15 research outputs found
An introduction to decoupling and harmonic analysis over
The goal of this expository paper is to provide an introduction to decoupling
by working in the simpler setting of decoupling for the parabola over
. Over , commonly used heuristics in decoupling are
significantly easier to make rigorous over than over
and such decoupling theorems over are still strong
enough to derive interesting number theoretic conclusions.Comment: 26 pages; revised version incorporating referee comment
Arithmetic Properties of Picard-Fuchs Equations and Holonomic Recurrences
The coefficient series of the holomorphic Picard-Fuchs differential equation
associated with the periods of elliptic curves often have surprising
number-theoretic properties. These have been widely studied in the case of the
torsion-free, genus zero congruence subgroups of index 6 and 12 (e.g. the
Beauville families). Here, we consider arithmetic properties of the
Picard-Fuchs solutions associated to general elliptic families, with a
particular focus on the index 24 congruence subgroups. We prove that elliptic
families with rational parameters admit linear reparametrizations such that
their associated Picard-Fuchs solutions lie in Z[[t]]. A sufficient condition
is given such that the same holds for holomorphic solutions at infinity. An
Atkin-Swinnerton-Dyer congruence is proven for the coefficient series attached
to \Gamma_1(7). We conclude with a consideration of asymptotics, wherein it is
proved that many coefficient series satisfy asymptotic expressions of the form
u_n \sim \ell \lambda^n/n. Certain arithmetic results extend to the study of
general holonomic recurrences.Comment: 21 pages, to appear in the Journal of Number Theor
Strichartz inequalities: some recent developments
Strichartz inequalities, originating from Fourier restriction theory, play a
central role in the analysis of dispersive partial differential equations. They
serve as a cornerstone for many subsequent developments. We survey some of them
in memory of Strichartz, highlighting connections to recent developments in
Fourier decoupling.Comment: 23 pages, expository pape
A bilinear proof of decoupling for the cubic moment curve
Using a bilinear method that is inspired by the method of efficient
congruencing of Wooley [Woo16], we prove a sharp decoupling inequality for the
moment curve in .Comment: 28 pages; revised version incorporating referee comment
A short proof of decoupling for the moment curve
We give a short and elementary proof of the decoupling inequality
for the moment curve in , using a bilinear approach inspired by
the nested efficient congruencing argument of Wooley (arXiv:1708.01220).Comment: v2: 10 pages, minor correction