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On automorphism groups of positive definite binary quaternion Hermitian lattices and new mass formula. (English) Zbl 0703.11019

Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math. 15, 301-349 (1989).
[For the entire collection see Zbl 0688.00008.]
This article gives a refinement of results achieved by the author and K. Hashimoto in a series of papers [J. Fac. Sci., Univ. Tokyo, Sect. IA 27, 549-601 (1980; Zbl 0452.10029), 28, 695-699 (1981; Zbl 0493.10030), 30, 393-401 (1983; Zbl 0533.10019)] on class numbers of positive definite binary quaternion Hermitian forms. More specifically, for one (the “non-principal”) of the two genera of maximal binary positive definite Hermitian lattices over the maximal order of a definite quaternion algebra B over \({\mathbb{Q}}\) of prime discriminant the author completely determines the number of classes in that genus having a fixed group of units (for each of the finitely many finite groups that can occur as such a unit group). This generalizes results of the author, T. Katsura and F. Oort [Compos. Math. 57, 127-152 (1986; Zbl 0589.14028)].
The main tool is a “new mass formula” which allows to compute the weighted average over the lattices \(L_ i\) (with unit groups \(\Gamma_ i)\) in the genus of the number of r-tuples \(g_ 1,...,g_ r\in M_ 2(B)^ r\) (up to conjugacy by an element of \(GL_ 2(B))\) contained in \(\Gamma^ r_ i\) (choosing \(g_ 1,...,g_ r\) as a generating set of the fixed finite group \(\Gamma\) under investigation). The method seems to be applicable to other genera of Hermitian forms in principle, but the explicit evaluation of the formula becomes quite complicated already in the special case studied here.
Reviewer: R.Schulze-Pillot

MSC:

11E41 Class numbers of quadratic and Hermitian forms
11H56 Automorphism groups of lattices