Supersingular abelian surfaces and Eichler’s class number formula

Abstract

Let F be a totally real field with ring of integers O F , and D be a totally definite quaternion algebra over F . A well-known formula established by Eichler and then extended by Körner computes the class number of any O F -order in D. In this paper we generalize the Eichler class number formula so that it works for arbitrary Z-orders in D. Our motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a finite prime field Fp. We give explicit formulas for the number of these isomorphism classes for all primes p.

On superspecial abelian surfaces over finite fields II

“…In [26] we calculated explicitly the number of superspecial abelian surfaces over an arbitrary finite field F q of odd degree over F p . This extended our earlier works [24,25] and [31], which contributed to the study of superspecial abelian varieties over finite fields. In this paper we treat the even degree case.…”

On Counting Certain Abelian Varieties Over Finite Fields

“…In a series of papers [32,31,33,34], the current authors and T.-C. Yang obtain an explicit formula for the number of F q -isomorphism classes of superspecial abelian surfaces in each isogeny class over F q . Our next step is to compute explicitly that for each isogeny of supersingular abelian surfaces.…”

“…When the center F = Q(π) is a CM field, we show that |A π | is the sum of certain explicit ray class numbers of F (see Proposition 2.2). For the other case where F is totally real, we discuss the classification of "genera" in details, and one can apply the generalized Eichler trace formula in [31] to compute the class numbers of these genera. For the reader's convenience, we describe the extended trace formula in Section 3.…”

“…

This paper contains two parts toward studying abelian varieties from the classification point of view. In a series of papers [32,31,33,34], the current authors and T.-C. Yang obtain explicit formulas for the numbers of superspecial abelian surfaces over finite fields. In this paper, we give an explicit formula for the size of the isogeny class of simple abelian surfaces with real Weil number √ q.

…”

“…Thus, the Eichler trace formula developed in [16] for an arbitrary O F -order is not sufficiently general for computing h(O X ). In [31] we generalize the Eichler trace formula for an arbitrary Z-order in any totally definite quaternion over a totally real field. For the reader's convenience, we describe this extended formula in Section 3.…”