105 research outputs found
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 for which
a given non-zero rational number is a primitive root modulo can be
written as an infinite product of local factors
reflecting the degree of the splitting field of at the primes ,
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-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
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