On superspecial abelian surfaces over finite fields
Let \(q = p^a\) be a prime power. An abelian variety over a field \(k\) is called superspecial if it is isomorphic to a product of supersingular elliptic curves over an algebraic closure of \(k\) and superspecial if it is isogenous to a product of supersingular elliptic curves over an algebraic closure of \(k\). The authors denote the set of isomorphism classes of \(d\)-dimensional superspecial abelian varieties over \(\mathbb{F}_q\) by \(\mathrm{Sp}_d(\mathbb{F}_q)\). The purpose of this article is to determine \(\mathrm{Sp}_d(\mathbb{F}_q)\) for \(d=1\) and \(d=2\) and \(a\) odd. Furthermore, it is proved that for \(q, q'\) powers of \(p\) of the same exponent parity, there is a natural bijection \(\mathrm{Sp}_d(\mathbb{F}_q) \cong \mathrm{Sp}_d(\mathbb{F}_{q'})\) preserving isogeny classes. This allows to reduce the case \(a\) odd to \(q = p\) using Galois cohomology; Proposition 5.1 only works for \(q = p\). \[ \mathrm{Sp}_d(\mathbb{F}_p) \cong \mathrm{H}^1_{\text{ét}}(\mathbb{F}_p, \mathrm{Aut}(X_0 \otimes \overline{\mathbb{F}}_p)), \quad d > 1, \] where \(X_0\) is a fixed \(d\)-dimensional supersingular abelian variety over \(\mathbb{F}_p\). The method of computing the cardinality of \(\mathrm{Sp}_d(\mathbb{F}_p)\) is as follows: One divides the set of isomorphism classes of superspecial abelian varieties into isogeny classes and classifies them. Isogeny classes of supersingular elliptic curves are in bijection with supersingular Weil \(p\)-numbers by Honda-Tate theory (these have been studied by Deuring, see [\textit{W. C. Waterhouse}, Ann. Sci. Éc. Norm. Supér. (4) 2, 521--560 (1969; Zbl 0188.53001), Section 4], so isogeny classes of superspecial abelian varieties are in bijection with supersingular multiple Weil \(p\)-numbers. Proposition 4.4. Denote the class number of \(\mathbb{Q}(\sqrt{m})\) by \(h(\sqrt{m})\). (1) If \(a\) is odd, \[ |\mathrm{Sp}_1(\mathbb{F}_q)| = \begin{cases} 3, & p=2, \\ 4, & p=3, \\ h(\sqrt{-p}), &p \equiv 1 \pmod{4},\\ \big(3-(\frac{2}{p})\big)h(\sqrt{-p}), &p \equiv 3 \pmod{4},\quad p > 3. \end{cases} \] (2) If \(a\) is even, \[ |\mathrm{Sp}_1(\mathbb{F}_q)| = \frac{p-1}{6} + \frac{8}{3}\Big(1-\Big(\frac{-3}{p}\Big)\Big) + \frac{3}{2}\Big(1-\Big(\frac{-4}{p}\Big)\Big). \] Theorem 1.1. Let \(K_{m,j} = \mathbb{Q}(\sqrt{m},\sqrt{-j})\), \(w_m = 3[O^\times_{\mathbb{Q}(\sqrt{m})}:\mathbb{Z}[\sqrt{m}]^\times]^{-1}\) for \(m \equiv 1 \pmod{4}\) and let \(H(\sqrt{p})\) be the number of \(\mathbb{F}_p\)-isomorphism classes of abelian varieties in the simple isogeny class corresponding to the Weil \(p\)-number \(\pi = \sqrt{p}\) and let \(F = \mathbb{Q}(\sqrt{p})\). Then (1) \(H(\sqrt{p}) = 1,2,3\) for \(p=2,3,5\), respectively. (2) For \(p > 5\) and \(p \equiv 3 \pmod{4}\), we have \[ H(\sqrt{p}) = \frac{1}{2}h(F)\zeta_F(-1) + \Big(\frac{3}{8} + \frac{5}{8}\Big(2-\Big(\frac{2}{p}\Big)\Big)h(K_{p,1}) + \frac{1}{4}h(K_{p,2}) + \frac{1}{3}h(K_{p,3}), \] where \(\zeta_F(s)\) is the Dedekind zeta function of \(F\) and \(h\) is the class number. (3) For \(p > 5\) and \(p \equiv 1 \pmod{4}\), we have \[ H(\sqrt{p}) = \begin{cases} 8\zeta_F(-1)h(F) + h(K_{p,1}) + \frac{4}{3}h(K_{p,3}), & p \equiv 1 \pmod{8},\\ \frac{1}{2}(15w_p+1)\zeta_F(-1)h(F) + \frac{1}{4}(3w_p+1)h(K_{p,1}) + \frac{4}{3}h(K_{p,3}), & p \equiv 5 \pmod{8}. \end{cases} \] Theorem 1.2. We have \(|\mathrm{Sp}_2(\mathbb{F}_p)| = H(\sqrt{p}) + \Delta(p)\) with (1) \(\Delta(p) = 15,20,9\) for \(p = 2,3,5\), respectively. (2) For \(p > 5\) and \(p \equiv 1 \pmod{4}\), we have \[ \Delta(p) = (w_p+1)h(K_{p,3}) + h(K_{2p,1}) + h(K_{3p,3}) + h(\sqrt{-p}). \] (3) For \(p > 5\) and \(p \equiv 3 \pmod{4}\), we have \[ \Delta(p) = h(K_{p,3}) + h(K_{2p,1}) + (w_p+1)h(K_{3p,3}) + \Big(4-\Big(\frac{2}{p}\Big)\Big)h(\sqrt{-p}). \]
scientific article; zbMATH DE number 6674986