On superspecial abelian surfaces and type numbers of totally definite quaternion algebras

Abstract

In this paper we determine the number of endomorphism rings of superspecial abelian surfaces over a field Fq of odd degree over Fp in the isogeny class corresponding to the Weil q-number ± √ q . This extends earlier works of T.-C. Yang and the present authors on the isomorphism classes of these abelian surfaces, and also generalizes the classical formula of Deuring for the number of endomorphism rings of supersingular elliptic curves. Our method is to explore the relationship between the type and class numbers of the quaternion orders concerned. We study the Picard group action of the center of an arbitrary Z-order in a totally definite quaternion algebra on the ideal class set of said order, and derive an orbit number formula for this action. This allows us to prove an integrality assertion of Vignéras [Enseign. Math.(2), 1975] as follows. Let F be a totally real field of even degree over Q, and D be the (unique up to isomorphism) totally definite quaternion F -algebra unramified at all finite places of F . Then the quotient h(D)/h(F ) of the class numbers is an integer.

Polarized superspecial simple abelian surfaces with real Weil numbers

“…Compared to the previous case, End(X π ) is no longer necessarily a maximal order in End 0 (X π ) even in the case n = 1 [57, Theorem 6.2], which causes new difficulties. The number of endomorphism rings (up to isomorphism) of members of Isog( √ p ) is calculated in [63]. We produce the counterpart of (1.11) for the set PPAV( √ p ).…”

Unit groups of maximal orders in totally definite quaternion algebras over real quadratic fields

“…n O F [ √ −1 ], O F [ √ −ε ] O F ( , B1,2 Here B 1,2 ⊂ F ( √ −1 ) is the order defined in (7.3), and O F [ √ −ε ] coincides with the ring of integers of F ( √ −ε ) = F ( √ −2 ) by Lemma 7.2.5. For each B ∈ B n , we sett(C n , B) := #{ O ′ ∈ Tp(H) | O ′⋆ ⋆ ⋆ ≃ C n , and Emb(B, O ′ ) = ∅}.It is shown in[48, Corollary 3.5] (see also Remark 3.6) that(8.8) h(C n , B) = h(F )t(C n , B) ∀B ∈ B n .The main tool to compute h(C n , B) (and in turn t(C n , B)) is equation(3.19). In the current setting, we haved(H) = O F and hence ω(H) = 0.…”