Pseudofinite difference fields and counting dimensions

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.

Pseudofinite difference fields

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we establish a partial connection between coarse dimension and transformal transcendence degree in these difference fields.

Computing the Trace of Frobenius

This chapter aims to compute the trace Tr(Frob-1 ¦H* (BunG(X);Zℓ)), where ℓ is a prime number which is invertible in F q. It follows the strategy outlined in Chapter 1. If X is an algebraic curve over the field C of complex numbers and G is a smooth affine group scheme over X whose fibers are semisimple and simply connected, then Theorem 1.5.4.10 (and Example 1.5.4.15) supply a quasi-isomorphism whose right-hand side is the continuous tensor product of Construction 1.5.4.8. The remainder of this chapter is devoted to explaining how Theorem 4.1.2.1 can be used to compute the trace of the arithmetic Frobenius automorphism on the ℓ-adic cohomology of BunG(X).

The Average of the Smallest Prime in a Conjugacy Class

Let C be a conjugacy class of $S_{n}$ and K an $S_{n}$-field. Let $n_{K,C}$ be the smallest prime, which is ramified or whose Frobenius automorphism Frob$_{p}$ does not belong to C. Under some technical conjectures, we show that the average of $n_{K,C}$ is a constant. We explicitly compute the constant. For $S_{3}$- and $S_{4}$-fields, our result is unconditional. Let $N_{K,C}$ be the smallest prime for which Frob$_{p}$ belongs to C. We obtain the average of $N_{K,C}$ under some technical conjectures. For n = 3 and C = [(12)], we have the average value of $N_{K,C}$ unconditionally.

NONCOMMUTATIVE POLYNOMIAL MAPS

Polynomial maps attached to polynomials of an Ore extension are naturally defined. In this setting we show the importance of pseudo-linear transformations and give some applications. In particular, factorizations of polynomials in an Ore extension over a finite field 𝔽q[t;θ], where θ is the Frobenius automorphism, are translated into factorizations in the usual polynomial ring 𝔽q[x].

GAUSSIAN LAWS ON DRINFELD MODULES

Let A = 𝔽q[T] be the polynomial ring over the finite field 𝔽q, k = 𝔽q(T) the rational function field, and K a finite extension of k. Let ϕ be a Drinfeld A-module over K of rank r. For a place 𝔓 of K of good reduction, write [Formula: see text], where [Formula: see text] is the valuation ring of 𝔓 and [Formula: see text] its maximal ideal. Let P𝔓, ϕ(X) be the characteristic polynomial of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ. Let χϕ(𝔓) = P𝔓, ϕ(1), and let ν(χϕ(𝔓)) be the number of distinct primes dividing χϕ(𝔓). If ϕ is of rank 2 with [Formula: see text], we prove that there exists a normal distribution for the quantity [Formula: see text] For r ≥ 3, we show that the same result holds under the open image conjecture for Drinfeld modules. We also study the number of distinct prime divisors of the trace of the Frobenius automorphism of 𝔽𝔓acting on a Tate module of ϕ and obtain similar results.

ON DIFFERENCE FIELDS WITH QUANTIFIER ELIMINATION

This paper proves that a difference field (E, σ) admits quantifier elimination if and only if E is an algebraically closed field, and σ is an integer power of the Frobenius automorphism.