Related papers: The shape of cubic fields
We prove the thin set version of Manin's conjecture for the chordal (or: determinantal) cubic fourfold, which is the secant variety of the Veronese surface. We reduce this counting problem to a result of Schmidt for quadratic points in the…
The aim of the research presented in this paper is to derive the systems of ordinary differential equations (ODEs) satisfied by modular forms of level six and to construct extensions of the differential field of the cubic theta functions,…
We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy,…
Every conformal field theory has the symmetry of taking each field to its adjoint. We consider here the quotient (orbifold) conformal field theory obtained by twisting with respect to this symmetry. A general method for computing such…
Roth's theorem is extended to finitely generated field extensions of $\Bbb Q$, using Moriwaki's framework for heights.
We study equivariant birationality from the perspective of derived categories. We produce examples of nonlinearizable but stably linearizable actions of finite groups on smooth cubic fourfolds.
In this exploratory article, we present a constructive method for scattering points on the surface of $d$ dimensional spheres which we believe is new and of interest. Indeed, the problem of uniformly distributing points on spheres is an…
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.
Given a number field, it is an important question in algorithmic number theory to determine all its subfields. If the search is restricted to abelian subfields, one can try to determine them by using class field theory. For this, it is…
We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations…
For any integer $k\ge 1$, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order $2^k$. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class…
We study space-time symmetries in scalar quantum field theory (including interacting theories) on static space-times. We first consider Euclidean quantum field theory on a static Riemannian manifold, and show that the isometry group is…
The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito-Ito-Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between…
In this paper we give a new proof of the Ohno-Nakagawa Theorem using the techniques of $L$-series. By applying Eisenstein's parametrization of binary cubic forms on the one hand, and a class field theory interpretation of Datskovsky \&…
In this paper we use tools from set theory and the uncountable categoricity of Zilber's pseudo-exponential field to show that Zilber's field is isomorphic to the complex field with (standard) exponentiation and hence Schanuel's conjecture…
The moduli space of cubic surfaces in complex projective space is known to be isomorphic to the quotient of the complex 4-ball by a certain arithmetic group. We apply Borcherds' techniques to construct automorphic forms for this group and…
In this paper, we give a dimension formula for spaces of Siegel cusp forms of general degree with respect to neat arithmetic subgroups. The formula was conjectured before by several researchers. The dimensions are expressed by special…
We present a new structure theorem for finite fields of odd order that relates multiplicative and additive structure in an interesting way. This theorem has several applications, including an improved understanding of Dickson and Chebyshev…
We derive the Cardano formula of cubic equations by completing the cube, and provide radical solutions to some algebraic equations of higher degree by completing powers. The main idea of completing powers arises from Harrison's center…
We study elements of small norm in quadratic fields of Riachaud-Degert type.