Hilbert 90 for KnM

This chapter formulates a norm-trace relation for the Milnor 𝐾-theory and étale cohomology of a cyclic Galois extension, herein called Hilbert 90 for 𝐾𝑀 𝑛. To begin, the chapter uses condition BL(n) to establish a related exact sequence in Galois cohomology. It then establishes that condition BL(n − 1) implies the particular case of condition H90(n) for 𝓁-special fields 𝑘 such that 𝐾𝑀 𝑛(𝑘) is 𝓁-divisible. This case constitutes the first part of the inductive step in the proof of Theorem A. The remainder of this chapter explains how to reduce the general case to this particular one. The chapter concludes with some background on the Hilbert 90 for 𝐾𝑀 𝑛.

ON THE DIMENSION OF PERMUTATION VECTOR SPACES

Let $K$ be a field that admits a cyclic Galois extension of degree $n\geq 2$. The symmetric group $S_{n}$ acts on $K^{n}$ by permutation of coordinates. Given a subgroup $G$ of $S_{n}$ and $u\in K^{n}$, let $V_{G}(u)$ be the $K$-vector space spanned by the orbit of $u$ under the action of $G$. In this paper we show that, for a special family of groups $G$ of affine type, the dimension of $V_{G}(u)$ can be computed via the greatest common divisor of certain polynomials in $K[x]$. We present some applications of our results to the cases $K=\mathbb{Q}$ and $K$ finite.

On the lower bound for diameter of commuting graph of prime-square sized matrices

It is known that the diameter of commuting graph of n-by-n matrices is bounded above by six if the graph is connected. In the commuting graph of p2-by-p2 matrices over a sufficiently large field which admits a cyclic Galois extension of degree p2 we construct two matrices at distance at least five. This shows that five is the lower bound for its diameter. Our results are applicable for all sufficiently large finite fields as well as for the field of rational numbers.

FACTORIZATION OF AUTOMORPHIC L-FUNCTIONS AND THEIR ZERO STATISTICS

In this paper we define a Rankin–Selberg L-function attached to two Galois invariant automorphic cuspidal representations of GL m(𝔸E) and GL m′(𝔸F) over cyclic Galois extensions E and F of prime degree. This differs from the classical case in that the two extension fields E and F could be completely unrelated to one another, and we exploit the existence of the automorphic induction functor over cyclic extensions (see [J. Arthur and L. Clozel, Simple Algebras, Base Change, and the Advanced Theory of the Trace Formula, Annals of Mathematics Studies, No. 120 (Princeton University Press, Princeton, NJ, 1989)]) to define the L-function. Using a result proved by C. S. Rajan, we prove a prime number theorem for this L-function, and proceed to calculate the n-level correlation function of high nontrivial zeros of a product L(s, π1)L(s, π2)…L(s, πk) where πi is a Galois invariant cuspidal representation of GL ni(𝔸Fi) with Fi a cyclic Galois extension of prime degree ℓi for i = 1,…,k, thus generalizing the results of Liu and Ye [Functoriality of automorphic L-functions through their zeros, Sci. China Ser. A51(1) (2008) 1–16].