Proofs of Ibukiyama’s conjectures on Siegel modular forms of half-integral weight and of degree 2

[1] J. Adams, The theta correspondence over R, Harmonic analysis, group representations, automorphic forms and invariant theory, pp. 1–39, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 12, World Sci. Publ., Hackensack, NJ, (2007).

[2] J. Adams, D. Barbasch, Reductive dual pair correspondence for complex groups, J. Funct. Anal. 132 , no. 1, pp. 1-42, (1995).

[3] J. Adams, D. Barbasch, Genuine representations of the metaplectic group, Compos. Math. 113, no. 1, pp. 23-66, (1998).

[4] A. N. Andrianov, V. G. Zhuravlev, Modular forms and Hecke operators, Translations of Mathematical Mono- graphs, vol. 145, Amer. Math. Soc., Providence, RI, (1995).

[5] J. Arthur, The endoscopic classification of representations: orthogonal and symplectic groups, American Math- ematical Society Colloquium Publications, 61, (2013).

[6] M. Asgari, R. Schmidt, Siegel modular forms and representations, Manuscripta Math. 104, no. 2, pp. 173–200, (2001).

[7] R. Berndt, R. Schmidt, Elements of the representation theory of the Jacobi group, Birkh¨auser, (1998).

[8] G. Chenevier, J. Lannes, Automorphic forms and even unimodular lattices, Ergeb. Math. Grenz. (3), 69, Springer Verlag, (2019).

[9] W. T. Gan, A Langlands Program for Covering Groups?, Proceedings of the Sixth International Congress of Chinese Mathematicians, vol. I, pp. 57–78. Adv. Lect. Math. 36. Somerville, MA: Int. Press, (2017).

[10] W. T. Gan, A. Ichino, The Shimura–Waldspurger correspondence for Mp2n, Ann. Math. (2) 188, no. 3, pp. 965–1016, (2018).

[11] W. T. Gan, A. Ichino, The automorphic discrete spectrum of Mp4, Int. Math. Res. Not., rnaa 121, (2020).

[12] W. T. Gan, G. Savin, Representations of metaplectic groups I: epsilon dichotomy and local Langlands corre- spondence, Compos. Math. 148, no. 6, pp. 1655-1694, (2012).

[13] M. Harris, R. Taylor, On the Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Studies 151, Princeton Univ. Press, Princeton, N.J., (2001).

[14] G. Henniart, Une preuve simple des conjectures de Langlands de GL(n) sur un corps p-adique, Invent. Math. 139, pp. 439-455, (2000).

[15] K. Hiraga, T. Ikeda, On the Kohnen plus space for Hilbert modular forms of half-integral weight I, Compos. Math. 149, pp. 1963-2010, (2013).

[16] T. Ibukiyama, A conjecture on a Shimura type correspondence for Siegel modular forms, and Harder’s conjecture on congruences, Modular Forms on Schiermonnikoog Edited by Bas Edixhoven, Gerard van der Geer and Ben Moonen, Cambridge University Press, pp. 107-144, (2008).

[17] T. Ibukiyama, Conjectures of Shimura type and of Harder type revisited, Comment. Math. Univ. St. Paul. 41 pp. 79-103, (2014).

[18] W. Kohnen, Modular forms of half integral weight on Γ0(4), Math. Ann., 248, pp. 249-266, (1980).

[19] R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr. 31, Amer. Math. Soc., Prov- idence, RI, pp. 101-170, (1989).

[20] C. Mœglin, D. Renard, Sur les paquets d’Arthur des groupes classiques r´eels, J. Eur. Math. Soc., vol. 22, Issue 6, pp. 1827-1892, (2020).

[21] C. Mœglin, D. Renard, Sur les paquets d’Arthur aux places r´eelles, translation, Geometric Aspects of the Trace Formula, Edited by W. Mu¨ller, S.-W. Shin, N. Templier, Simons Symposia, Springer, Cham., pp.299-320, (2018).

[22] A. Murase, L-functions attached to Jacobi forms of degree n. Part I: The basic identity, J. reine ang. Math. 401, pp. 122-156, (1989).

[23] R. Ranga Rao, On some explicit formulas in the theory of Weil representation, Pacific Journal of Math., 157, no. 2, pp. 335-371,(1993).

[24] R. Schmidt, Spherical representations of the Jacobi group, Abh. Math. Sem. Univ. Hamburg 68, pp. 273-296, (1998).

[25] G. Shimura, On modular forms of half integral weight, Ann. of Math. (2) 97, pp. 440–481, (1973).

[26] R. H. Su, The Kohnen plus space for Hilbert-Siegel modular forms, J. Number Theory, 163, pp. 267-295, (2016).

[27] B. Sun, On representations of real Jacobi groups, Sci. China Math. 55 (3), pp.541-555, (2012).

[28] D. Szpruch, Computation of the local coefficients for principal series representations of the metaplectic double cover of SL2(F), Journal of Number Theory 129, no. 9, pp. 2180-2213, (2009).

[29] J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl. (9) 59, no. 1, pp. 1–132, (1980).

[30] J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3, no. 3, pp. 219-307, (1991).

[31] V. G. Zhuravlev, Hecke rings for a covering of the symplectic group, Math. Sbornik. 121(163) No.3 pp.381-402, (1983).