Isogeny class and Frobenius root statistics for abelian varieties over finite fields (Q2732624)

Let \(A\) be an abelian variety of dimension \(g\) over a number field \(k\). Let \(v\) be a finite place of \(k\) of good reduction for \(A\). Then by standard conjectures, the characteristic polynomial of the Frobenius at \(v\), acting on the \(\ell\)-adic Tate module of \(A\), does not depend on \(\ell\), and its roots have absolute value \(q_v^{1/2}\), where \(q_v\) is the size of the residue class field. The eigenvalues, multiplied by \(q_v^{-1/2}\), then can be specified uniquely as \(\{e^{\pi i \theta_1}, e^{-\pi i \theta_1}, \dots, e^{\pi i \theta_g}, e^{- \pi i \theta_g}\}\) with \(0 \leq \theta_1 \leq \dots \leq \theta_g \leq 1\). Write \(\Theta_{A,v} = \theta(A_{k_v}) = (\theta_1, \dots, \theta_g) \in \Sigma_g\) with \(\Sigma_g = \{\theta \in {\mathbb R}^g :0\leq \theta_1 \leq \dots \leq \theta_g \leq 1\}\). NEWLINENEWLINENEWLINEThe Sato-Tate conjecture for \(A\) then asserts that the family \(\Theta_A = \bigcup_v A_{A,v}\) is uniformly distributed in \(\Sigma_g\) with respect to the probability measure induced by the Haar measure of \({\text{USP}}_{2g}({\mathbb C})\), explicitly NEWLINE\[NEWLINE\mu_{ST}(g)=2^{g^2} \left(\prod_{j<k} (\cos \pi\theta_k - \cos \pi\theta_j)^2\right) \prod_i (\sin^2(\pi\theta_i) d\theta_i).NEWLINE\]NEWLINE Write NEWLINE\[NEWLINE\Theta_q = \{\theta(A) : A/{\mathbb F}_q {\text{ up to isomorphism}}\}NEWLINE\]NEWLINE (as a family, i.e., with multiplicities). In the case of elliptic curves, the assertion is known when one replaces \(\Theta_A\) by the family \(\bigcup_p \Theta_p\) [\textit{B. Birch}, J. Lond. Math. Soc. 43, 57-60 (1968; Zbl 0183.25503)] or by \(\bigcup_m \Theta_{p^m}\) (\textit{H. Yoshida} [Invent. Math. 19, 261-177 (1973; Zbl 0292.14011)], \textit{V. K. Murty} [Progr. Math. 26, 195-205 (1982; Zbl 0526.14011)]). NEWLINENEWLINENEWLINEIn the paper under review, a similar result is proved with respect to \textit{isogeny classes} of abelian varieties instead of isomorphism classes. Write \(\Xi_{g,q} = \{\theta(A) : A {\mathbb F}_q{ \text{ up to isogeny}}\}\) and \(\Xi_g = \bigcup_q \Xi_{g,q} \subset \Sigma_g\). Then \(\Xi_g\) is uniformly distributed in \(\Sigma_g\) with respect to the probability measure NEWLINENEWLINE\qquad \(\upsilon_g^{-1} \left(\prod_{j<k} (\cos \pi\theta_j - \cos \pi\theta_k)\right) \prod_i (\sin(\pi\theta_i) d\theta_i)\) NEWLINENEWLINENEWLINEwith \(\upsilon_g = (2^g/g!) \prod_{j=1}^g (2j/(2j-1))^{g+1-j}\). The author points out that, together with heuristic arguments due to E. Howe, this would imply uniform distribution of \(\Theta_g\) with respect to a measure different from \(\mu_{ST}\), if \(g > 1\). NEWLINENEWLINENEWLINEThe other main result in the paper is about the number of isogeny classes of abelian varieties (of dimension \(g \geq 2\)) over \({\mathbb F}_q\) with a given number of \({\mathbb F}_q\)-rational points. The author proves that for \(q \to \infty\), this distribution (suitably normalized) tends to a continuous limit measure, which is piecewise given by a polynomial of degree at most \((g-1)(g+2)/2\). For \(g = 2\) and \(g = 3\), these measures are given explicitly. NEWLINENEWLINENEWLINEThe proofs build upon results of \textit{S. A. DiPippo} and \textit{E. Howe} [J. Number Theory 73, 426-450 (1998; Zbl 0931.11023), Corrigendum in J. Number Theory 83, 182 (2000)].