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تسجيل مستخدم جديدCounting abelian varieties over finite fields via Frobenius densities
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Let $[X,lambda]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor $ u_v([X,lambda])$ for each place $v$ of $mathbb Q$, and show that the product of these factors essentially computes the size of the isogeny class of $[X,lambda]$. The derivation of this mass formula depends on a formula of Kottwitz and on analysis of measures on the group of symplectic similitudes, and in particular does not rely on a calculation of class numbers.
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These are notes of my lectures at the summer school Higher-dimensional geometry over finite fields in Goettingen, June--July 2007. We present a proof of Tates theorem on homomorphisms of abelian varieties over finite fields (including the $ell=p$ c
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In the paper [On superspecial abelian surfaces over finite fields II. J. Math. Soc. Japan, 72(1):303--331, 2020], Tse-Chung Yang and the first two current authors computed explicitly the number $lvert mathrm{SSp}_2(mathbb{F}_q)rvert$ of isomorphism c
lasses of superspecial abelian surfaces over an arbitrary finite field $mathbb{F}_q$ of even degree over the prime field $mathbb{F}_p$. There it was assumed that certain commutative $mathbb{Z}_p$-orders satisfy an etale condition that excludes the primes $p=2, 3, 5$. We treat these remaining primes in the present paper, where the computations are more involved because of the ramifications. This completes the calculation of $lvert mathrm{SSp}_2(mathbb{F}_q)rvert$ in the even degree case. The odd degree case was previous treated by Tse-Chung Yang and the first two current authors in [On superspecial abelian surfaces over finite fields. Doc. Math., 21:1607--1643, 2016]. Along the proof of our main theorem, we give the classification of lattices over local quaternion Bass orders, which is a new input to our previous works.
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