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Computing the number of ways of representing primes by a norm form. (English) Zbl 0643.10022

In an earlier paper [J. Indian Math. Soc., New. Ser. 40, 87-122 (1976; Zbl 0437.10010)], the author obtained formulae for the number of different integral solutions of the norm forms of some definite quaternion algebras D by making explicit the arithmetic content of the Jacquet-Langlands correspondence between automorphic representations of \(D^{\times}\) and GL(2). The purpose of the present note is to show how these results may be extended to the norm forms of other definite quaternion algebras ramifying at a single finite prime.
The crucial extra information is provided by local character formulas of H. Shimizu for some of the supercuspidal representations of GL(2) (over a local field) [J. Fac. Sci., Univ. Tokyo, Sect. I A 24, 97-113 (1977; Zbl 0359.10023)]. The author uses these formulas to make completely explicit (a part of) the Jacquet-Langlands correspondence \(\pi\leftrightarrow \pi '\), so that an analysis of the corresponding Hecke operators (at the unramified places) yields the desired formulae for representing these primes by the quaternary form.
Reviewer: S.Gelbart

MSC:

11F11 Holomorphic modular forms of integral weight
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11E45 Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
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References:

[1] Gelbart, C., Automorphic forms on adele groups (1975), Princeton: University Press, Princeton · Zbl 0329.10018
[2] Hecke E, Werke,Vandenhoeck and Ruprecht, Gottingen 1959 · Zbl 0092.00102
[3] Jacqet H and Langlands R P, Automorphic forms onGL(2),Lecture notes in mathematics 114 (Berlin: Springer Verlag)
[4] Serre J P,Corps Locaux (Paris: Herman)
[5] Shimizu H, Some examples of new forms,J. Fac. Sci Univ. Tokyo Sec. IA24 97-113 · Zbl 0359.10023
[6] Tandon, R., The Hecke theory ofGL(2) and quadratic forms, J. Indian Math. Soc., 40, 87-122 (1976) · Zbl 0437.10010
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