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Moduli spaces and modular forms. Abstracts from the workshop held January 31 – February 6, 2021 (hybrid meeting). (English) Zbl 1487.00034

Summary: The relation between moduli spaces and modular forms goes back to the theory of elliptic curves. On the one hand both topics experience their own growth and development, but from time to time new unexpected links show up and usually these lead to progress on both sides. One subject where there has been a lot of progress concerns the moduli of abelian varieties and K3 surfaces and especially on the Kodaira dimension of these spaces. The idea of the workshop was to bring together the experts of the two areas in the hope that discussion, interaction and lectures would spur the development of new ideas. The lectures of the workshop gave ample evidence of the interaction and provided opportunities for further interaction. Besides the lectures participants interacted via zoom in smaller groups.

MSC:

00B05 Collections of abstracts of lectures
00B25 Proceedings of conferences of miscellaneous specific interest
11-06 Proceedings, conferences, collections, etc. pertaining to number theory
11Fxx Discontinuous groups and automorphic forms
14J15 Moduli, classification: analytic theory; relations with modular forms
Full Text: DOI

References:

[1] J. Arthur, The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publ. 61, Amer. Math. Soc. (2013). · Zbl 1310.22014
[2] R. Bacher, Tables de réseaux entiers unimodulaires construits comme k-voisins de Z n , Journal de Th. des Nombres de Bordeaux 9, 479-497 (1997). · Zbl 0906.11033
[3] R. Bacher & B. Venkov, Réseaux entiers unimodulaires sans racine en dimension 27 et 28, in J. Martinet (ed.), Réseaux euclidiens, designs sphériques et formes modulaires, Monogr. Enseign. Math. (in French) 37, L’Enseignement Mathématique, 212-267 (2001). · Zbl 1139.11319
[4] J .H. Conway & N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der Math. Wissenschaften 290, Springer-Verlag, New York (1999). · Zbl 0915.52003
[5] J. Hsia & M. Jöchner, Almost strong approximations for definite quadratic spaces, In-ventiones math 129, 471-487 (1997). · Zbl 0894.11013
[6] O. King, A mass formula for unimodular lattices with no roots, Mathematics of Com-putation 72 (242), 839-863 (2003). · Zbl 1099.11035
[7] Barros, I. Geometry of the moduli space of n-pointed K3 surfaces of genus 11. Bull. Lond. Math. Soc. 50 (2018), no. 6, 1071-1084. · Zbl 1410.14028
[8] Barros, I.; Mullane, S. Two moduli spaces of Calabi-Yau type. Int. Math. Res. Notices (2019)
[9] Beauville, A.; Donagi, R. La variétés des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 703-706. · Zbl 0602.14041
[10] Borcherds, R.; Katzarkov, L.; Pantev, T.; Shepherd-Barron, N. I. Families of K3 surfaces. J. Algebraic Geom. 7 (1998), no. 1, 183-193. · Zbl 0946.14021
[11] Debarre, O.; Voisin, C. Hyper-Kähler fourfolds and Grassmann geometry. J. Reine Angew. Math. 649 (2010), 63-87. · Zbl 1217.14028
[12] Farkas, G.; Verra, A. The universal K3 surface of genus 14 via cubic fourfolds. J. Math. Pures Appl. (9) 111 (2018), 1-20. · Zbl 1458.14053
[13] Farkas, G.; Verra, A. The unirationality of the moduli space of K3 surfaces of degree 42. Math. Ann. (2020)
[14] Gritsenko, V. A.; Hulek, K.; Sankaran, G. K. The Kodaira dimension of the moduli of K3 surfaces. Invent. Math. 169 (2007), no. 3, 519-567. · Zbl 1128.14027
[15] Iliev, A.; Kapustka, G.; Kapustka, M.; Ranestad, K. EPW cubes. J. Reine Angew. Math. 748 (2019), 241-268. · Zbl 1423.14220
[16] Iliev, A.; Ranestad, K. K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds. Trans. Amer. Math. Soc. 353 (2001), no. 4, 1455-1468. · Zbl 0966.14027
[17] Kondō, S. On the Kodaira dimension of the moduli space of K3 surfaces. II. Compositio Math. 116 (1999), no. 2, 111-117. · Zbl 0948.14007
[18] Lehn, C.; Lehn, M.; Sorger, C.; van Straten, D. Twisted cubics on cubic fourfolds. J. Reine Angew. Math. 731 (2017), 87-128. · Zbl 1376.53096
[19] Ma, S.Mukai models and Borcherds products. arXiv:1909.03946
[20] Ma, S. Kodaira dimension of universal holomorphic symplectic varieties. arXiv:2001.04328
[21] Mukai, S. Curves, K3 surfaces and Fano 3-folds of genus ≤ 10. in “Algebraic geometry and commutative algebra”, Vol. I, 357-377, Kinokuniya, 1988. · Zbl 0701.14044
[22] O’Grady, K. G. Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics. Duke Math. J. 134 (2006), no. 1, 99-137. · Zbl 1105.14051
[23] H. Clemens: Double Solids, Adv. in Math. 47 ( 1983) 107-230 · Zbl 0509.14045
[24] M. Dittmann, R. Salvati Manni, N. Scheithauer: Harmonic theta series and the Kodaira dimension of A 6 , preprint arXiv:1909.07062
[25] R. Donagi:The unirationality of A 5 , Annals of Math. 119 (1984), 269-307. · Zbl 0589.14043
[26] E. Freitag: Die Kodairadimension von Korpern automorpher Funktionen. J. Reine Angew. Math. 296 (1977), 162-170. · Zbl 0366.10023
[27] E. Freitag: Holomorphe Differentialformen zu Kongruenzgruppen der Siegelschen Modul-gruppe. Invent. Math., 30(2):181-196, 1975 · Zbl 0314.32017
[28] E. Freitag: Siegelsche Modulfunktionen. Grundlehren der Mathematischen Wissenschaften 254. Springer-Verlag, Berlin, 1983 · Zbl 0498.10016
[29] Mori, S., Mukai, S.: The uniruledness of the moduli space of curves of genus 11. Algebraic geometry (Tokyo/Kyoto, 1982), 334-353, Lecture Notes in Math. 1016, Springer, Berlin, 1983. · Zbl 0557.14015
[30] D. Mumford: On the Kodaira dimension of the Siegel modular variety, Algebraic geometry-open problems (Ravello, 1982), Lecture Notes in Math., vol. 997, Springer, 1983, 348-375. · Zbl 0527.14036
[31] Y.-S. Tai. On the Kodaira dimension of the moduli space of abelian varieties. Invent. Math., 68(3):425-439, 1982. · Zbl 0508.14038
[32] A. Verra: A short proof of the unirationality of A 5 , Indagationes Math. 46 (1984), 339-355. References · Zbl 0553.14010
[33] S. Kudla, Algebraic cycles on Shimura varieties of orthogonal type, Duke Math. J. 86 (1997), no. 1, 39-78. · Zbl 0879.11026
[34] , Remarks on generating series for special cycles, arXiv:1908.08390v1.
[35] , On the subring of special cycles, arXiv:2001.09068v1.
[36] S. Kudla and J. Millson, The theta correspondence and harmonic forms I, Math. Annalen, 274 (1986), 353-378. · Zbl 0594.10020
[37] , The theta correspondence and harmonic forms II, Math. Annalen, 277 (1987), 267-314. · Zbl 0618.10022
[38] , Intersection numbers for cycles in locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several variables, Publ. math. IHES, 71 (1990), 121-172. · Zbl 0722.11026
[39] S. Kudla and S. Rallis, On the Weil-Siegel formula, J. Reiine Angew. Math. 387 (1988), 1-68. · Zbl 0644.10021
[40] E. Arbarello and M. Cornalba, Footnotes to a paper of Beniamino Segre, Mathematische Annalen 256 (1981), 341-362. · Zbl 0454.14023
[41] D. Eisenbud and J. Harris, The Kodaira dimension of the moduli space of curves of genus ≥ 23 Inventiones Math. 90 (1987), 359-387. · Zbl 0631.14023
[42] G. Farkas, The geometry of the moduli space of curves of genus 23, Math. Annalen 318 (2000), 43-65. · Zbl 0992.14006
[43] G. Farkas, Syzygies of curves and the effective cone of Mg , Duke Math. Journal 135 (2006), 53-98. · Zbl 1107.14019
[44] G. Farkas, Koszul divisors on moduli spaces of curves, American Journal of Math. 131 (2009), 819-869. · Zbl 1176.14006
[45] G. Farkas, D. Jensen and S. Payne, The Kodaira dimension of M 22 and M 23 , arXiv:2005.00622.
[46] J. Harris, On the Kodaira dimension of the moduli space of curves II: The even genus case, Inventiones Math. 75 (1984), 437-466. · Zbl 0542.14014
[47] J. Harris and D. Mumford, On the Kodaira dimension of Mg, Inventiones Math. 67 (1982), 23-88. · Zbl 0506.14016
[48] F. Liu, B. Osserman, M. Teixidor and N. Zhang, The strong maximal rank conjecture and moduli of curves, preprint.
[49] F. Severi, Sulla classificazione delle curve algebriche e sul teorema d’esistenza di Riemann, Rendiconti della R. Accad. Naz. Lincei 24 (1915), 877-888. · JFM 45.1375.02
[50] A. Verra, The unirationality of the moduli space of curves of genus ≤ 14, Compositio Mathematica 141 (2005), 1425-1444. · Zbl 1095.14024
[51] G. Codogni, Hyperelliptic Schottky problem and stable modular forms, Doc. Math. 21 (2016), 445-466. · Zbl 1346.14079
[52] G. Codogni, Vertex algebras and Teichmüller modular forms, Arxiv, 2020
[53] G. Codogni and N. I. Shepherd-Barron The non-existence of stable Schottky forms, Compos. Math. 150 (2014), no. 4, 679-690 · Zbl 1304.14037
[54] G. Farkas, Birational aspects of the geometry of Mg, Surveys in differential geometry. Vol. XIV. Geometry of Riemann surfaces and their moduli spaces, Surv. Differ. Geom., vol. 14, Int. Press, Somerville, MA, 2009, pp. 57-110 · Zbl 1215.14024
[55] E. Freitag, Stabile Modulformen, Math. Ann. 230 (1977), no. 3, 197-211. · Zbl 0359.10026
[56] A. N. Schellekens, Meromorphic c=24 conformal field theories, Comm. Math. Phys., 153 (1):159-185, 1993 · Zbl 0782.17014
[57] M. R. Gaberdiel, C. A. Keller, and R. Volpato, Genus two partition functions of chiral conformal field theories, Commun.Num.Theor.Phys (2010), no. 4, 295-364 · Zbl 1218.81092
[58] M. R. Gaberdiel and R. Volpato, Higher genus partition functions of meromorphic conformal field theories, J. High Energy Phys. (2009), no. 6, 048, 43.
[59] G. Mason and M. P. Tuite, Torus chiral n-point functions for free boson and lattice vertex operator algebras, Comm. Math. Phys. 235 (2003), no. 1, 47-68 · Zbl 1020.17020
[60] J. van Ekeren,C. H. Lam, S. Möller and H. Shimakura, Schellekens’ List and the Very Strange Formula, Arxiv 2021 · Zbl 1492.17027
[61] Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), no. 1, 237-302 · Zbl 0854.17034
[62] J.H. Bruinier and B. Howard, Arithmetic volumes of unitary Shimura varieties. In prepa-ration.
[63] B. Howard, Arithmetic volumes of unitary Shimura curves. Preprint.
[64] S. Kudla, M. Rapoport, T. Yang, Modular forms and Special Cycles on Shimura Curves. Annals of Mathematics Studies, Princeton University Press, 2006. · Zbl 1157.11027
[65] U. Kühn, Generalized arithmetic intersection numbers. J. Reine Angew. Math., 534:209-236, 2001. · Zbl 1084.14028
[66] W. L. Baily, A. Borel, Compactification of arithmetic quotients of bounded symmetric do-mains. Ann. of Math. (2), 84 (1966), 442-528. · Zbl 0154.08602
[67] J. H. Bruinier, S. Ehlen, E. Freitag, Lattices with many Borcherds products. Math. Comp. 85 (2016), 1953-1981. · Zbl 1404.11042
[68] O. V. Shvartsman, E. B. Vinberg, A criterion of smoothness at infinity for an arithmetic quotient of the future tube. Funkts. Anal. Prilohzen. 51:1 (2017), 40-59; · Zbl 1371.32016
[69] English transl.: Functional Anal. Appl., 51:1 (2017), 32-47. · Zbl 1371.32016
[70] E. B. Vinberg, On the algebra of Siegel modular forms of genus 2. Trudy Moskov. Mat. Obshch. 74:1, 1-16; English transl.: Trans. Moscow Math. Soc. 74 (2013), 1-13. · Zbl 1302.05005
[71] H. Wang, The classification of 2-reflective modular forms. arXiv: 1906.10459.
[72] H. Wang, The classification of free algebras of orthogonal modular forms. arXiv: 2006.02291.
[73] K. Wirthmüller, Root systems and Jacobi forms. Compos. Math. 82 (1992) 293-354. · Zbl 0780.17006
[74] H. Wang, B. Williams, On some free algebras of orthogonal modular forms. Adv. Math. 373 (2020), 107332. · Zbl 1464.11052
[75] H. Wang, B. Williams, Projective spaces as orthogonal modular varieties. arXiv: 2008.08392.
[76] H. Wang, B. Williams, Simple lattices and free algebras of modular forms. arXiv: 2009.13343. References
[77] J. H. Bruinier and M. Westerholt-Raum, Kudla’s modularity conjecture and formal Fourier-Jacobi series, Forum Math. Pi 3 (2015), e7, 30 pages. · Zbl 1376.11032
[78] G. Faltings and C.-L. Chai, Degenerations of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3rd series, vol. 22, Springer-Verlag, Berlin-Heidelberg, 1990. · Zbl 0744.14031
[79] J. Kramer, An arithmetic theory of Jacobi forms in higher dimensions, J. Reine Angew. Math. 458 (1995), 157-182. · Zbl 0814.11031
[80] J. Kramer, On formal Fourier-Jacobi expansions revisited, Preprint 2020.
[81] S. S. Kudla, Algebraic cycles on Shimura varieties of orthogonal type, Duke Math. J. 86 (1997), 39-78. · Zbl 0879.11026
[82] W. Zhang, Modularity of generating functions of special cycles on Shimura varieties, PhD Thesis, Columbia University, 2009.
[83] Alfes, Claudia;
[84] Griffin, Michael; Ono, Ken; Rolen, Larry. Weierstrass mock modular forms and elliptic curves. Res. Number Theory 1 (2015), 31 pp. · Zbl 1388.11019
[85] Alfes-Neumann, Claudia; Schwagenscheidt, Markus. On a theta lift related to the Shintani lift. Adv. Math. 328 (2018), 858-889. · Zbl 1448.11090
[86] Bruinier, Jan. Harmonic Maass forms and periods. Math. Ann. 357 (2013), no. 4, 1363-1387. · Zbl 1328.11050
[87] Bruinier, Jan Hendrik;
[88] Funke, Jens. On two geometric theta lifts. Duke Math. J. 125 (2004), no. 1, 45-90. · Zbl 1088.11030
[89] Bruinier, Jan Hendrik;
[90] Funke, Jens. Traces of CM values of modular functions. J. Reine Angew. Math. 594 (2006), 1-33. · Zbl 1104.11021
[91] Bruinier, Jan; Ono, Ken. Heegner divisors, L-functions and harmonic weak Maass forms. Ann. of Math. (2) 172 (2010), no. 3, 2135-2181. · Zbl 1244.11046
[92] Gross, Benedict ; Kohnen, Winfried ; Zagier, Don. Heegner points and derivatives of L-series. II. Math. Ann. 278 (1987), no. 1-4, 497-562. · Zbl 0641.14013
[93] Kohnen, Winfried.; Zagier, Don. Values of L-series of modular forms at the center of the critical strip. Invent. Math. 64 (1981), no. 2, 175-198. · Zbl 0468.10015
[94] Waldspurger, Jean-Loup. Sur les coefficients de Fourier des formes modulaires de poids demi-entier. (French) [On the Fourier coefficients of modular forms of half-integral weight] J. Math. Pures Appl. (9) 60 (1981), no. 4, 375-484 · Zbl 0431.10015
[95] Zagier, Don. Traces of singular moduli. Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), 211-244, Int. Press Lect. Ser., 3, I, Int. Press, Somerville, MA, 2002. References · Zbl 1048.11035
[96] K. Bringmann, J. Kaszian, and A. Milas, Higher depth quantum modular forms, multiple Eichler integrals, and sl 3 false theta functions Research in the Mathematical Sciences 6:20 (2019). · Zbl 1419.11077
[97] K. Bringmann, J. Kaszian, A. Milas, and C. Nazaroglu, Integral representations of rank two false theta functions and their modularity properties, submitted for publication. · Zbl 1482.11069
[98] K. Bringmann, J. Kaszian, A. Milas, and S. Zwegers, Rank two false theta functions and Ja-cobi forms of negative definite matrix index, Advances in Applied Mathematics 112 (2020), 101946. · Zbl 1468.11111
[99] K. Bringmann and A. Milas, W-algebras, higher rank false theta functions and quantum dimensions, Selecta Mathematica 23 (2017), 1249-1278. · Zbl 1405.17052
[100] K. Bringmann and C. Nazaroglu, A framework for modular properties of false theta func-tions, Research in the Mathematical Sciences 6:30 (2019). · Zbl 1456.11074
[101] K. Bringmann, L. Rolen, and S. Zwegers, On the Fourier coefficients of negative index meromorphic Jacobi forms, Research in the Mathematical Sciences 3 (2016), 1-9. · Zbl 1405.11046
[102] M. Eichler and D. Zagier, The theory of Jacobi forms 3, Progress in Mathematics 55 (1985), Birkhäuser Boston, Inc., Boston MA. · Zbl 0554.10018
[103] D. Zagier, Quantum modular forms, Clay Math. Proc. 11, Amer. Math. Soc., Providence, RI, 2010. · Zbl 1294.11084
[104] G. Almkvist, C. van Enckevort, D. van Straten, W. Zudilin, Tables of Calabi-Yau operators, arXiv:math/0507430.
[105] R. Apéry, Irrationalité de ζ(2) et ζ(3), Luminy Conference on Arithmetic. Astérisque No. 61 (1979), 11-13. · Zbl 0401.10049
[106] A. Ash, P.Gunnels, M. McConnell, Cohomology of congruence subgroups of SL(4, Z) II, J. Number Theory (2008), no. 8, 2263-2274. · Zbl 1213.11113
[107] F. Beukers, Irrationality of ζ(2), periods of an elliptic curve and Γ 1 (5). In: Diophan-tine approximations and transcendental numbers (Luminy, 1982), 47-66, Progr. Math., 31, Birkhäuser, Boston, Mass., 1983. · Zbl 0518.10040
[108] V. Batyrev, I. Ciocan-Fontanine, B. Kim, D. van Straten, Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians, Nuclear Phys. B (1998), no. 3, 640-666. · Zbl 0896.14025
[109] B.Birch, H. P. F. Swinnerton-Dyer, Notes on elliptic curves. I, J. Reine Angew. Math. 212 (1963), 7-25. · Zbl 0118.27601
[110] B. Birch and H. P. F Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math. (1965), 79-108. · Zbl 0147.02506
[111] Birch, B. J. Elliptic curves over Q: A progress report. 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 396-400. Amer. Math. Soc., Providence, R.I., 1971.
[112] B.J. Birch and W. Kuyk, [eds.], Modular Functions of One Variable IV, Springer Lecture Notes 476, Berlin-Heidelberg-New York, 1975, Table I, pp. 81-113.
[113] C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the Modularity of Elliptic Curves Over Q: Wild 3-Adic Exercises., J. Amer. Math. Soc. 14, 843-939, (2001). · Zbl 0982.11033
[114] A. Brumer and K. Kramer, Paramodular abelian varieties of odd conductor, Trans. Amer. Math. Soc. 366 (2014), no. 5, 2463 -2516. · Zbl 1285.11087
[115] A. Brumer, A. Pacetti, C. Poor, G. Tornaría, J. Voight, D. Yuen, On the paramodularity of typical abelian surfaces, Algebra Number Theory 13 (2019), no. 5, 1145-1195. · Zbl 1466.11019
[116] P. Candelas, X. de la Ossa, D. van Straten, Local Zeta-functions from Calabi-Yau operators, in preparation.
[117] P. Candelas, X. de la Ossa, P. Green, L. Parkes,A pair of Calabi-Yau manifolds as an exactly soluble superconformal theor. Nucl. Phys., B359 (1991), 21-74. · Zbl 1098.32506
[118] H. Cohen, Computing L-functions: A survey J. Th. Nombres de Bordeaux, 27: 699-726, (2015). · Zbl 1397.11079
[119] J. Cremona, The L-functions and modular forms database project. Found. Comput. Math. 16 (2016), no. 6, 1541-1553. · Zbl 1354.11005
[120] S. Cynk, E. Freitag, R. Salvati-Manni, The geometry and arithmetic of a Calabi-Yau Siegel threefold, Internat. J. Math. 22 (2011), no. 11, 1585 -1602. · Zbl 1244.11048
[121] L. Dieulefait, On the modularity of rigid Calabi-Yau threefolds: epilogue J. Math. Sci. 171 (2010) 725-727. · Zbl 1290.14029
[122] N. Dummigan, Congruneces of Saito-Kurokawa lifts and denominators of central spionr L-values, preprint 2020.
[123] E. Freitag, R. Salvati-Manni On Siegel three-folds with a projective Calabi-Yau model Com-mun. Number Theory Phys. 5 (2011), no. 3, 713-750. · Zbl 1260.11036
[124] F. Gouvêa, N. Yui, Rigid Calabi-Yau Threefolds over Q Are Modular, Expo. Math. 29 (2011), no. 1, 142-149. · Zbl 1230.14056
[125] V. Gritsenko, The geometrical genus of the moduli space of abelian varieties. Mathematics in St. Petersburg, 9 -19, Amer. Math. Soc. Transl. Ser. 2, 174, Adv. Math. Sci., 30 Amer. Math. Soc., Providence, RI, (1996). · Zbl 0886.11028
[126] V. Gritsenko, K. Hulek, Minimal Siegel modular threefolds. Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 3, 461-485. · Zbl 0930.11028
[127] B. Gross, On the Langlands correspondence for symplectic motives (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 4, 49-64; translation in Izv. Math. 80 (2016), no. 4, 678-692 · Zbl 1391.11075
[128] M. Gross, S. Popescu, The moduli space of (1,11)-polarized abelian surfaces is unirational. Compositio Math. 126 (2001), no. 1, 1-23. · Zbl 1015.14022
[129] M. Gross and S. Popescu, Calabi-Yau threefolds and moduli of abelian surfaces. I, Compo-sitio Math. 127 (2001), no. 2, 169-228. · Zbl 1063.14051
[130] M. Gross, S. Popescu, Calabi-Yau three-folds and moduli of abelian surfaces II, Trans. Amer. Math. Soc. 363 (2011), no. 7, 3573-3599. · Zbl 1228.14039
[131] B.van Geemen, N. Nygaard, On the geometry and arithmetic of some Siegel modular three-folds J. Number Theory 53 (1995), no. 1, 45-87. · Zbl 0838.11047
[132] B. van Geemen, D. van Straten, The cusp forms of weight 3 on Γ 2 (2, 4, 8) Math. Comp. 61 (1993), no. 204, 849-872. · Zbl 0787.11018
[133] J. Hein, Orthogonal Modular Forms: An Application to a Conjecture of Birch, Algotithms and Computations, Thesis, Darthmouth College (2016).
[134] F. Klein,Über die Transformationen elfter Ordnung der elliptischen Functionen, Math. Ann. 15 (1879). · JFM 11.0299.06
[135] F. Klein, R. Fricke, Vorlesungenüber die Theorie der elliptischen Modulfunktionen, Bd.II, B. G. Teubner, Leipzig (1892). · JFM 24.0412.01
[136] W. Ladd, Algebraic Modular Forms on SO 5 (Q) and the Computation of Paramodular Forms, Thesis, Berkeley, (2018).
[137] C. Meyer, Modular Calabi-Yau threefolds, Fields Institute Monographs 22, AMS (2005). · Zbl 1096.14032
[138] C. Poor, D. Yuen, Paramodular cusp forms, Math. Comp 84 (2015), no. 293, 1401-1438. · Zbl 1392.11028
[139] G. Rama and G. Tornaría, Computation of paramodular forms Fourteenth Algorithmic Number Theory Symposium The Open Book Series 4 (2020), https://doi.org/10.2140/obs.2020.4.353. (see also: http://www.cmat.edu.uy/cnt/omf5/). · Zbl 1460.11149 · doi:10.2140/obs.2020.4.353
[140] B. Roberts and R. Schmidt, Local Newforms for GSp(4), Lecture Notes in Mathematics 1918, Berlin: Springer, 2007.
[141] R.Taylor, R. and A. Wiles, Ring-Theoretic Properties of Certain Hecke Algebras, Ann. Math. 141, 553-572, (1995). · Zbl 0823.11030
[142] J. Tingley, Elliptic curves uniformized by modular functions, Thesis, University of Oxford, (1975).
[143] P. Tsai, On Newforms for Split Special Odd Orthogonal Groups, Thesis, Harvard, (2013).
[144] J. Vélu, Courbes elliptiques sur Q ayant bonne réduction en dehors de {11}, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A73-A75. · Zbl 0225.14013
[145] A. Wiles, Modular Elliptic-Curves and Fermat’s Last Theorem, Ann. Math. 141, 443-551, (1995). · Zbl 0823.11029
[146] A. Weil,Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149-156. · Zbl 0158.08601
[147] W. Zudilin, Approximations to -, di-and tri-logarithms, Journal of Computational and Applied Mathematics 202 (2007) 450-459. · Zbl 1220.65028
[148] Tobias Berger, and Lassina Dembélé, and Ariel Pacetti, and Mehmet Haluk Sengun, Theta lifts of Bianchi modular forms and applications to paramodularity, J. Lond. Math. Soc. (2) 92 (2015) · Zbl 1396.11074
[149] Tobias Berger and Krzysztof Klosin, Deformations of Saito-Kurokawa Type and the Paramodular Conjecture-, American Journal of Mathematics, 142 (2020) 1821-1875. (with appendix by Cris Poor, Jerry Shurman, David S. Yuen) · Zbl 1469.11151
[150] Armand Brumer, Ariel Pacetti, Cris Poor, Gonzalo Tornaría, John Voight, and David S. Yuen. On the paramodularity of typical abelian surfaces, Journal of Algebra and Number Theory, 13 (2019), 1145-1195. · Zbl 1466.11019
[151] Armand Brumer and Kenneth Kramer, Paramodular abelian varieties of odd conductor, Trans. Amer. Math. Soc. 366 (2014), 2463-2516. · Zbl 1285.11087
[152] Tomoyoshi Ibukiyama, On relations of dimensions of automorphic forms of Sp(2, R) and its compact twist Sp(2), Automorphic forms and number theory, (Sendai, 1983), volume 7 of Advanced Studies in Pure Mathematics, 7-30, North Holland, Amsterdam. 1985. · Zbl 0609.10018
[153] Jennifer Johnson-Leung and Brooks Roberts. Siegel modular forms of degree two attached to Hilbert modular forms. Int. J. Number Theory, 132 (2012). · Zbl 1272.11063
[154] Jennifer Johnson-Leung and Brooks Roberts. Twisting of Siegel Paramodular Forms, Int. J. Number Theory, 13 (2017). · Zbl 1376.11034
[155] Cris Poor, Jerry Shurman, and David S. Yuen. Siegel paramodular forms of weight 2 and squarefree level, Int. J. Number Theory, 13 (2017), 2627-2652. · Zbl 1428.11087
[156] Cris Poor, Jerry Shurman, and David S. Yuen. Nonlift weight two paramodular eigenform constructions, Journal of the Korean Mathematical Society, 57 (2020), 507-522. · Zbl 1452.11055
[157] Cris Poor and David S. Yuen. Paramodular cusp forms, Math. Comp., 84 (2015), 1401-1438. · Zbl 1392.11028
[158] Brooks Roberts and Ralf Schmidt. Local newforms for GSp(4), Lecture Notes in Mathemat-ics, 1918. Springer, Berlin, 2007.
[159] Brooks Roberts and Ralf Schmidt. On modular forms for the paramodular groups, Auto-morphic forms and zeta functions, 334-364. World Scientific Publishers, Hackensack, NJ, 2006. References · Zbl 1161.11340
[160] Tobias Berger and Krzysztof Klosin, Deformations of Saito-Kurokawa Type and the Paramodular Conjecture-, American Journal of Mathematics, 142 (2020) 1821-1875. (with appendix by Cris Poor, Jerry Shurman, David S. Yuen) · Zbl 1469.11151
[161] Jeffery Breeding, II, Cris Poor, Jerry Shurman, and David S. Yuen. Using Restriction to Humbert surfaces to compute paramodular forms, in preparation.
[162] Jeffery Breeding, II, Cris Poor, and David S. Yuen. Computations of spaces of paramodular forms of general level, Journal Korean Math. Soc., 53 (2016) 645-689. · Zbl 1416.11063
[163] Armand Brumer, Ariel Pacetti, Cris Poor, Gonzalo Tornaría, John Voight, and David S. Yuen. On the paramodularity of typical abelian surfaces, Journal of Algebra and Number Theory, 13 (2019), 1145-1195. · Zbl 1466.11019
[164] Armand Brumer and Kenneth Kramer, Paramodular abelian varieties of odd conductor, Trans. Amer. Math. Soc. 366 (2014), 2463-2516. · Zbl 1285.11087
[165] Valeri A. Gritsenko and Viacheslav V. Nikulin. Automorphic forms and Lorentzian Kac-Moody algebras. II, Internat. J. Math., 9 (1998), 201-275. · Zbl 0935.11016
[166] Valeri Gritsenko, Cris Poor, and David S. Yuen, Antisymmetric paramodular forms of weights 2 and 3, International Mathematics Research Notices, (2019).
[167] Valeri A. Gritsenko, Nils-Peter Skoruppa, and Don Zagier, Theta Blocks, https://arxiv.org/abs/1907.00188
[168] Tomoyoshi Ibukiyama, Dimension formulas of Siegel modular forms of weight 3 and super-singular abelian surfaces, Siegel Modular Forms and Abelian Varieties, Proceedings of the 4-th Spring Conference on Modular Forms and Related Topics (2007), 39-60.
[169] Tomoyoshi Ibukiyama and Hidetaka Kitayama, Dimension formulas of paramodular forms of squarefree level and comparison with inner twist, J. Math. Soc. Japan 69 (2017), 597-671. · Zbl 1422.11110
[170] Cris Poor, Jerry Shurman, and David S. Yuen. Siegel paramodular forms of weight 2 and squarefree level, Int. J. Number Theory, 13 (2017), 2627-2652. · Zbl 1428.11087
[171] Cris Poor, Jerry Shurman, and David S. Yuen. Finding all Borcherds lift paramodular cusp forms of a given weight and level, Mathematics of Computation 89 (2020), 2435-2480. · Zbl 1443.11067
[172] Cris Poor, Jerry Shurman, and David S. Yuen. Nonlift weight two paramodular eigenform constructions, Journal of the Korean Mathematical Society, 57 (2020), 507-522. · Zbl 1452.11055
[173] Cris Poor and David S. Yuen. Paramodular cusp forms, Math. Comp., 84 (2015), 1401-1438. · Zbl 1392.11028
[174] Gerard van der Geer, Hilbert modular surfaces, volume 16 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin, 1988. · Zbl 0634.14022
[175] I. Barros, Uniruledness of strata of holomorphic differentials on low genus, Adv. Math. 233 (2018), 670-693 · Zbl 1406.32004
[176] A. Bud, Maximal gonality on strata and uniruledness of strata of low genus, Preprint, arXiv: 2008.02813 (2020)
[177] Q. Gendron, The Deligne-Mumford and the Incidence Variety Compactifications of the Strata of the moduli space of Abelian differentials, Ann. Inst. Fourier 32 (2018), 1169-1220
[178] M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, M. Möller, The moduli space of multi-scale differntials, Preprint, arXiv: 1910.13492 (2019)
[179] M. Costantini, M, Möller, J. Zachhuber, The Chern class and the Euler characteristic of the moduli spaces of Abelian differentials, Preprint, arXiv: 2006.12803 (2020)
[180] M. Kontsevich, A. Zorich, Connected components of the moduli spaces of Abelian differen-tials with prescribed singularities, Invent. Math. 153 (2003), 631-678 · Zbl 1087.32010
[181] J. Harris, D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23-88 · Zbl 0506.14016
[182] G. van der Geer, The Chow ring of the moduli space of abelian threefolds, J. Algebr. Geom. 7 (1998), 753-770. · Zbl 0952.14003
[183] S. Grushevsky, K. Hulek, The cone of effective surfaces on A 3 . arXiv:2011.06785. · Zbl 1189.14050
[184] K. Hulek, O. Tommasi Cohomology of the toroidal compactification of A 3 , In: Vector bun-dles and complex geometry. Eds. O. Garcia-Prada et al., Contemporary Mathematics 552 (2010), 89-103.
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