Arithmetic of Hermitian Forms

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

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Non unimodular Hermitian forms

Inventiones mathematicae ◽

10.1007/bf01232374 ◽

1996 ◽

Vol 123 (1) ◽

pp. 233-240 ◽

Cited By ~ 2

Author(s):

Eva Bayer-Fluckiger ◽

Laura Fainsilber

Keyword(s):

Hermitian Forms

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On the representation of integers by indefinite binary Hermitian forms

Bulletin of the London Mathematical Society ◽

10.1112/blms/bdr041 ◽

2011 ◽

Vol 43 (6) ◽

pp. 1048-1058 ◽

Cited By ~ 2

Author(s):

Jouni Parkkonen ◽

Frédéric Paulin

Keyword(s):

Hermitian Forms ◽

Representation Of Integers

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Indefinite Binary Hermitian Forms

Proceedings of the London Mathematical Society ◽

10.1112/plms/s3-6.2.205 ◽

1956 ◽

Vol s3-6 (2) ◽

pp. 205-223 ◽

Cited By ~ 1

Author(s):

K. Rogers

Keyword(s):

Hermitian Forms

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Spherical functions and local densities on the space of p-adic quaternion Hermitian forms

International Journal of Number Theory ◽

10.1142/s1793042122500324 ◽

2021 ◽

pp. 1-38

Author(s):

Yumiko Hironaka

Keyword(s):

Laurent Polynomial ◽

Spherical Functions ◽

Plancherel Formula ◽

Explicit Formulas ◽

Hermitian Forms ◽

Spherical Fourier Transform ◽

Local Densities ◽

Laurent Polynomial Ring ◽

Injective Map ◽

Residual Characteristic

We introduce the space [Formula: see text] of quaternion Hermitian forms of size [Formula: see text] on a [Formula: see text]-adic field with odd residual characteristic, and define typical spherical functions [Formula: see text] on [Formula: see text] and give their induction formula on sizes by using local densities of quaternion Hermitian forms. Then, we give functional equation of spherical functions with respect to [Formula: see text], and define a spherical Fourier transform on the Schwartz space [Formula: see text] which is Hecke algebra [Formula: see text]-injective map into the symmetric Laurent polynomial ring of size [Formula: see text]. Then, we determine the explicit formulas of [Formula: see text] by a method of the author’s former result. In the last section, we give precise generators of [Formula: see text] and determine all the spherical functions for [Formula: see text], and give the Plancherel formula for [Formula: see text].

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