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

We establish an explicit lower bound for the N\'eron-Tate height on elliptic curves with complex multiplication, for nontorsion points defined over the maximal abelian extension of a number field. Building on a strategy developed by…

Let $X$ be a simply connected space with finite-dimensional rational homotopy groups. Let $p_\infty \colon UE \to \mathrm{Baut}_1(X)$ be the universal fibration of simply connected spaces with fibre $X$. We give a DG Lie model for the…

Starting from computer experiments with the fundamental group of the Cartwright--Steger surface, we construct an infinite tower $(X_n)_{n\ge 1}$ of normal projective surfaces obtained by successive $\mathbb Z/3$-Galois covers $X_{n}\to…

We study the orchard problem on cubic surfaces. We classify possibly reducible cubic surfaces $X\subseteq \mathbb{P}^3(\C)$ with smooth components on which there exist families of finite sets (of unbounded size) with quadratically many…

Let $K$ be an imaginary quadratic field and $p$ be an odd prime number. Let $E/\mathbb{Q}$ be an elliptic curve with good ordinary reduction at $p$. We study the Iwasawa theory of $E$ over the anticyclotomic $\mathbb{Z}_p$-extension of $K$…

The enhanced power graph of a finite group $G$, denoted by $\mathcal{P}_E(G)$, is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this…

The interplay between equivariant stable homotopy theory and spectral algebraic geometry is used to construct a derived Tate curve over $\mathrm{KU}((q))$, a lift of the classical elliptic curve of Tate over $\mathbf{Z}((q))$. Applications…

We construct a new family of flat connections generalising the KZ connection, the Casimir connection and the dynamical connection. These new connections are attached to simply-laced graphs, and are obtained via quantisation of…

The 102581 flat toric elliptic fibrations over P^2 are identified among the Calabi-Yau hypersurfaces that arise from the 473800776 reflexive 4-dimensional polytopes. In order to analyze their elliptic fibration structure, we describe the…

Let $E/\mathbb{Q}$ be an elliptic curve, let $\overline{\mathbb{Q}}$ be a fixed algebraic closure of $\mathbb{Q}$, and let $G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. The…

Let $K$ be a number field and let $E$ be an elliptic curve defined and of rank one over $K$. For a set $\calW_K$ of primes of $K$, let $O_{K,\calW_K}=\{x\in K: \ord_{\pp}x \geq 0, \forall \pp \not \in \calW_K\}$. Let $P \in E(K)$ be a…

We analyze general F-theory compactifications with U(1) x U(1) x U(1) Abelian gauge symmetry by constructing the general elliptically fibered Calabi-Yau manifolds with a rank three Mordell-Weil group of rational sections. The general…

Starting with a $\mathbb{C}^*$-valued cocycle on the global quotient orbifold $X // G$, we apply transgression techniques for 2-gerbes, as developed by Lupercio and Uribe, to construct a gerbe on the orbifold loop space $\mathcal{L}(X//G)$.…

We investigate a notion of "higher modularity" for elliptic curves over function fields. Given such an elliptic curve $E$ and an integer $r\geq 1$, we say that $E$ is $r$-modular when there is an algebraic correspondence between a stack of…

For a simple linear algebraic group $G$, the chiral universal centralizer $\mathbf{I}_{G,k}$ is a vertex operator algebra, which is the chiralization of the universal centralizer $\mathfrak{Z}_G$. The variety $\mathfrak{Z}_G$ is identified…

Given a local ring $(R,\mathfrak{m})$ and an elliptic curve $E(R/\mathfrak{m})$, we define elliptic loops as the points of $\mathbb{P}^2(R)$ projecting to $E$ under the canonical modulo-$\mathfrak{m}$ reduction, endowed with an operation…

The category of rational mixed Hodge-Tate structures is a mixed Tate category. So thanks to the Tannakian formalism, it is equivalent to the category of finite dimensional graded comodules over a graded commutative Hopf algebra H over Q.…

Building upon the Bloch-Kato conjecture in Milnor K-theory, we relate the third unramified cohomology group with Q/Z coefficients with a group which measures the failure of the integral Hodge conjecture in degree 4. As a first consequence,…

Let $M$ be a closed manifold and $\alpha : \pi_1(M)\to U_n$ a representation. We give a purely $K$-theoretic description of the associated element $[\alpha]$ in the $K$-theory of $M$ with $\R/\Z$-coefficients. To that end, it is convenient…

In this article, a novel description of the hypergeometric differential equation found from Gel'fand-Kapranov-Zelevinsky's system (referred to GKZ equation) for Givental's $J$-function in the Gromov-Witten theory will be proposed. The GKZ…