Related papers: Counting hyperelliptic curves that admit a Koblitz…

Let $G$ be a subgroup of the three dimensional projective group $\mathrm{PGL}(3,q)$ defined over a finite field $\mathbb{F}_q$ of order $q$, viewed as a subgroup of $\mathrm{PGL}(3,K)$ where $K$ is an algebraic closure of $\mathbb{F}_q$.…

Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C:y^2=f(x)$ the corresponding genus $g$…

We introduce the 2-sorted counting logic $GC^k$ that expresses properties of hypergraphs. This logic has available k variables to address hyperedges, an unbounded number of variables to address vertices, and atomic formulas E(e,v) to…

In his previous papers (Math. Res. Letters 7 (2000), 123--13; Progress in Math. 195 (2001), 473--490; Math. Res. Letters 8 (2001), 429--435; Moscow Math. J. 2 (2002), issue 2, 403-431; Proc. Amer. Math. Soc. 131 (2003), no. 1, 95--102) the…

We present three families of pairs of geometrically non-isomorphic curves whose Jacobians are isomorphic to one another as unpolarized abelian varieties. Each family is parametrized by an open subset of P^1. The first family consists of…

We give a classification of all possible $2$-adic images of Galois representations associated to elliptic curves over $\mathbb{Q}$. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop…

In his previous papers (Math. Res. Letters 7 (2000), 123--13; Math. Res. Letters 8 (2001), 429--435; Moscow Math. J. 2 (2002), issue 2, 403-431) the author proved that in characteristic $\ne 2$ the jacobian $J(C)$ of a hyperelliptic curve…

We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of…

Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly…

In this paper we generalize results of P. Le Duff to genus n hyperelliptic curves. More precisely, let C/Q be a hyperelliptic genus n curve and let J(C) be the associated Jacobian variety. Assume that there exists a prime p such that J(C)…

We consider genus $g$ hyperelliptic curves over $\mathbb{Q}$ with a rational Weierstrass point, ordered by height. If $d < g$ is odd, we prove, under an assumption, that there exists $B_d$ such that a positive proportion of these curves…

In the previous work (J. Geom. Phys. {\bf{39}} (2001) 50-61), the closed loop solitons in a plane, \it i.e., loops whose curvatures obey the modified Korteweg-de Vries equations, were investigated for the case related to algebraic curves…

We establish the existence of hyperelliptic curves of genus $g\ge 2$ defined over $\mathbb{Q}$ whose Jacobians possess rational torsion points of order $N$ where $N=4g^2+2g-2$ or $4g^2+ 2g -4$. For $N=2g^2+7g+1$, we introduce a…

Let $\mathcal{F}_g$ be the family of monic odd-degree hyperelliptic curves of genus $g$ over $\mathbb{Q}$. Poonen and Stoll have shown that for every $g \geq 3$, a positive proportion of curves in $\mathcal{F}_g$ have no rational points…

In this paper we show a method for computing the set of twists of a non-singular projective curve defined over an arbitrary (perfect) field $k$. The method is based on a correspondence between twists and solutions to a Galois embedding…

We study the chromatic number of the curve graph of a surface. We show that the chromatic number grows like k log k for the graph of separating curves on a surface of Euler characteristic -k. We also show that the graph of curves that…

Let $k$ be a number field, and let $S$ be a finite set of maximal ideals of the ring of integers of $k$. In his 1962 ICM address, Shafarevich asked if there are only finitely many $k$-isomorphism classes of algebraic curves of a fixed genus…

In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-$5$ curves that are…

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice…

Let $C: y^2=f(x)$ be a hyperelliptic curve of genus $g\geq 1$, defined over a complete discretely valued field $K$, with ring of integers $O_K$. Under certain conditions on $C$, mild when residue characteristic is not $2$, we explicitly…