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Let $A$ be an abelian variety over $mathbb{Q}$ of dimension $g$ such that the image of its associated absolute Galois representation $rho_A$ is open in $operatorname{GSp}_{2g}(hat{mathbb{Z}})$. We investigate the arithmetic of the traces $a_{1, p}$ of the Frobenius at $p$ in $operatorname{Gal}(overline{mathbb{Q}}/mathbb{Q})$ under $rho_A$, modulo varying primes $p$. In particular, we obtain upper bounds for the counting function $#{p leq x: a_{1, p} = t}$ and we prove an Erdos-Kac type theorem for the number of prime factors of $a_{1, p}$. We also formulate a conjecture about the asymptotic behaviour of $#{p leq x: a_{1, p} = t}$, which generalizes a well-known conjecture of S. Lang and H. Trotter from 1976 about elliptic curves.
We compute some arithmetic path integrals for BF-theory over the ring of integers of a totally imaginary field, which evaluate to natural arithmetic invariants associated to $mathbb{G}_m$ and abelian varieties.
We exhibit a probabilistic algorithm which computes a rational point of an absolutely irreducible variety over a finite field defined by a reduced regular sequence. Its time--space complexity is roughly quadratic in the logarithm of the cardinality o
If $pi: Y to X$ is an unramified double cover of a smooth curve of genus $g$, then the Prym variety $P_pi$ is a principally polarized abelian variety of dimension $g-1$. When $X$ is defined over an algebraically closed field $k$ of characteristic $p$
Suppose ( rho_1 ) and ( rho_2 ) are two pure Galois representations of the absolute Galois group of a number field $K$ of weights ( k_1 ) and ( k_2 ) respectively, having equal normalized Frobenius traces ( Tr(rho_1(sigma_v)) /Nv^{k_1/2}) and ( Tr(rh
In this paper, we classified the surfaces whose canonical maps are abelian covers over $mathbb{P}^2$. Moveover, we construct a new Campedelli surface with fundamental group $mathbb{Z}_2^{oplus 3}$ and give defining equations for Persssons surface and