Let $z=x+iy \in \mathbb{H}:=\{z= x+ i y\in\mathbb{C}: y>0\}$ and $ \theta (s;z)=\sum_{(m,n)\in\mathbb{Z}^2 } e^{-s \frac{\pi }{y }|mz+n|^2}$ be the theta function associated with the lattice $\Lambda ={\mathbb Z}\oplus z{\mathbb Z}$. In this paper we consider the following pair of minimization problems \begin{equation}\aligned \min_{ \mathbb{H} } \theta (2;\frac{z+1}{2})+\rho\theta (1;z),\;\;\rho\in[0,\infty), \\ \min_{ \mathbb{H} } \theta (1; \frac{z+1}{2})+\rho\theta (2; z),\;\;\rho\in[0,\infty), \endaligned\end{equation} where the parameter $\rho\in[0,\infty)$ represents the competition of two intertwining lattices, the particular selection of the parameters $s=1,2$ is determined by the physical model, which can be generalized by our strategy and method proposed here. We find that as $\rho$ varies the optimal lattices admit a novel pattern: they move from rectangular (the ratio of long and short sides changes from $\sqrt3$ to 1 continuously), square, rhombus (the angle changes from $\pi/2$ to $\pi/3$ continuously) to hexagonal continuously; geometrically, up to an invariant group(a subgroup of the classical modular group), they move continuously on a special curve ; furthermore, there exists a closed interval of $\rho$ such that the optimal lattices is always a square lattice. This is the first, novel and also the complete result on the minimizer problem for theta functions with parameter $\rho$. This is in sharp contrast to optimal lattice shapes for single theta function ($\rho=\infty$ case), for which the hexagonal lattice prevails. As a consequence, we give a partial and positive answer to optimal lattice arrangements of vortices in competing systems of Bose-Einstein condensates as conjectured (and numerically and experimentally verified) by Mueller-Ho \cite{Mue2002}, this is the first progress on Mueller-Ho conjecture.

This paper provides a new construction of a supply of Darmon-like points on elliptic curves E defined over a quadratic extension of totally real number fields.

We study special cycles on integral models of Shimura varieties associated with unitary similitude groups of signature (n.1, 1).We construct an arithmetic theta lift from harmonic Maass forms of weight 2 . n to the arithmetic Chow group of the integral model of a unitary Shimura variety, by associating to a harmonic Maass form f a linear combination of Kudla– Rapoport divisors, equipped with the Green function given by the regularized theta lift of f . Our main result is an equality of two complex numbers: (1) the height pairing of the arithmetic theta lift of f against a CM cycle, and (2) the central derivative of the convolution L-function of a weight n cusp form (depending on f ) and the theta function of a positive definite hermitian lattice of rank n − 1. When specialized to the case n = 2, this result can be viewed as a variant of the Gross–Zagier formula for Shimura curves associated to unitary groups of signature (1, 1). The proof relies on, among other things, a new method for computing improper arithmetic intersections.

This project was sponsored through the Schiff Fellowship program of Brandeis University. This project involved using the power series method to construct a third order nonlinear ordinary differential equation, a Schwarzian equation, for each of the "genus zero" modular functions, described in the Conway-Norton paper. We first use the Borcherd recursion formuli to generate, in each case, a modular function up to whatever degree we desire, and then use the fact that there is a Schwarzian equation, determined by a single rational function we call a Q-value. By similar power series methods, we compute the coefficients of our rational function, and hence have all the necessary data to create a Schwarzian differential equation for each modular function. This equation can, in turn, be used to recover the modular function itself.

We call a finite set ${\mathcal{D}}\subset {\Bbb Z}^s$ a {\it (self-affine) tile digit set} with respect to an expanding integral matrix ${\bf A}$ if the self-affine set $T({\bf A}, \D)$ is a tile in ${\Bbb R}^s$. It has been a widely open problem to characterize the tile digit sets for a given ${\bf A}$. While there are substantial investigations on ${\Bbb R}$, there is no result on ${\Bbb R}^s$ other than the case where $|\det {\bf A}| =p$ with $p$ a prime. In this paper, we make an initiation to study a basic case ${\bf A} = p{\bf I}_2$ in ${\Bbb R}^2$. We characterize the tile digit sets by making use of the zeros of the mask polynomial of ${\mathcal{D}}$ associated with a tile criterion of Kenyon [K], together with a recent result of Iosevich {\it et al} on factorization of sets in ${\Bbb Z}_p \times {\Bbb Z}_p$ [IMP].