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Conjectures on correspondence of symplectic modular forms of middle parahoric type and Ihara lifts

  • Tomoyoshi Ibukiyama
Research
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Part of the following topical collections:
  1. Modular Forms are Everywhere: Celebration of Don Zagier’s 65th Birthday

Abstract

By Ihara (J Math Soc Jpn 16:214–225, 1964) and Langlands (Lectures in modern analysis and applications III, lecture notes in math, vol 170. Springer, Berlin, pp 18–61, 1970), it is expected that automorphic forms of the symplectic group \(Sp(2,{\mathbb {R}})\subset GL_{4}({\mathbb {R}})\) of rank 2 and those of its compact twist have a good correspondence preserving L functions. Aiming to give a neat classical isomorphism between automorphic forms of this type for concrete discrete subgroups like Eichler (J Reine Angew Math 195:156–171, 1955) and Shimizu (Ann Math 81(2):166–193, 1965) (and not aiming the general representation theory), in our previous papers Hashimoto and Ibukiyama (Adv Stud Pure Math 7:31–102, 1985) and Ibukiyama (J Fac Sci Univ Tokyo Sect IA Math 30:587–614, 1984; Adv Stud Pure Math 7:7–29 1985; in: Furusawa (ed) Proceedings of the 9-th autumn workshop on number theory, 2007), we have given two different conjectures on precise isomorphisms between Siegel cusp forms of degree two and automorphic forms of the symplectic compact twist USp(2), one is the case when subgroups of both groups are maximal locally, and the other is the case when subgroups of both groups are minimal parahoric. We could not give a good conjecture at that time when the discrete subgroups for Siegel cusp forms are middle parahoric locally (like \(\Gamma _0^{(2)}(p)\) of degree two). Now a subject of this paper is a conjecture for such remaining cases. We propose this new conjecture with strong evidence of relations of dimensions and also with numerical examples. For the compact twist, it is known by Ihara that there exist liftings of Saito–Kurokawa type and of Yoshida type. It was not known about the description of the image of these liftings, but we can give here also a very precise conjecture on the image of the Ihara liftings.

Keywords

Siegel modular form Middle parahoric Compact twist Ihara lift 

Mathematics Subject Classification

Primary 11F46 Secondly 11F66 

Notes

References

  1. 1.
    Aoki, H.: On Siegel paramodular forms of degree 2 with small levels. Int. J. Math. 27, 1650011 (2016). 20 ppMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aoki, H., Ibukiyama, T.: Simple graded rings of Siegel modular forms, differential operators and Borcherds products. Int. J. Math. 16, 249–279 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boecherer, S., Schulze-Pillot, R.: Siegel modular forms and theta series attached to quaternion algebras II. Nagoya Math. J. 147, 71–106 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dern, T.: Paramodular forms of degree 2 and level 3. Comment. Math. Univ. St. Paul. 51, 157–194 (2002)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Eichler, M.: Über die Darstellbarkeit von Modulformen durch Thetareihen. J. Reine Angew. Math. 195, 156–171 (1955)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Eichler, M., Zagier, D.: The theory of Jacobi forms. Progress in Mathematics 55, pp. v+148. Birkhäuser Boston, Inc., Boston, MA, (1985)Google Scholar
  7. 7.
    Evdokimov, S.A.: Euler products for congruence subgroups of the Siegel group of genus 2. Math. USSR Sbornik 28 (1976)Google Scholar
  8. 8.
    Freitag, E., Salvati Manni, R.: The Burkhardt group and modular forms. Transform. Groups 9, 25–45 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gritsenko, V.: Arithmetical lifting and its applications, Number theory (Paris, 1992–1993), London Math. Soc. Lecture Note Series 215, pp. 103–126. Cambridge Univ. Press, Cambridge (1995)Google Scholar
  10. 10.
    Hashimoto, K.: On Brandt matrices associated with the positive definite quaternion Hermitian forms. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 227–245 (1980)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Hashimoto, K.: The dimension of the spaces of cusp forms on Siegel upper half-plane of degree two I. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30, 403–488 (1983)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hashimoto, K., Ibukiyama, T.: On class numbers of positive definite binary quaternion harmitian forms. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 549–601 (1980); 28, 695–699 (1982); 30, 393–401 (1983)Google Scholar
  13. 13.
    Hashimoto, K., Ibukiyama, T.: On relations of dimensions of automorphic forms of \(Sp(2,{\mathbb{R}})\) and its compact twist \(Sp(2)\) II, Automorphic forms and number theory (Sendai 1983). Adv. Stud. Pure Math. 7, 31–102 (1985)zbMATHGoogle Scholar
  14. 14.
    Ibukiyama, T.: On symplectic Euler factors of genus two. J. Fac. Sci. Univ. Tokyo Sec. IA Math. 30, 587–614 (1984)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ibukiyama, T.: On automorphic forms of \(Sp(2,{\mathbb{R}})\) and its compact form \(Sp(2)\). Séminaire de Théorie des nomble de Paris, 1982–1983 . Progr. Math. 51, pp. 125–134. Birkhäuser Boston Inc. (1984)Google Scholar
  16. 16.
    Ibukiyama, T.: On relations of dimensions of automorphic forms of \(Sp(2,{\mathbb{R}})\) and its compact twist \(Sp(2)\) I, Automorphic forms and number theory (Sendai, 1983). Adv. Stud. Pure Math. 7, 7–29 (1985)zbMATHGoogle Scholar
  17. 17.
    Ibukiyama, T.: On Siegel modular varieties of level three. Int. J. Math. 2, 17–35 (1991)CrossRefzbMATHGoogle Scholar
  18. 18.
    Ibukiyama, T.: On some alternating sums of dimemsions of Siegel modular forms of general degree and cusp configurations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40, 245–283 (1993)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ibukiyama, T.: Paramodular forms and compact twist. Automorphic Forms on \(GSp(4)\). In: Furusawa, M. (ed.)Proceedings of the 9-th Autumn Workshop on Number Theory, pp. 37–48 (2007)Google Scholar
  20. 20.
    Ibukiyama, T.: Dimension formulas of Siegel modular forms of weight 3 and supersingular abelian surfaces. In: Proceedings of the 4-th Spring Conference. Abelian Varieties and Siegel Modular Forms, pp. 39–60 (2007)Google Scholar
  21. 21.
    Ibukiyama, T.: Saito-Kurokawa liftings of level \(N\) and practical construction of Jacobi forms. Kyoto J. Math. 52, 141–178 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ibukiyama, T., Ihara, Y.: On automorphic forms on the unitary symplectic group \(Sp(n)\) and \(SL_2({\mathbb{R}})\). Math. Ann. 278, 307–327 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ibukiyama, T., Kitayama, H.: Dimension formulas of paramodular forms of squarefree level and comparison with inner twist. J. Math. Soc. Jpn. 69, 597–671 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ibukiyama, T., Onodera, F.: On the graded ring of modular forms of the Siegel paramodular group of level 2. Abh. Math. Semin. Univ. Hamburg. 67, 297–305 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ibukiyama, T., Zagier, D.: Higher Spherical Polynomials/ MPIM preprint series 2014–16, p. 97 (2014)Google Scholar
  26. 26.
    Igusa, J.: On Siegel modular forms genus two II. Am. J. Math. 86, 392–412 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ihara, T.: On certain arithmetical Dirichlet series. J. Math. Soc. Jpn. 16, 214–225 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Langlands, R.P.: Problems in the theory of automorphic forms. In: Lectures in Modern Analysis and Applications III, pp. 18–61. Lecture Notes in Math. 170, Springer, Berlin (1970)Google Scholar
  29. 29.
    Löschel, R.: Thetakorrespondenz automorpher Formen. Inaugural Dissertation zur Erlangung des Doktorgrades der Mathematisch- Naturwissenshaftlichen Fakultät der Universität zu Köln., pp. ii+102 (1997)Google Scholar
  30. 30.
    Matsuda, I.: Dirichlet series corresponding to Siegel modular forms of degree \(2\), level \(N\). Sci. Pap. Coll. Gen. Ed. Univ. Tokyo 28, 21–49 (1978)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Matsuda Makino, I.: On the meromorphy of Dirichlet series corresponding to Siegel cusp form of degree 2 with respect to \(\Gamma _0(N)\). Comment. Math. Univ. St. Pauli 34, 105–134 (1985)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Poor, C., Schmidt, R., Yuen, D.S.: Paramodular forms of level 8 and weights 10 and 12. Int. J. Number Theory 14, 417–467 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Roberts, B., Schmidt, R.: Local new forms for \(GSp(4)\). In: Lecture Notes in Mathematics, vol. 1918, pp. viii+307. Springer, Berlin (2007)Google Scholar
  34. 34.
    Satake, I.: Surjectivité globale de l’opérateur \(\Phi \). Séminaire H. Cartan 1957/58, Fonctions Automorphes Exposé 16, École Normale Supérieure (1958)Google Scholar
  35. 35.
    Schmidt, R.: Iwahori-spherical representations of \(GSp(4)\) and Siegel modular forms of degree \(2\) with square-free level. J. Math. Soc. Jpn. 57, 259–293 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Schmidt, R.: On classical Saito–Kurokawa liftings. J. Reine Angew. Math. 604, 211–236 (2007)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Saha, A., Schmidt, R.: Yoshida lifts and simultaneous non-vanishing of dihedral twists of modular \(L\)-functions. J. Lond. Math. Soc. 88(2), 251–270 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Shimizu, H.: On zeta functions of quaternion algebras. Ann. Math. 81(2), 166–193 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Shimura, G.: Arithmetic of alternating forms and quaternion hermitian forms. J. Math. Soc. Jpn. 15, 33–65 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Skoruppa, N.-P., Zagier, D.: Jacobi forms and a certain space of modular forms. Invent. Math. 94, 113–146 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Wakatsuki, S.: Dimension formulas for spaces of vector-valued Siegel cusp forms of degree two. J. Number Theory 132, 200–253 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Weyl, H.: The Classical Groups. Their Invariants and Representations, p. ), xiv+320. Princeton University Press, Princeton (1946)zbMATHGoogle Scholar
  43. 43.
    Yamauchi, T.: On the traces of Hecke operators for a normalizer of \(\Gamma _0(N)\). J. Math. Kyoto Univ. 13, 403–411 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Yoshida, H.: Siegel’s modular forms and the arithmetic of quadratic forms. Invent. Math. 60, 193–248 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonakaJapan

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