Related papers: Binary Forms and the Hyperelliptic Superstring Ans…
We study a proposal of D'Hoker and Phong for the chiral superstring measure for genus three. A minor modification of the constraints they impose on certain Siegel modular forms leads to a unique solution. We reduce the problem of finding…
We complete the program indicated by the Ansatz of D'Hoker and Phong in genus ~4 by proving the uniqueness of the restriction to Jacobians of the weight 8 Siegel cusp forms satisfying the Anstaz. We prove $\dim [\Gamma_4(1,2),8]_0=2$ and…
Very recently, Grushevsky continued D'Hoker and Phong's program of finding the chiral superstring measure from first principles by constructing modular forms satisfying certain factorization constraints. He has proposed an ansatz in genus 4…
The finite symplectic group Sp(2g) over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this…
A main issue in superstring theory are the superstring measures. D'Hoker and Phong showed that for genus two these reduce to measures on the moduli space of curves which are determined by modular forms of weight eight and the bosonic…
Waring problem for forms is important and classical in mathematics. It has been widely investigated because of its wide applications in several areas. In this paper, we consider the Waring problem for binary forms with complex coefficients.…
In this paper we continue the program pioneered by D'Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain…
Let $F$ be an irreducible binary form attached to a number field $K$ of degree $\geq 3$. Let $\epsilon\not\in \{-1, 1\}$ be a totally real unit of $K$. By twisting $F$ with the powers $\epsilon^a$ of $\epsilon$, ($a\in{\mathbf Z}$), we…
Let $p$ be an unramified prime in a totally real field $L$ such that $h^+(L)=1$. Our main result shows that Hilbert modular newforms of parallel weight two for $\Gamma_0(p)$ can be constructed naturally, via classical theta series, from…
We present an algebraic theory of orthogonal polynomials in several variables that includes classical orthogonal polynomials as a special case. Our bottom line is a straightforward connection between apolarity of binary forms and the inner…
We develop a new strategy for studying low weight specializations of $p$-adic families of ordinary modular forms. In the elliptic case, we give a new proof of a result of Ghate--Vatsal which states that a Hida family contains infinitely…
In this paper we describe how representation theory of groups can be used to shorten the derivation of two loop partition functions in string theory, giving an intrinsic description of modular forms appearing in the results of D'Hoker and…
We extend our previous results on the number of integers which are values of some cyclotomic form of degree larger than a given value (see \cite{FW1}), to more general families of binary forms with integer coefficients. Our main ingredient…
The low energy expansion of Type II superstring amplitudes at genus one is organized in terms of modular graph functions associated with Feynman graphs of a conformal scalar field on the torus. In earlier work, surprising identities between…
We classify four dimensional $\mathcal{N}=2$ SCFTs whose Seiberg-Witten (SW) geometries can be written as hyperelliptic families. By using special K\"ahler condition of SW geometry, we reduce the problem to one parameter quasi-homogeneous…
After a brief introduction to the classical theory of binary quadratic forms we use these results for proving (most of) the claims made by P\'epin in a series of articles on unsolvable quartic diophantine equations, and for constructing…
For a field of characteristic $\ne 2$ we study vector spaces that are graded by the weight lattice of a root system, and are endowed with linear operators in each simple root direction. We show that these data extend to a graded semisimple…
We elucidate the correspondence between a particular class of superconformal field theories in six dimensions and homomorphisms from discrete subgroups of $SU(2)$ into $E_8$, as predicted from string dualities. We show how this match works…
Following the new gauging fixing method of D'Hoker and Phong, we study two-loop superstrings in hyperelliptic language. By using hyperelliptic representation of genus 2 Riemann surface we derive a set of identities involving the Szeg\"o…
In this paper, we study contragredient duals and invariant bilinear forms for modular vertex algebras (in characteristic $p$). We first introduce a bialgebra $\mathcal{H}$ and we then introduce a notion of $\mathcal{H}$-module vertex…