Hyperelliptic curve
View article
Computing Hasse–Witt matrices of hyperelliptic curves in average polynomial time, II Open
We present an algorithm that computes the Hasse-Witt matrix of given\nhyperelliptic curve over Q at all primes of good reduction up to a given bound\nN. It is simpler and faster than the previous algorithm developed by the\nauthors.\n
View article
Rational points on hyperelliptic curves having a marked non-Weierstrass point Open
In this paper, we consider the family of hyperelliptic curves over $\mathbb{Q}$ having a fixed genus $n$ and a marked rational non-Weierstrass point. We show that when $n\geqslant 9$ , a positive proportion of these curves have exactly two…
View article
Constructing genus-3 hyperelliptic Jacobians with CM Open
Given a sextic CM field $K$ , we give an explicit method for finding all genus- $3$ hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approx…
View article
Arithmetic of hyperelliptic curves over local fields Open
We study hyperelliptic curves $$y^2 = f(x)$$ over local fields of odd residue characteristic. We introduce the notion of a “cluster picture” associated to the curve, that describes the p -adic distances between the roots of f ( x ), and sh…
View article
FROM HYPERELLIPTIC TO SUPERELLIPTIC CURVES Open
In this long survey article we show that the theory of elliptic and\nhyperelliptic curves can be extended naturally to all superelliptic curves. We\nfocus on automorphism groups, stratification of the moduli space\n$\\mathcal{M}_g$, binary…
View article
Arithmetic of hyperelliptic curves over local fields Open
We study hyperelliptic curves y^2=f(x) over local fields of odd residue characteristic. We introduce the notion of a cluster picture associated to the curve, that describes the p-adic distances between the roots of f(x), and show that this…
View article
The integral Chow ring of the stack of smooth non-hyperelliptic curves of genus three Open
We compute the integral Chow ring of the stack of smooth, nonhyperelliptic curves of genus 3. We obtain this result by computing the integral Chow ring of the stack of smooth plane quartics, by means of equivariant intersection theory.
View article
Spectral Curves of the Hyperelliptic Hitchin Systems Open
A description of the class of spectral curves, and explicit formulas for\nalgebraic-geometric action-angle coordinates are obtained for the Hitchin\nsystems on hyperelliptic curves, for any complex simple Lie algebra of the\ntypes $A_l$, $…
View article
GL2ℝ–invariant measures in marked strata :generic marked points, Earle–Kra for strata and illumination Open
We show that nontrivial [math] –invariant point markings for translation surfaces in strata of abelian differentials exist only when the translation surface belongs to a hyperelliptic component. As an application, we establish constraints …
View article
From topological recursion to wave functions and PDEs quantizing hyperelliptic curves Open
Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of degree $2$ spectral curves with a global involution satisfy a system of partial differential equations, whose equations ca…
View article
A Study of Hyperelliptic Curves in Cryptography Open
Elliptic curves are some specific type of curves known as hyper elliptic curves.Compared to the integer factorization problem(IFP) based systems, using elliptic curve based cryptography will significantly decrease key size of the encryptio…
View article
A unique pair of triangles Open
A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. …
View article
A non-hyperelliptic curve with torsion Ceresa class Open
We exhibit a non-hyperelliptic curve of genus 3 such that the class of the Ceresa cycle in the intermediate Jacobian of is torsion.
View article
A Non-Hyperelliptic Curve with Torsion Ceresa Cycle Modulo Algebraic Equivalence Open
We exhibit a non-hyperelliptic curve $C$ of genus $3$ such that the class of the Ceresa cycle $[C]-[C^-]$ in $JC$ modulo algebraic equivalence is torsion.
View article
On Reducible Degeneration of Hyperelliptic Curves and Soliton Solutions Open
In this paper we consider a reducible degeneration of a hyperelliptic curve\nof genus $g$. Using the Sato Grassmannian we show that the limits of\nhyperelliptic solutions of the KP-hierarchy exist and become soliton solutions\nof various t…
View article
Schottky–Kronecker forms and hyperelliptic polylogarithms Open
Elliptic polylogarithms can be defined as iterated integrals on a genus-one Riemann surface of a set of integration kernels whose generating series was already considered by Kronecker in the 19th century. In this article, we employ the Sch…
View article
Numerical Verification of the Birch and Swinnerton-Dyer Conjecture for Hyperelliptic Curves of Higher Genus over ℚ up to Squares Open
The Birch and Swinnerton-Dyer conjecture has been numerically verified for the Jacobians of 32 modular hyperelliptic curves of genus 2 by Flynn, Leprévost, Schaefer, Stein, Stoll and Wetherell, using modular methods. In the calculation of …
View article
On certain maximal hyperelliptic curves related to Chebyshev polynomials Open
We study hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs (q,d) such that the hyperelliptic curve C over a finite field Fq given by y2=φd(x) is maximal over the finite field Fq of …
View article
2‐Selmer parity for hyperelliptic curves in quadratic extensions Open
We study the 2‐parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. The proof proceeds via a genera…
View article
On different expressions for invariants of hyperelliptic curves of genus 3 Open
International audience
View article
Models and integral differentials of hyperelliptic curves Open
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 exp…
View article
From topological recursion to wave functions and PDEs quantizing hyperelliptic curves Open
Starting from loop equations, we prove that the wave functions constructed from topological recursion on families of degree $2$ spectral curves with a global involution satisfy a system of partial differential equations, whose equations ca…
View article
The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3 Open
We show that the average size of the $2$-Selmer group of the family of\nJacobians of non-hyperelliptic genus-$3$ curves with a marked rational\nhyperflex point, when ordered by a natural height, is bounded above by $3$. We\nachieve this by…
View article
Multi-variable sigma-functions: old and new results Open
We consider multi-variable sigma function of a genus $g$ hyperelliptic curve as a function of two group of variables -jacobian variables and parameters of the curve. In the theta-functional representation of sigma-function, the second grou…
View article
Abyss and Game of Polynomials and Coefficients : HyperElliptic Curves Open
In this paper 100 hyperelliptic curve equations are studied with varying coefficients. Random polynomial equations with even degree of 64 and the interpolation variants of cubic curves like elliptic curve, Ochoa curve, Mordell curve, proje…
View article
Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class Open
Let l be a prime and G a pro-l group with torsion-free abelianization. We produce group-theoretic analogues of the Johnson/Morita cocycle for G -- in the case of surface groups, these cocycles appear to refine existing constructions when l…
View article
Hyperelliptic Curves with Maximal Galois Action on the Torsion Points of their Jacobians Open
In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. Thi…
View article
Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell–Weil rank Open
We show that there is a bound depending only on g, r and [ K : \mathbb Q ] for the number of K -rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell–Weil rank r of its Jacobian is at most g–3 . …
View article
Moduli of Hyperelliptic Curves and Multiple Dirichlet Series Open
In this paper we provide an explicit construction of a $distinctive$ multiple Dirichlet series associated to products of quadratic Dirichlet L-series, which we believe should be tightly connected to a generalized metaplectic Whittaker func…
View article
Prym varieties of genus four curves Open
Double covers of a generic genus four curve C are in bijection with Cayley cubics containing the canonical model of C. The Prym variety associated to a double cover is a quadratic twist of the Jacobian of a genus three curve X. The curve X…