Related papers: An Ultradiscrete QRT mapping from Tropical Ellipti…
Let $E$ be an elliptic curve, defined over a quartic extension $K$ of $\mathbb{Q}$, with $j(E) \in \mathbb{Q}$. In this paper, we classify the possible torsion subgroup structures $E(K)_{\text{tors}}$.
The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how…
We consider the question of when points in tropical affine space uniquely determine a tropical hypersurface. We introduce a notion of multiplicity of points so that this question may be meaningful even if some of the points coincide. We…
In their book Katz and Mazur discuss the moduli problem of the subgroup-schemes of elliptic curves. We give the classification of the finite subgroups of the Tate curve before. Moreover, Katz and Mazur define the universal finite subgroup…
We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows…
We develop some algebraic structure notions such as composition series and convexity degree, along with some notions holding a geometric interpretation, like reducibility and hyperdimension, with the main objective being a tropical…
We prove that a semiring isomorphism between the rational function semifields of two tropical curves induces an expansive map between those tropical curves. This semiring isomorphism and the expansive map respect zeros and poles of rational…
We introduce a novel formulation for geometry on discrete points. It is based on a universal differential calculus, which gives a geometric description of a discrete set by the algebra of functions. We expand this mathematical framework so…
In this paper, we survey and study definitions and properties of tropical polynomials, tropical rational functions and in general, tropical meromorphic functions, emphasizing practical techniques that can really carry out computations. For…
A QRT map is the composition of two involutions on a biquadratic curve: one switching the $x$-coordinates of two intersection points with a given horizontal line, and the other switching the $y$-coordinates of two intersections with a…
An elliptic version of quantum groups is proposed. It comes form the quantization of the Knizhnik-Zamolodchikov- Bernard equation on the torus. The relation with elliptic IRF models is explained.
We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related…
We describe methods to determine all the possible torsion groups of an elliptic curve that actually appear over a fixed quadratic field. We use these methods to find, for each group that can appear over a quadratic field, the field with the…
We study the tropical lines contained in smooth tropical surfaces in R^3. On smooth tropical quadric surfaces we find two one-dimensional families of tropical lines, like in classical algebraic geometry. Unlike the classical case, however,…
We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed $j$-invariant, refining results of Mikhalkin and Nishinou--Siebert. As an application, we determine a formula for such counts…
We define arroids as an abstract axiom set encoding the intersection properties of arrangements of curves. The tropicalization of the complement of arrangement of curves meeting pairwise transversely is shown to be determined by the…
A topological group is (openly) almost-elliptic if it contains a(n open) dense subset of elements generating relatively-compact cyclic subgroups. We classify the (openly) almost-elliptic connected locally compact groups as precisely those…
In this paper, we give some explicit Diophantine parameters of the cyclic torsion subgroups of odd order of elliptic curves over $\mathbb{Q}$.
We revisit the Universal Approximation Theorem(UAT) through the lens of the tropical geometry of neural networks and introduce a constructive, geometry-aware initialization for sigmoidal multi-layer perceptrons (MLPs). Tropical geometry…
In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves…