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

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  1. Modular Forms are Everywhere: Celebration of Don Zagier’s 65th Birthday


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.


Siegel modular form Middle parahoric Compact twist Ihara lift 

Mathematics Subject Classification

Primary 11F46 Secondly 11F66 



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Authors and Affiliations

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

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