Related papers: The dynamics of the Hesse derivative on the $j$-in…
Each elliptic curve can be embedded uniquely in the projective plane, up to projective equivalence. The hessian curve of the embedding is generically a new elliptic curve, whose isomorphism type depends only on that of the initial elliptic…
The Hesse curve or Hesse derivative Hess$(\Gamma_f)$ of a cubic curve $\Gamma_{f}$ given by a homogeneous polynomial $f$ is the set of points $P$ such that $\det \left(H_f (P)\right)=0$, where $H_f (P)$ is the Hesse matrix of $f$ evaluated…
We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.
We consider the geometric map $ \mathfrak C$, called Cayleyan, associating to a plane cubic $E$ the adjoint of its dual curve. We show that $ \mathfrak C$ and the classical Hessian map $ \mathfrak H$ generate a free semigroup. We begin the…
We prove that the Hessian transformation of elliptic curves, both as an action on $j$-invariants and on the Hesse pencil, is a Latt\`es map, namely it ascends to a degree-3 endomorphism $\psi$ of a prescribed elliptic curve $E$. This result…
In this paper we relate some classical normal forms for complex elliptic curves in terms of 4-point sets in the Riemann sphere. Our main result is an alternative proof that every elliptic curve is isomorphic as a Riemann surface to one in…
We show that the results we had obtained on diagonals of nine and ten parameters families of rational functions using creative telescoping, yielding modular forms expressed as pullbacked $ _2F_1$ hypergeometric functions, can be obtained,…
We construct some integral elements in the motivic cohomology of the Hesse cubic curves and express their regulators in terms of generalized hypergeometric functions and Kamp\'e de F\'eriet hypergeometric functions. By using these…
We describe deterministic and probabilistic algorithms to determine whether or not a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given…
The blown up complex projective plane in the twelve triple points of the dual Hesse arrangement has an infinite number of irreducible rational curves of self-intersection $-1$, for short, $(-1)$-curves. In the preprint version of [Dumnicki,…
We study invariants defined by count of charged, elliptic $J$-holomorphic curves in locally conformally symplectic manifolds. We use this to define $\mathbb{Q} $-valued deformation invariants of certain complete Riemann-Finlser manifolds…
We determine the possible degrees of cyclic isogenies defined over quadratic fields for non-CM elliptic curves with rational $j$-invariant.
In this article, we study how to compute the number of $K$-rational points with a given $j$-invariant on an arbitrary modular curve. As an application, for each positive integer $n$, we determine the list of possible numbers of cyclic…
Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complex-analytic, so a broad spectrum of techniques can contribute to their study (quasiconformal…
We show that two ordinary isogenous elliptic curves have isomorphic groups of rational points if they have the same $j$-invariant and we extend this result to certain isogenous supersingular elliptic curves, namely those with equal…
We study the $J-$holomorphic curves in the symplectization of the contact manifolds and prove that there exists at least one periodic Reeb orbits in any closed contact manifold with any contact form by using the well-known Gromov's…
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…
We characterize the possible reductions of $j$-invariants of elliptic curves which admit complex multiplication by an order $\mathcal{O}$ where the curve itself is defined over $\mathbb{Z}_p$. In particular, we show that the distribution of…
We construct invariants under deformation of real symplectic 4-manifolds. These invariants are obtained by counting three different kinds of real rational J-holomorphic curves which realize a given homology class and pass through a given…
We compare real and complex dynamics for automorphisms of rational surfaces that are obtained by lifting \chg{some} quadratic birational maps of the plane. In particular, we show how to exploit the existence of an invariant cubic curve to…