Divisibility by 2-Powers of Certain Quadratic Class Numbers

AbstractWe study the divisibility of the strict class numbers of the quadratic fields of discriminant 8p, −8p, and −4p by powers of 2 for p ≡ 1 mod 4 a prime number. Various criteria for divisibility by 8 are discussed, and an analogue of the relation 8|h+8p ↔ 8|h−8p and 8|h−4p is given for divisibility by 16. We present numerical data related to the known and conjectured densities of primes p giving rise to specific 2-power divisibilities

Über das Fortsetzen von Bewertungen in vollständigen Korpern

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Character sums for primitive root densities

It follows from the work of Artin and Hooley that, under assumption of the generalized Riemann hypothesis, the density of the set of primes $q$ for which a given non-zero rational number $r$ is a primitive root modulo $q$ can be written as an infinite product $\prod_p \delta_p$ of local factors $\delta_p$ reflecting the degree of the splitting field of $X^p-r$ at the primes $p$, multiplied by a somewhat complicated factor that corrects for the `entanglement' of these splitting fields. We show how the correction factors arising in Artin's original primitive root problem and some of its generalizations can be interpreted as character sums describing the nature of the entanglement. The resulting description in terms of local contributions is so transparent that it greatly facilitates explicit computations, and naturally leads to non-vanishing criteria for the correction factors. The method not only applies in the setting of Galois representations of the multiplicative group underlying Artin's conjecture, but also in the GL$_2$-setting arising for elliptic curves. As an application, we compute the density of the set of primes of cyclic reduction for Serre curves.Comment: 23 pages. This version is to appear in the Mathematical Proceedings of the Cambridge Philosophical Societ

Primes of degree one and algebraic cases of fi Cebotarev's theorem

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Singular values of some modular functions

We study the properties of special values of the modular functions obtained from Weierstrass P-function at imaginary quadratic points.Comment: 19 pages,corrected typo

The road to deterministic matrices with the restricted isometry property

The restricted isometry property (RIP) is a well-known matrix condition that provides state-of-the-art reconstruction guarantees for compressed sensing. While random matrices are known to satisfy this property with high probability, deterministic constructions have found less success. In this paper, we consider various techniques for demonstrating RIP deterministically, some popular and some novel, and we evaluate their performance. In evaluating some techniques, we apply random matrix theory and inadvertently find a simple alternative proof that certain random matrices are RIP. Later, we propose a particular class of matrices as candidates for being RIP, namely, equiangular tight frames (ETFs). Using the known correspondence between real ETFs and strongly regular graphs, we investigate certain combinatorial implications of a real ETF being RIP. Specifically, we give probabilistic intuition for a new bound on the clique number of Paley graphs of prime order, and we conjecture that the corresponding ETFs are RIP in a manner similar to random matrices.Comment: 24 page