Related papers: Appell-Lerch Sums and N=2 Moduli
The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$…
Consider the moduli space of framed flat $U(2)$ connections with fixed odd determinant over a surface. Newstead combined some fundamental facts about this moduli space with the Mayer-Vietoris sequence to compute its betti numbers over any…
We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kaehler manifold X. These solutions are known to be related to polystable triples via a Kobayashi-Hitchin type…
We investigate nonregular elliptic problems with boundary conditions of higher orders. We prove that these problems are Fredholm on appropriate pairs of inner product H\"ormander spaces that form a two-sided refined Sobolev scale. We also…
We study the semilinear indefinite elliptic problem \[ -\Delta u = Q_\Omega |u|^{p-2}u \quad \text{in } \mathbb{R}^N, \] where $Q_\Omega = \chi_\Omega - \chi_{\mathbb{R}^N \setminus \Omega}$, $\Omega \subset \mathbb{R}^N$ is a bounded…
It has been conjectured that every algebraic curve may be defined either over its field of moduli or over an extension of degree two of it. In this paper we provide a negative answer to it by giving examples of hyperelliptic curves which…
We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with…
We study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all cases, the modular…
Genus two curves covering elliptic curves have been the object of study of many articles. For a fixed degree $n$ the subloci of the moduli space $\mathcal M_2$ of curves having a degree $n$ elliptic subcover has been computed for $n=3, 5$…
We show that elliptic curves with complex multiplication (CM) naturally emerge in the spectral geometry of Hermitian one-matrix models in the two-cut phase. Focusing on a symmetric quartic potential, we derive the corresponding genus-one…
For every normalized newform f in S_2(Gamma_1(N)) with complex multiplication, we study the modular parametrizations of elliptic curves C from the abelian variety A_f. We apply the results obtained when C is Gross's elliptic curve A(p).
We study Appell functions associated to an arbitrary positive definite lattice $\Lambda$ and a choice of $M\leq {\rm dim}(\Lambda)$ linearly independent vectors $d_r\in \Lambda$, $r=1,\dots,M$. These functions are instances of…
We provide a generalization of an algebraic linear combination for the trace of certain elliptic modular forms, and through specializing the expression at a suitable pair consisting of an elliptic curve over algebraic number fields and its…
We are concerned with fully nonlinear elliptic equations on complex manifolds and search for technical tools to overcome difficulties in deriving a priori estimates which arise due to the nontrivial torsion and curvature, as well as the…
For the system of semilinear elliptic equations \[ \Delta V_i = V_i \sum_{j \neq i} V_j^2, \qquad V_i > 0 \qquad \text{in $\mathbb{R}^N$} \] we devise a new method to construct entire solutions. The method extends the existence results…
We study natural D-modules on the moduli stack of elliptic curves over a field of characteristic zero. We use this to produce an algebro-geometric version of the algebra of higher depth mock modular forms, studied from a Conformal Field…
We consider second order elliptic divergence form systems with complex measurable coefficients $A$ that are independent of the transversal coordinate, and prove that the set of $A$ for which the boundary value problem with $L_2$ Dirichlet…
This note gives explicit equations for the elliptic curves (in characteristic not 2 or 3) with mod 2 representation isomorphic to that of a given one.
We study Severi curves parametrizing rational bisections of elliptic fibrations associated to general pencils of plane cubics. Our main results show that these Severi curves are connected and reduced, and we give an upper bound on their…
In this paper we study we study a Dirichlet optimal control prob- lem associated with a linear elliptic equation the coefficients of which we take as controls in the class of integrable functions. The characteristic feature of this control…