Mappings of least Dirichlet energy and their Hopf differentials

The paper is concerned with mappings between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in the domain) of the energy-minimal mappings is established within the class $\bar{\mathscr H}_2(X, Y)$ of strong limits of homeomorphisms in the Sobolev space $W^{1,2}(X, Y)$, a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation leads to the Hopf differential $h_z \bar{h_{\bar{z}}} dz \otimes dz$ and its trajectories. For a pair of doubly connected domains, in which $X$ has finite conformal modulus, we establish the following principle: A mapping $h \in \bar{\mathscr H}_2(X, Y)$ is energy-minimal if and only if its Hopf-differential is analytic in $X$ and real along the boundary of $X$. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in $X$. Nevertheless, cracks are triggered only by the points in the boundary of $Y$ where $Y$ fails to be convex. The general law of formation of cracks reads as follows: Cracks propagate along vertical trajectories of the Hopf differential from the boundary of $X$ toward the interior of $X$ where they eventually terminate before making a crosscut.Comment: 51 pages, 4 figure

Surpassing the Ratios Conjecture in the 1-level density of Dirichlet LL-functions

We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions in the family of all characters modulo $q$, with $Q/2 < q\leq Q$. For test functions whose Fourier transform is supported in $(-3/2, 3/2)$, we calculate this quantity beyond the square-root cancellation expansion arising from the $L$-function Ratios Conjecture of Conrey, Farmer and Zirnbauer. We discover the existence of a new lower-order term which is not predicted by this powerful conjecture. This is the first family where the 1-level density is determined well enough to see a term which is not predicted by the Ratios Conjecture, and proves that the exponent of the error term $Q^{-\frac 12 +\epsilon}$ in the Ratios Conjecture is best possible. We also give more precise results when the support of the Fourier Transform of the test function is restricted to the interval $[-1,1]$. Finally we show how natural conjectures on the distribution of primes in arithmetic progressions allow one to extend the support. The most powerful conjecture is Montgomery's, which implies that the Ratios Conjecture's prediction holds for any finite support up to an error $Q^{-\frac 12 +\epsilon}$.Comment: Version 1.2, 30 page

On the Ranks of the 2-Selmer Groups of Twists of a Given Elliptic Curve

We extend work of Swinnerton-Dyer on the density of the number of twists of a given elliptic curve that have 2-Selmer group of a particular rank

n-Harmonic mappings between annuli

The central theme of this paper is the variational analysis of homeomorphisms h\colon \mathbb X \onto \mathbb Y between two given domains $\mathbb X, \mathbb Y \subset \mathbb R^n$. We look for the extremal mappings in the Sobolev space $\mathscr W^{1,n}(\mathbb X,\mathbb Y)$ which minimize the energy integral $$\mathscr E_h=\int_{\mathbb X} ||Dh(x)||^n dx.$$ Because of the natural connections with quasiconformal mappings this $n$-harmonic alternative to the classical Dirichlet integral (for planar domains) has drawn the attention of researchers in Geometric Function Theory. Explicit analysis is made here for a pair of concentric spherical annuli where many unexpected phenomena about minimal $n$-harmonic mappings are observed. The underlying integration of nonlinear differential forms, called free Lagrangians, becomes truly a work of art.Comment: 120 pages, 22 figure