Related papers: Abelian functions associated with genus three alge…
We study the geometry and arithmetic of the curves $C \colon y^3 = x^4 + ax^2 + b$ and their associated Prym abelian surfaces $P$. We prove a Torelli theorem in this context and give a geometric proof of the fact that $P$ has quaternionic…
In this paper, we show that there exist families of curves (defined over an algebraically closed field $k$ of characteristic $p >2$) whose Jacobians have interesting $p$-torsion. For example, for every $0 \leq f \leq g$, we find the…
By combining tools from different areas of mathematics, we obtain 3D visualizations of elliptic curves over different fields that faithfully capture the underlying algebra and geometry.
For several applications in the arithmetic of abelian varieties it is important to compute canonical heights. Following Faltings and Hriljac, we show how the canonical height on the Jacobian of a smooth projective curve can be computed…
We study the problem of modeling a binary operation that satisfies some algebraic requirements. We first construct a neural network architecture for Abelian group operations and derive a universal approximation property. Then, we extend it…
Expanding upon recent work, a new class of $A$-functions is introduced that can be viewed as an appropriate generalization of the class of regular $A$-functions, the class of structured $A$-functions, and the class of perfect $A$-functions.…
The connection between q-analogs of special functions and representations of quantum algebras has been developed recently. It has led to advances in the theory of q-special functions that we here review.
We continue our development of the invariant theory of genus one curves with the aim of computing certain twists of the universal family of elliptic curves parametrised by the modular curve X(n) for n = 2,3,4,5. Our construction makes use…
The Birch and Swinnerton-Dyer conjecture predicts that the parity of the algebraic rank of an abelian variety over a global field should be controlled by the expected sign of the functional equation of its $L$-function, known as the global…
We construct indecomposable cycles in the motivic cohomology group $H^3_{{\mathcal M}}(A,{\mathbb Q}(2))$ where $A$ is an Abelian surface over a number field or the function field of a base. When $A$ is the self product of the universal…
In this paper, we explore three combinatorial descriptions of semistable types of hyperelliptic curves over local fields: dual graphs, their quotient trees by the hyperelliptic involution, and configurations of the roots of the defining…
Elliptic curves are planar curves which can be used to define an abelian group. The efficient computation of discrete logarithms over this group is a longstanding problem relevant to cryptography. It may be possible to efficiently compute…
Based on Burnside's parametrization of the algebraic curve $y^2=x^5-x$ we provide remaining attributes of its uniformization: Fuchsian equations and their solutions, accessory parameters, monodromies, conformal maps, fundamental polygons,…
In this paper, we develop a new method to classify abelian automorphism groups of hypersurfaces. We use this method to classify (Theorem 4.2) abelian groups that admit a liftable action on a smooth cubic fourfold. A parallel result (Theorem…
Haile, Han and Kuo have studied certain non-commutative algebras associated to a binary quartic or ternary cubic form. We extend their construction to pairs of quadratic forms in four variables, and conjecture a further generalisation to…
A particular solution to the equations of motion of the Abelian Higgs model is given. The solution involves the Jacobi elliptic functions as well as the Heun functions.
We formulate a problem called \emph{Generalized Root Extraction} in finite Abelian groups that have more than one generator. We then study this problem for the specific case of the torsion subgroups of elliptic curves. We give a necessary…
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…
In the abelian case (the subject of several beautiful books) fixing some combinatorial structure (so called theta structure of level k) one obtains a special basis in the space of sections of canonical polarization powers over the…
We study the set of isomorphism classes of principal polarizations on abelian varieties of GL2-type. As applications of our results, we construct examples of curves C, C'/\Q of genus two which are nonisomorphic over \bar \Q and share…