An explicit formula for the Siegel series of a quadratic form over a non-archimedean local field

References

[1] S. Böcherer, Über die Funktionalgleichung automorpher L-Funktionen zur Siegelschen Modulgruppe, J. reine angew. Math. 362 (1985), 146–168. 10.1515/crll.1985.362.146Suche in Google Scholar

[2] S. Böcherer and R. Schulze-Pillot, On the central critical value of the triple product L-function, Number theory (Paris 1993–1994), London Math. Soc. Lecture Note Ser. 235, Cambridge University, Cambridge (1996), 1–46. 10.1017/CBO9780511662003.002Suche in Google Scholar

[3] S. Breulmann and M. Kuss, On a conjecture of Duke–Imamoḡlu, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1595–1604. 10.1090/S0002-9939-00-05586-6Suche in Google Scholar

[4] S. Cho, T. Ikeda, H. Katsurada, C.-H. Lee and T. Yamauchi, An explicit formula for the extended Gross–Keating datum of a quadratic form, preprint (2020), https://arxiv.org/abs/1709.02772v2. Suche in Google Scholar

[5] S. Cho and T. Yamauchi, A reformulation of the Siegel series and intersection numbers, Math. Ann. 377 (2020), no. 3–4, 1757–1826. 10.1007/s00208-020-01999-2Suche in Google Scholar

[6] H. Cohen, Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann. 217 (1975), no. 3, 271–285. 10.1007/BF01436180Suche in Google Scholar

[7] N. Dummigan, T. Ibukiyama and H. Katsurada, Some Siegel modular standard L-values, and Shafarevich–Tate groups, J. Number Theory 131 (2011), no. 7, 1296–1330. 10.1016/j.jnt.2011.01.012Suche in Google Scholar

[8] B. H. Gross and K. Keating, On the intersection of modular correspondences, Invent. Math. 112 (1993), no. 2, 225–245. 10.1007/BF01232433Suche in Google Scholar

[9] T. Ibukiyama and H. Katsurada, Exact critical values of the symmetric fourth L function and vector valued Siegel modular forms, J. Math. Soc. Japan 66 (2014), no. 1, 139–160. 10.2969/jmsj/06610139Suche in Google Scholar

[10] T. Ibukiyama, H. Katsurada, C. Poor and D. S. Yuen, Congruences to Ikeda–Miyawaki lifts and triple L-values of elliptic modular forms, J. Number Theory 134 (2014), 142–180. 10.1016/j.jnt.2013.07.013Suche in Google Scholar

[11] T. Ikeda, On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2⁢n, Ann. of Math. (2) 154 (2001), no. 3, 641–681. 10.2307/3062143Suche in Google Scholar

[12] T. Ikeda, Pullback of the lifting of elliptic cusp forms and Miyawaki’s conjecture, Duke Math. J. 131 (2006), no. 3, 469–497. 10.1215/S0012-7094-06-13133-2Suche in Google Scholar

[13] T. Ikeda, On the functional equation of the Siegel series, J. Number Theory 172 (2017), 44–62. 10.1016/j.jnt.2016.08.002Suche in Google Scholar

[14] T. Ikeda and H. Katsurada, On the Gross–Keating invariant of a quadratic form over a non-archimedean local field, Amer. J. Math. 140 (2018), no. 6, 1521–1565. 10.1353/ajm.2018.0046Suche in Google Scholar

[15] T. Ikeda and S. Yamana, On the lifting of Hilbert cusp forms to Hilbert–Siegel cusp forms, Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), no. 5, 1121–1181. 10.24033/asens.2442Suche in Google Scholar

[16] H. Katsurada, An explicit formula for Siegel series, Amer. J. Math. 121 (1999), no. 2, 415–452. 10.1353/ajm.1999.0013Suche in Google Scholar

[17] H. Katsurada, Special values of the standard zeta functions for elliptic modular forms, Exp. Math. 14 (2005), no. 1, 27–45. 10.1080/10586458.2005.10128910Suche in Google Scholar

[18] H. Katsurada, Exact standard zeta values of Siegel modular forms, Exp. Math. 19 (2010), no. 1, 65–77. 10.1080/10586458.2010.10129062Suche in Google Scholar

[19] H. Katsurada and H.-A. Kawamura, On the Andrianov-type identity for power series attached to Jacobi forms and its application, Acta Arith. 145 (2010), no. 3, 233–265. 10.4064/aa145-3-3Suche in Google Scholar

[20] H. Katsurada and H.-A. Kawamura, Ikeda’s conjecture on the period of the Duke–Imamoḡlu–Ikeda lift, Proc. Lond. Math. Soc. (3) 111 (2015), no. 2, 445–483. 10.1112/plms/pdv011Suche in Google Scholar

[21] H. Katsurada and S. Mizumoto, Congruences for Hecke eigenvalues of Siegel modular forms, Abh. Math. Semin. Univ. Hambg. 82 (2012), no. 2, 129–152. 10.1007/s12188-012-0069-4Suche in Google Scholar

[22] Y. Kitaoka, A note on local densities of quadratic forms, Nagoya Math. J. 92 (1983), 145–152. 10.1017/S0027763000020626Suche in Google Scholar

[23] Y. Kitaoka, Arithmetic of quadratic forms, Cambridge Tracts in Math. 106, Cambridge University, Cambridge 1993. 10.1017/CBO9780511666155Suche in Google Scholar

[24] N. Kozima, On special values of standard L-functions attached to vector valued Siegel modular forms, Kodai Math. J. 23 (2000), no. 2, 255–265. 10.2996/kmj/1138044215Suche in Google Scholar

[25] S. S. Kudla, Central derivatives of Eisenstein series and height pairings, Ann. of Math. (2) 146 (1997), no. 3, 545–646. 10.2307/2952456Suche in Google Scholar

[26] S. S. Kudla, Special cycles and derivatives of Eisenstein series, Heegner points and Rankin L-series, Math. Sci. Res. Inst. Publ. 49, Cambridge University, Cambridge (2004), 243–270. 10.1017/CBO9780511756375.009Suche in Google Scholar

[27] S. S. Kudla, M. Rapoport and T. Yang, Modular forms and special cycles on Shimura curves, Ann. of Math. Stud. 161, Princeton University, Princeton 2006. 10.1515/9781400837168Suche in Google Scholar

[28] T. Orloff, Special values and mixed weight triple products (with an appendix by Don Blasius), Invent. Math. 90 (1987), no. 1, 169–188. 10.1007/BF01389036Suche in Google Scholar

[29] A. Otsuka, On the image of the Saito–Kurokawa lifting over a totally real number field and the Maass relation, Abh. Math. Semin. Univ. Hambg. 84 (2014), no. 1, 49–65. 10.1007/s12188-013-0086-ySuche in Google Scholar

[30] M. Rapoport and T. Wedhorn, The connection to Eisenstein series, Astérisque 312 (2007), 191–208. Suche in Google Scholar

[31] I. Satake, Linear algebra, Marcel Dekker, New York 1975. Suche in Google Scholar

[32] F. Sato and Y. Hironaka, Local densities of representations of quadratic forms over p-adic integers (the non-dyadic case), J. Number Theory 83 (2000), no. 1, 106–136. 10.1006/jnth.1999.2505Suche in Google Scholar

[33] G. Shimura, Euler products and Fourier coefficients of automorphic forms on symplectic groups, Invent. Math. 116 (1994), no. 1–3, 531–576. 10.1007/BF01231572Suche in Google Scholar

[34] G. Shimura, Euler products and Eisenstein series, CBMS Reg. Conf. Ser. Math. 93, American Mathematical Society, Providence 1997. 10.1090/cbms/093Suche in Google Scholar

[35] G. Shimura, Arithmetic of quadratic forms, Springer Monogr. Math., Springer, New York 2010. 10.1007/978-1-4419-1732-4Suche in Google Scholar

[36] T. Wedhorn, Calculation of representation densities, Astérisque 312 (2007), 179–190. Suche in Google Scholar

[37] T. Yang, Local densities of 2-adic quadratic forms, J. Number Theory 108 (2004), no. 2, 287–345. 10.1016/j.jnt.2004.05.002Suche in Google Scholar