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We study a class of complex polynomial equations on a finite graph with a view to understanding how holistic phenomena emerge from combinatorial structure. Particular solutions arise from orthogonal projections of regular polytopes,…

Given a real elliptic curve $E$ with non-empty real part and $[D]\in \mbox{Pic}^2 E$ its $g_2^1$, we study the real inflection points of distinguished subseries of the complete real linear series $|\mathcal{L}_\mathbb{R}(kD)|$ for $k\geq…

A prime number $p$ is said to be irregular if it divides the class number of the $p$-th cyclotomic field $\mathbb{Q}(\zeta_{p}) = \mathbb{Q}(\mathbb{G}_m[p])$. In this paper, we study its elliptic analogue for the division fields of an…

We are concerned with fully nonlinear elliptic equations on complex manifolds and search for technical tools to overcome difficulties in deriving a priori estimates which arise due to the nontrivial torsion and curvature, as well as the…

We study negative curves on surfaces obtained by blowing up special configurations of points in the complex projective palne. Our main results concern the following configurations: very general points on a cubic, 3-torsion points on an…

We study Extremal Combinatorics problems where local properties are used to derive global properties. That is, we consider a given configuration where every small piece of the configuration satisfies some restriction, and use this local…

In this paper the number of $\mathbb{F}_q$-isomorphism classes of Legendre elliptic curves over the finite fields $\mathbb{F}_q$ is enumerated.

For a global field K and an elliptic curve E_eta over K(T), Silverman's specialization theorem implies that rank(E_eta(K(T))) <= rank(E_t(K)) for all but finitely many t in P^1(K). If this inequality is strict for all but finitely many t,…

We establish existence, uniqueness and optimal regularity results for very weak solutions to certain nonlinear elliptic boundary value problems. We introduce structural asymptotic assumptions of Uhlenbeck type on the nonlinearity, which are…

For a fixed $j$-invariant $j_0$ of an elliptic curve without complex multiplication we bound the number of $j$-invariants $j$ that are algebraic units and such that elliptic curves corresponding to $j$ and $j_0$ are isogenous. Our bounds…

We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic as a special case of…

An elliptic curve may be immersed in ${\mathbf{P}}^{N-1}$ as a degree $N$ curve using level $N$ structure. In the case where $N$ is odd, there are well known classical results dating back to Bianchi and Klein. In this paper we study the…

In this work, we study elliptic curves of the form $E_L: y^2 = x(x-1)(x-L)$, where $L^2-L+1 \in \left(\mathbb{Q}^{\times}\right)^2$ and $L \in \mathbb{Q} \setminus \{0,1\}$. We will show that for almost all quadratic extensions…

We consider some bilinear recurrences that have applications in number theory. The explicit solution of a general three-term bilinear recurrence relation of fourth order is given in terms of the Weierstrass sigma function for an associated…

We investigate not only the associated curves of regular plane curves, but also those of Legendre curves. As associated curves, we consider Bertrand regular plane curves and Bertrand Legendre curves. These curves contain parallel, evolute…

We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of…

We consider an obstacle problem for elastic curves with fixed ends. We attempt to extend the graph approach provided in [8]. More precisely, we investigate nonexistence of graph solutions for special obstacles and extend the class of…

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\rho_E\colon {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\rm GL}(2,\widehat{ \mathbb{Z} })$ be the adelic representation associated to the natural action of Galois on the…

Let $[K:\mathbb{Q}]=p$ be a prime number and let $E/K$ be an elliptic curve with $j(E) \in \mathbb{Q}$. We determine the all possibilities for $E(K)_{tors}$. We obtain these results by studying Galois representations of $E$ and of it's…

Our study concerns the Euclidean distance function in case of complex plane curves. We decompose the ED discriminant into components which are responsible for three types of behavior of the Morse points. Besides the traditional focal…