Integral points on hyperelliptic curves

We study the arithmetic of a family of non-hyperelliptic curves of genus 3 over the field$\mathbb{Q}$of rational numbers. These curves are the nearby fibers of the semi-universal deformation of a simple singularity of type$E_{6}$. We show that average size of the 2-Selmer sets of these curves is finite (if it exists). We use this to show that a positive proposition of these curves (when ordered by height) has integral points everywhere locally, but no integral points globally.

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Quadratic Chabauty: p-adic heights and integral points on hyperelliptic curves

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2014-0048 ◽

2016 ◽

Vol 2016 (720) ◽

Cited By ~ 4

Author(s):

Jennifer S. Balakrishnan ◽

Amnon Besser ◽

J. Steffen Müller

Keyword(s):

Hyperelliptic Curves ◽

Integral Points

AbstractWe give a formula for the component at

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S-INTEGRAL POINTS ON HYPERELLIPTIC CURVES

International Journal of Number Theory ◽

10.1142/s1793042111004435 ◽

2011 ◽

Vol 07 (03) ◽

pp. 803-824 ◽

Cited By ~ 2

Author(s):

HOMERO R. GALLEGOS-RUIZ

Keyword(s):

Upper Bound ◽

Hyperelliptic Curve ◽

Rational Point ◽

Hyperelliptic Curves ◽

Integral Points ◽

Finite Set ◽

Genus 2 ◽

The Right ◽

Model C

Let C : Y2 = an Xn + ⋯ + a0 be a hyperelliptic curve with the ai rational integers, n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. Let S be a finite set of rational primes. We give a completely explicit upper bound for the size of the S-integral points on the model C, provided we know at least one rational point on C and a Mordell–Weil basis for J(ℚ). We use a refinement of the Mordell–Weil sieve which, combined with the upper bound, is capable of determining all the S-integral points. The method is illustrated by determining the S-integral points on the genus 2 hyperelliptic model Y2 - Y = X5 - X for the set S of the first 22 primes.

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Coleman integration for even-degree models of hyperelliptic curves

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157015000029 ◽

2015 ◽

Vol 18 (1) ◽

pp. 258-265 ◽

Cited By ~ 2

Author(s):

Jennifer S. Balakrishnan

Keyword(s):

Number Theory ◽

Computer Science ◽

Hyperelliptic Curves ◽

Open Book ◽

Line Integral ◽

Book Series ◽

Numerical Examples ◽

Lecture Notes ◽

Integration Algorithms ◽

Mathematical Sciences

The Coleman integral is a $p$-adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Harrison [J. Symbolic Comput. 47 (2012) no. 1, 89–101], we extend the Coleman integration algorithms in Balakrishnan et al. [Algorithmic number theory, Lecture Notes in Computer Science 6197 (Springer, 2010) 16–31] and Balakrishnan [ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Sciences Publishers, 2013) 41–61] to even-degree models of hyperelliptic curves. We illustrate our methods with numerical examples computed in Sage.

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Uniformization of hyperelliptic curves as a systematic approach to establishing decision regions of hyperbolic signal sets

Computational and Applied Mathematics ◽

10.1007/s40314-021-01518-2 ◽

2021 ◽

Vol 40 (4) ◽

Author(s):

Érika Patricia Dantas de Oliveira Guazzi ◽

Reginaldo Palazzo

Keyword(s):

Systematic Approach ◽

Hyperelliptic Curves ◽

Signal Sets

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On the integer part of the reciprocal of the Riemann zeta function tail at certain rational numbers in the critical strip

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2230-4 ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

WonTae Hwang ◽

Kyunghwan Song

Keyword(s):

Zeta Function ◽

Rational Number ◽

Riemann Zeta Function ◽

Rational Numbers ◽

Critical Strip ◽

Integer Part ◽

Integral Points ◽

The Riemann Zeta Function ◽

Riemann Zeta

Abstract We prove that the integer part of the reciprocal of the tail of $\zeta (s)$ ζ ( s ) at a rational number $s=\frac{1}{p}$ s = 1 p for any integer with $p \geq 5$ p ≥ 5 or $s=\frac{2}{p}$ s = 2 p for any odd integer with $p \geq 5$ p ≥ 5 can be described essentially as the integer part of an explicit quantity corresponding to it. To deal with the case when $s=\frac{2}{p}$ s = 2 p , we use a result on the finiteness of integral points of certain curves over $\mathbb{Q}$ Q .

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The monodromy problem for hyperelliptic curves

Bulletin des Sciences Mathématiques ◽

10.1016/j.bulsci.2021.102998 ◽

2021 ◽

pp. 102998

Author(s):

Daniel López Garcia

Keyword(s):

Hyperelliptic Curves

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Hyperelliptic curves with reduced automorphism group A 5

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-006-0030-9 ◽

2006 ◽

Vol 18 (1-2) ◽

pp. 3-20 ◽

Cited By ~ 1

Author(s):

David Sevilla ◽

Tanush Shaska

Keyword(s):

Automorphism Group ◽

Hyperelliptic Curves ◽

Group A

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The surfaces whose prime-sections contain a

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100016583 ◽

1934 ◽

Vol 30 (2) ◽

pp. 170-177 ◽

Cited By ~ 2

Author(s):

J. Bronowski

Keyword(s):

Hyperelliptic Curves ◽

Ruled Surfaces ◽

Minimum Order ◽

Rational Surfaces ◽

Pencil Of Conics

The surfaces whose prime-sections are hyperelliptic curves of genus p have been classified by G. Castelnuovo. If p > 1, they are the surfaces which contain a (rational) pencil of conics, which traces the on the prime-sections. Thus, if we exclude ruled surfaces, they are rational surfaces. The supernormal surfaces are of order 4p + 4 and lie in space [3p + 5]. The minimum directrix curve to the pencil of conics—that is, the curve of minimum order which meets each conic in one point—may be of any order k, where 0 ≤ k ≤ p + 1. The prime-sections of these surfaces are conveniently represented on the normal rational ruled surfaces, either by quadric sections, or by quadric sections residual to a generator, according as k is even or odd.

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