Soc. (N.S.), Volume 5 (1981), pp. 173-175 | MR | Zbl | DOI

[TW95.] R. Taylor; A. Wiles Ring-theoretic properties of certain Hecke algebras, Ann. Math. (2), Volume 141 (1995), pp. 553-572 | MR | Zbl | DOI

[TY07.] R. Taylor; T. Yoshida Compatibility of local and global Langlands correspondences, J. Am. Math. Soc., Volume 20 (2007), pp. 467-493 (electronic) | MR | Zbl | DOI

[Urb11.] É. Urban Eigenvarieties for reductive groups, Ann. Math. (2), Volume 174 (2011), pp. 1685-1784 | MR | Zbl | DOI

[Vig98.] M.-F. Vignéras Induced r-representations of p-adic reductive groups, Sel. Math., Volume 4 (1998), pp. 549-623 | MR | Zbl | DOI

[Vig05.] M.-F. Vignéras Pro- $p$ -Iwahori Hecke ring and supersingular ${\bar{𝐅}}_{p}$ -representations, Math. Ann., Volume 331 (2005), pp. 523-556 | MR | Zbl | DOI

[VW13.] E. Viehmann; T. Wedhorn Ekedahl-Oort and Newton strata for Shimura varieties of PEL type, Math. Ann., Volume 356 (2013), pp. 1493-1550 | MR | Zbl | DOI

[Wal84.] N. R. Wallach On the constant term of a square integrable automorphic form, Operator algebras and group representations, Vol. II, Volume 18 (1984), pp. 227-237 | MR

[Wei52.] A. Weil Number-theory and algebraic geometry, Proceedings of the International Congress of Mathematicians, Volume 2 (1952), pp. 90-100 | MR

[Wie14.] G. Wiese On Galois representations of weight one, Doc. Math., Volume 19 (2014), pp. 689-707 | MR | Zbl | DOI

[Wil95.] A. Wiles Modular elliptic curves and Fermat’s last theorem, Ann. Math. (2), Volume 141 (1995), pp. 443-551 | MR | Zbl | DOI

[Yos80.] H. Yoshida Siegel’s modular forms and the arithmetic of quadratic forms, Invent. Math., Volume 60 (1980), pp. 193-248 | MR | Zbl | DOI

[Yos84.] H. Yoshida On Siegel modular forms obtained from theta series, J. Reine Angew. Math., Volume 352 (1984), pp. 184-219 | MR | Zbl

[Yu08.] C.-F. Yu Irreducibility and $p$ -adic monodromies on the Siegel moduli spaces, Adv. Math., Volume 218 (2008), pp. 1253-1285 | MR | Zbl | DOI

[Yu11.] C.-F. Yu Geometry of the Siegel modular threefold with paramodular level structure, Proc. Am. Math. Soc., Volume 139 (2011), pp. 3181-3190 | MR | Zbl | DOI

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