Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Split abelian surfaces over finite fields and reductions of genus-2
  curves   For prime powers q, let s(q) denote the probability that a randomly-chosen
principally-polarized abelian surface over the finite field F_q is not simple.
We show that there are positive constants B and C such that for all q,
  B (log q)^{-3}(log log q)^{-4} < s(q)sqrt(q) < C (log q)^4(log log q)^2,
  and we obtain better estimates under the assumption of the generalized
Riemann hypothesis. If A is a principally-polarized abelian surface over a
number field K, let pi_split(A/K, z) denote the number of prime ideals p of K
of norm at most z such that A has good reduction at p and A_p is not simple. We
conjecture that for sufficiently general A, the counting function pi_split(A/K,
z) grows like sqrt(z)/log z. We indicate why our theorem on the rate of growth
of s(q) gives us reason to hope that our conjecture is true.

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The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol