Orthogonal root numbers of tempered parameters

Root numbers

Fibonacci numbers and orthogonal polynomials

We prove that the sequence (1/Fn+2)n≥0 of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability, and we identify the orthogonal polynomials as little q-Jacobi polynomials with q = (1− √ 5)/(1+ √ 5). We prove that the corresponding kernel polynomials have integer coefficients, and from this we deduce that the inverse of the corresponding Hankel matrices (1/F...

The tempered spectrum of quasi-split classical groups III: The odd orthogonal groups

We continue our study of the tempered spectrum of quasi-split classical groups. Here we examine the case of the special orthogonal groups of odd dimension. While this is the last of the classical groups to be examined, it is the first for which our results address the tempered spectrum whose supercuspidal support is an arbitrary maximal parabolic subgroup, as we describe below. We continue to s...

Galois Invariance of Local Root Numbers

It is a standard remark that the group Aut(C) of abstract field automorphisms of C has uncountably many elements but only two that are are continuous. One consequence is that the analytic properties of a Dirichlet series ∑ n>1 a(n)n −s do not usually carry over to ∑ n>1 ι(a(n))n −s for ι ∈ Aut(C). Silly counterexamples abound: for instance, take a(n) = (1 − √ 2) and choose ι so that ι( √ 2) = −...

Least primitive root of prime numbers

Let p be a prime number. Fermat's little theorem [1] states that a^(p-1) mod p=1 (a hat (^) denotes exponentiation) for all integers a between 1 and p-1. A primitive root [1] of p is a number r such that any integer a between 1 and p-1 can be expressed by a=r^k mod p, with k a nonnegative integer smaller that p-1. If p is an odd prime number then r is a primitive root of p if and only if r^((p-...