Algebraic curves and their applications Beshaj, Lubjana; Shaska, Tony
Contemporary mathematics - American Mathematical Society,
2019, 2019, 2019-03-30, Volume:
724
eBook
Open access
This volume contains a collection of papers on algebraic curves and their applications. While algebraic curves traditionally have provided a path toward modern algebraic geometry, they also provide ...many applications in number theory, computer security and cryptography, coding theory, differential equations, and more.Papers cover topics such as the rational torsion points of elliptic curves, arithmetic statistics in the moduli space of curves, combinatorial descriptions of semistable hyperelliptic curves over local fields, heights on weighted projective spaces, automorphism groups of curves, hyperelliptic curves, dessins d'enfants, applications to Painlevé equations, descent on real algebraic varieties, quadratic residue codes based on hyperelliptic curves, and Abelian varieties and cryptography.This book will be a valuable resource for people interested in algebraic curves and their connections to other branches of mathematics.
This volume contains the proceedings of the AMS Special Session on Higher Genus Curves and Fibrations in Mathematical Physics and Arithmetic Geometry, held on January 8, 2016, in Seattle, ...Washington.Algebraic curves and their fibrations have played a major role in both mathematical physics and arithmetic geometry. This volume focuses on the role of higher genus curves; in particular, hyperelliptic and superelliptic curves in algebraic geometry and mathematical physics.The articles in this volume investigate the automorphism groups of curves and superelliptic curves and results regarding integral points on curves and their applications in mirror symmetry. Moreover, geometric subjects are addressed, such as elliptic $K$3 surfaces over the rationals, the birational type of Hurwitz spaces, and links between projective geometry and abelian functions.
We determine all genus 2 curves, defined over ℂ, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in ℳ
2
.... For each component, we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to ℂ-isomorphism) defined over ℚ, which have degree 2 and 3 elliptic subcovers also defined over ℚ.
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Let ... be any point in the moduli space of genus-two curves ... and ... its field of moduli. We provide a universal equation of a genus-two curve ... defined over ..., corresponding to ..., where ...... and ... satisfy a quadratic ... such that ... and ... are given in terms of ratios of Siegel modular forms. The curve ... is defined over the field of moduli ... if and only if the quadratic has a ...-rational point .... We discover some interesting symmetries of the Weierstrass equation of .... This extends previous work of Mestre and others. ProQuest: ... denotes formulae omitted.
On Isogenies Among Certain Abelian Surfaces Clingher, Adrian; Malmendier, Andreas; Shaska, Tony
The Michigan mathematical journal,
05/2022, Volume:
71, Issue:
2
Journal Article
ABSTRACT Let f ∈ ℝ(t) be given by f(t, x) = xn + t · g(x) and β1 < · · · < βm the distinct real roots of the discriminant ∆(f,x)(t) of f(t, x) with respect to x. Let γ be the number of real roots of .... For any ξ > |βm|, if n − s is odd then the number of real roots of f(ξ, x) is γ + 1, and if n − s is even then the number of real roots of f(ξ, x) is γ, γ + 2 if ts > 0 or ts < 0 respectively. A special case of the above result is constructing a family of degree n ≥ 3 irreducible polynomials over ℚ with many non-real roots and automorphism group Sn.
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