Related papers: The Universal Elliptic KZB Connection in Higher Le…

Let $X\rightarrow {\mathbb P}^1$ be an elliptically fibered $K3$ surface, admitting a sequence $\omega_{i}$ of Ricci-flat metrics collapsing the fibers. Let $V$ be a holomorphic $SU(n)$ bundle over $X$, stable with respect to $\omega_i$.…

We introduce a flat version of the KZB connection. This connection is defined on the complement of the locus of Weierstrass points on the moduli space of genus $g$ complex curves with marked points. We then give integral formulas for flat…

In the physics literature, Bilal--Fock--Kogan \cite{BFK} introduced the idea of parabolic reduced flat connections on a surface to give a geometric origin to $W$-algebras. In this paper, we combine these ideas with higher complex…

Let M_m be the generic fibre of the formal deformation space of a one-dimensional formal module X of finite height together with level-m-structures. We show that it is defined over the mth Lubin-Tate extension of the ground field and that…

The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb…

We present a 2-dimensional Peano continuum $\mathbb{T}\subseteq \mathbb{R}^3$ with the following properties: (1) There is a universal covering projection $q:\overline{\mathbb{T}}\rightarrow \mathbb{T}$ with uncountable fundamental group…

An elliptic orbifold is the quotient of an elliptic curve by a finite group. In 2001, Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular…

For a non-CM elliptic curve $E$ defined over a number field $K$, the Galois action on its torsion points gives rise to a Galois representation $\rho_E: Gal(\overline{K}/K)\to GL_2(\widehat{\mathbb{Z}})$ that is unique up to isomorphism. A…

Collino proved that the fundamental group of a certain Zariski open set of the symmetric square of a hyperelliptic curve is isomorphic to the integral Heisenberg group. We compute the mixed Hodge structure on this fundamental group, and…

Let $k$ be a field with separable closure $\bar{k}\supset k$, and let $X$ be a qcqs $k$-scheme. We use the theory of profinite Galois categories developed by Barwick-Glasman-Haine to provide a quick conceptual proof that the sequences…

We show that for an elliptic curve E defined over a number field K, the group E(A) of points of E over the adele ring A of K is a topological group that can be analyzed in terms of the Galois representation associated to the torsion points…

We construct an explicit bundle with flat connection on the configuration space of n points of a complex curve. This enables one to recover the `formality' isomorphism between the Lie algebra of the prounipotent completion of the pure braid…

Using the theory of Diophantine m-tuples, i.e. sets with the property that the product of its any two distinct elements increased by 1 is a perfect square, we construct an elliptic curve over Q(t) of rank at least 4 with three non-trivial…

Tate cohomology (as well as Borel homology and cohomology) of connective K-theory for $G=(\mathbb{Z}/2)^n$ was completely calculated by Bruner and Greenlees. In this note, we essentially redo the calculation by a different, more elementary…

Suppose $\mathcal{E} \to B$ is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve $B$. Let $k$ denote the function field $\overline{\mathbb{Q}}(B)$ and $E$ the associated elliptic curve over $k$. In…

We argue that the level-$1$ elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$ is a dynamical symmetry realized as a part of 2d/5d correspondence where the Drinfeld currents are the screening currents to the $q$-Virasoro/W block in the 2d…

In this paper, we establish the modularity of every elliptic curve $E/F$, where $F$ runs over infinitely many imaginary quadratic fields, including $\mathbb{Q}(\sqrt{-d})$ for $d=1,2,3,5$. More precisely, let $F$ be imaginary quadratic and…

Let G be an n-dimensional crystallographic group (n-space group). If G is a Z-reducible, then the flat n-orbifold E^n/G has a nontrivial fibered orbifold structure. We prove that this structure can be described by a generalized Calabi…

Let $k$ be a field, $\tilde{G}$ a connected reductive $k$-group, and $\Gamma$ a finite group. In a previous work, the authors defined what it means for a connected reductive $k$-group $G$ to be "parascopic" for $(\tilde{G},\Gamma)$.…

The paper introduces a new geometric interpretation of the quantum Knizhnik-Zamolodchikov equations introduced in 1991 by I.Frenkel and N.Reshetikhin. It turns out that these equations can be linked to certain holomorphic vector bundles on…