Related papers: Determinant Expressions for Hyperelliptic Function…
We show that a Frobenius-semisimple Weil representation over a local field K is determined by its Euler factors over the extensions of K. The construction is explicit, and we illustrate it for l-adic representations attached to elliptic and…
The well-known fact that all elliptic curves are modular, proven by Wiles, Taylor, Breuil, Conrad and Diamond, leaves open the question whether there exists a 'nice' representation of the modular form associated to each elliptic curve. Here…
Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the…
We investigate the simplest class of hyperdeterminants defined by Cayley in the case of Hankel hypermatrices (tensors of the form $A_{i_1i_2... i_k}=f(i_1+i_2+...+i_k)$). It is found that many classical properties of Hankel determinants can…
By introducing a class of meromorphic functions with certain ramification structures on $\Bbb{CP}^1$, a new method for the determination of the Legendre representation of elliptic curves with complex multiplication is introduced. These…
In this paper we give improved, probably not sharp, upper bounds of the Hankel determinant of third order for various classes of univalent functions and conjecture the sharp one.
The identities for elliptic gamma functions discovered by A. Varchenko and one of us are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in 3-dimensional space. The language of stacks…
Explicit determinant formulas are presented for the $\tau$ functions of the generalized Painlev\'e equations of type $A$. This result allows an interpretation of the $\tau$-functions as the Pl\"ucker coordinates of the universal Grassmann…
This paper contains an account of arbitrary cubic function fields of characteristic three. We define a standard form for an arbitrary cubic curve and consider its function field. By considering an integral basis for the maximal order of…
Let $J$ be the Jacobian of a superelliptic curve defined by the equation $y^{\ell} = f(x)$, where $f$ is a separable polynomial of degree non-divisible by $\ell$. In this article we study the "exponential" (i.e. $\ell$-power) torsion of…
Highest-weight representations of infinite dimensional Lie algebras and Hilbert schemes of points are considered, together with the applications of these concepts to partition functions, which are most useful in physics. Partition functions…
We study the geometry and partial differential equations arising from the consideration of Frobenius determinants, also called-group-determinants. This leads us to address some aspects of twistor theory as well as some extensions of Bessel…
This paper, a continuation of [3], involves a closer study of polynomials of supertropical semirings and their version of tropical geometry in which we introduce the concept of relatively prime polynomials and resultants, with the aid of…
For the elliptic curve defined by the most general form $y^2 + (\mu_1 x + \mu_3) y = x^3 + \mu_2 x^2 + \mu_4 x + \mu_6$, we show the power series expansion of Weierstsass sigma function $\sigma(u)$ at the origin is of Hurwitz integral over…
We give an explicit formula for wild conductor exponents of plane curves over $\mathbb{Q}_p$ in terms of standard invariants of explicit extensions of $\mathbb{Q}_p$, generalising a formula for hyperelliptic curves. To do so, we prove a…
We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These…
The Hilbert function of a module over a positively graded algebra is of quasi-polynomial type (Hilbert--Serre). We derive an upper bound for its grade, i.e. the index from which on its coefficients are constant. As an application, we give a…
We give an analog of Frobenius' theorem about the factorization of the group determinant on the group algebra of finite abelian groups and we extend it into dihedral groups and generalized quaternion groups. Furthermore, we describe the…
Let $g \geq 1$ be an integer and let $A/\mathbb{Q}$ be an abelian variety that is isogenous over $\mathbb{Q}$ to %the product $E_1 \times \ldots \times E_g$ of elliptic curves $E_1/\mathbb{Q}$, $\ldots$, $E_g/\mathbb{Q}$, without complex…
A square matrix is $k$-Toeplitz if its diagonals are periodic sequences of period $k$. We find universal formulas for the determinant, the characteristic polynomial, some eigenvectors, and the entries of the inverse of any tridiagonal…