On superspecial abelian surfaces over finite fields II

Abstract

Extending the results of [24, Asian J. Math.], in [26, Doc. Math. 21, 2016] we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of odd degree over the prime field Fp. A key step was to reduce the calculation to the prime field case, and we calculated the number of isomorphism classes in each isogeny class through a concrete lattice description. In the present paper we treat the even degree case by a different method. We first translate the problem by Galois cohomology into a seemingly unrelated problem of computing conjugacy classes of elements of finite order in arithmetic subgroups, which is of independent interest. We then explain how to calculate the number of these classes for the arithmetic subgroups concerned, and complete the computation in the case of rank two. This complements our earlier results and completes the explicit calculation of superspecial abelian surfaces over finite fields.Date: October 4, 2018. 2010 Mathematics Subject Classification. 11R52, 11G10. 1 This is well known when the ground field k is algebraically closed, and is proved in [29] for arbitrary k 1 Lemma 6.1. Suppose that S × ⊆ N ( R), the normalizer of R in B × . Then the suborder R J := x Rx −1 ∩ B of S J is independent of the choice of x ∈ B × for J, and.Proof. Suppose that J = x ′ S for x ′ ∈ B × as well. Then there exists u ∈ S × such that x ′ = xu. Since S × ⊆ N ( R), we havewhich proves the independence of R J of the choice of x. If I is a locally principal right R-ideal such that IS = J, then R J = O l (I), the associated left order of I.Conjugating by x ∈ B × on the right hand side of (6.3), we obtainThe assumption S × ⊆ N ( R) also implies that R × S × , and hence R × J S × J and R × J S × J . The left action of S × J on the quotient group S × J / R × J factors through S × J /R × J ⊆ S × J / R × J , and its orbits are the right cosets of S × JRemark 6.2. The condition S × ⊆ N ( R) implies that R × S × . However, the converse does not hold in general. It is enough to provide a counterexample locally Corollary 6.3. Keep the notation and assumption of Lemma 6.1. If the natural homomorphism S ×Proof. It is enough to show that π is injective. The surjectivity of S × J → S × J / R × J implies that the monomorphism S × J /R × J ֒→ S × J / R × J is an isomorphism, and hence |π −1 ([J])| = [ S × J / R × J : S × J /R × J ] = 1.

Counting isomorphism classes of superspecial curves

“…Igusa [16] also proved the same result as in Theorem 2.3 by directly computing the number of supersingular j-invariants from the Legendre form y 2 = x(x − 1)(x − λ) of an elliptic curve. We also refer to [30,Proposition 4.4] for results on the number of F q -isomorphism classes of supersingular elliptic curves over F q .…”

Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves

“…More importantly, isomorphism classes of products of isogenous supersingular elliptic curves over Fq are essentially trivial at least in a geometric sense. Indeed, according to a result of Deligne (see [39,Theorem 3.5] [41,Section 5]. Therefore, one could conceivably obtain a "commutative supersingular" version of the construction above, which would generalize the recent 2-party key exchange protocol CSIDH [11], assuming that Fp-isomorphism invariants can be computed in that setting.…”

On Counting Certain Abelian Varieties Over Finite Fields

“…The cardinality of Sp(π) is explicitly calculated in [33], and that of A 1 π will be calculated in Section 4.…”

“…The prototype of such descriptions is the Deuring-Eichler correspondence, which establishes the following bijection where p is a prime number and D p,∞ is the quaternion Q-algebra ramified exactly at {p, ∞}. We refer to Waterhouse [30], Deligne [7], Ekedahl [10], Katsura and Oort [15], C.-F. Yu [36,37,40,41], Centeleghe and Stix [3] J. Xue, T.-C. Yang and C.-F. Yu [31,33,34], Jordan-Keeton-Poonen-Shepherd-Barron-Tate [14] and others for various generalizations, and explicit formulas for numbers of certain abelian varieties.…”