Related papers: The shape of cubic fields
The geometry of algebraic curves over finite fields is a rich area of research. In previous work, the authors investigated a particular aspect of the geometry over finite fields of the classical unit circle, namely how the number of…
We compute and study two determinantal representations of the discriminant of a cubic quaternary form. The first representation is the Chow form of the $2$-uple embedding of $\mathbb{P}^3$ and is computed as the Pfaffian of the Chow form of…
The modularity of elliptic curves always intrigues number theorists. Recently, Thorne had proved a marvelous result that for a prime $ p $, every elliptic curve defined over a $ p $-cyclotomic extension of $ \mathbb{Q} $ is modular. The…
We prove a Cherednik style $p$-adic uniformization theorem for Shimura varieties associated to certain groups of unitary similitudes of size two over totally real fields. Our basic tool is the alternative modular interpretation of the…
It is shown that, under some mild technical conditions, representations of prime numbers by binary quadratic forms can be computed in polynomial complexity by exploiting Schoof's algorithm, which counts the number of $\mathbb F_q$-points of…
We construct extended Weil representations of unitary groups over finite fields geometrically, and show that they are Shintani lifts for Weil representations.
We extend to characteristic two recent results about isotropy of quadratic forms over function fields. In particular, we provide a characterization of function fields not only of quadratic forms but also more generally of polynomials in…
We define the QCD Husimi distribution as the phase space distribution of partons inside the nucleon. Compared to the more well-known Wigner distribution, the Husimi distribution is better behaved and positive. It thus allows for a…
This note contains a solution to the following problem: reconstruct the definition field and the equation of a projective cubic surface, using only combinatorial information about the set of its rational points. This information is encoded…
The pole shape optimization of an electromagnet typical of an MRI type application is investigated. We compare the use of different parameterizations of the pole shape and compare the convergence of the optimizations using a discrete…
We formulate Shintani's invariant in terms of the cyclic quantum dilogarithm. Building on earlier results that expressed Shintani's invariant using the $q$-Pochhammer symbol, we show how the cyclic quantum dilogarithm naturally arises in…
We study the bit complexity of two methods, related to the Euclidean algorithm, for computing cubic and quartic analogs of the Jacobi symbol. The main bottleneck in such procedures is computation of a quotient for long division. We give…
Using light-cone gauge formulation, massive arbitrary spin irreducible fields and massless (scalar and one-half spin) fields in three-dimensional flat space are considered. Both the integer spin and half-integer spin fields are studied. For…
We use Morris' theory of Eisenstein series for reductive groups over global function fields in order to extend Harder's computation of Tamagawa numbers to quasi-split groups.
We prove that any smooth cubic surface defined over any number field satisfies the lower bound predicted by Manin's conjecture possibly after an extension of small degree.
We generalize the explicit quadratic Chabauty techniques for integral points on odd degree hyperelliptic curves and for rational points on genus 2 bielliptic curves to arbitrary number fields using restriction of scalars. This is achieved…
Some upper bounds for the number of monogenizations of quartic orders are established by considering certain classical Diophantine equations, namely index form equations in quartic number fields, and cubic and quartic Thue equations.
We study a simple nonlinear model defined on the cubic lattice. We propose a bilinearization scheme for the field equations and demonstrate that the resulting system is closely related to the well-studied integrable models, such as the…
An elliptic orbifold is the quotient of an elliptic curve by a finite group. Eskin and Okounkov proved that generating functions for the number of branched covers of an elliptic curve with specified ramification are quasimodular forms for…
We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $|Q(x, y)| < \epsilon$, where $Q$ is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the…