Traces of powers of matrices over finite fields

Abstract

Let g be a random matrix distributed according to uniform probability measure on the finite general linear group GLn(Fq). We show that Tr(g k ) equidistributes on Fq as n → ∞ as long as log k = o(n 2 ) and that this range is sharp. We also show that nontrivial linear combinations of Tr(g 1 ), . . . , Tr(g k ) equidistribute as long as log k = o(n) and this range is sharp as well. Previously equidistribution of either a single trace or a linear combination of traces was only known for k ≤ cqn, where cq depends on q, due to work of the first-named author and Rodgers.We reduce the problem to exhibiting cancellation in certain short character sums in function fields. For the equidistribution of Tr(g k ) we end up showing that certain explicit character sums modulo T k+1 exhibit cancellation when averaged over monic polynomials of degree n in Fq[T ] as long as log k = o(n 2 ). This goes far beyond the classical range log k = o(n) due to Montgomery and Vaughan. To study these sums we build on the argument of Montgomery and Vaughan but exploit additional symmetry present in the considered sums.

Equidistribution of high traces of random matrices over finite fields and cancellation in character sums of high conductor

“…Theorem 1.4 is about 𝑝 𝑘 (𝑓) when we choose 𝑓 uniformly at random from  𝑛,𝑞 , while Theorems 1.1-1.3 are about 𝑝 𝑘 (𝑓) for a polynomial 𝑓 drawn from the space of possible characteristic polynomials of a matrix from GL 𝑛 (𝔽 𝑞 ) endowed with the uniform measure. As proved in [11,Theorem 1.4], the total variation distance of the law of the first 𝑘 next-to-leading coefficients of det(𝐼 𝑛 𝑇 − g) from the uniform distribution on 𝔽 𝑘 𝑞 tends to 0 as 𝑛 → ∞ for 𝑘 as large as 𝑛 − 𝑜(log 𝑛), so the setup of Theorem 1.4 is not that different from the setup of Theorems 1.1-1.3.…”

“…Let g ∈ GL 𝑛 (𝔽 𝑞 ) be an invertible 𝑛 × 𝑛 matrix over 𝔽 𝑞 chosen according to the uniform probability measure. The first author and Rodgers [11,Theorem 1.1] showed that Tr(g 𝑘 ) equidistributes in 𝔽 𝑞 as 𝑛 → ∞, uniformly for 𝑘 ⩽ 𝑐 𝑞 𝑛 for some sufficiently small 𝑐 𝑞 depending on 𝑞, and the rate of convergence is superexponential. However, nothing beyond 𝑘 = 𝑂(𝑛) was known.…”

“…Let $$g \in \mathrm{GL}_n(\mathbb {F}_q)$$ be an invertible $$n \times n$$ matrix over $$\mathbb {F}_q$$ chosen according to the uniform probability measure. The first author and Rodgers [11, Theorem 1.1] showed that $$\mathrm{Tr}(g^k)$$ equidistributes in $$\mathbb {F}_q$$ as $$n \rightarrow \infty$$, uniformly for $$k \leqslant c_q n$$ for some sufficiently small $$c_q$$ depending on $$q$$, and the rate of convergence is superexponential. However, nothing beyond $$k=O(n)$$ was known.…”

“…𝑛, P GL ⋅ 𝜉 𝐚,𝜓 )|. (2.7) In[11, Theorem 3.6], the first author and Rodgers showed that|𝑆(𝑛, P GL ⋅ 𝜉 𝐚,𝜓 )| ⩽ 𝑞 − 𝑛 22𝑘 +𝑂 𝑞 (𝑛) .Since P GL is a probability measure on  𝑗,𝑞 , the triangle inequality yields|𝑆(𝑛, P GL ⋅ 𝜉 𝐚,𝜓 )| ⩽ 𝑞 −𝑛 𝑛 ∑ 𝑖=0 |𝑆(𝑖, 𝜉 𝐚,𝜓 ⋅ 𝜒 0 )|.…”

On the distribution of equivalence classes of random symmetricp‐adic matrices

“…For an overview of random matrix theory over finite fields and in particular cycle index techniques see Fulman [24], and for an overview of rank problems and Stein's method, see Fulman and Goldstein [21]. One of the recent results exploring the similarities between theory over different fields is the work of Gorodetsky and Rodgers [27], on the finite field analogue of the fundamental result of Diaconis and Shahshahani [15] on the convergence of traces of unitary matrices to the normal distribution. Ideas used in [15] rely heavily on representation theory and symmetric function theory.…”