scholarly journals Computing integral points on hyperelliptic curves using quadratic Chabauty

2016 ◽  
Vol 86 (305) ◽  
pp. 1403-1434 ◽  
Author(s):  
Jennifer S. Balakrishnan ◽  
Amnon Besser ◽  
J. Steffen Müller
Keyword(s):  
Hyperelliptic Curves ◽  
Integral Points

scholarly journals Integral points on hyperelliptic curves

2008 ◽  
Vol 2 (8) ◽  
pp. 859-885 ◽  
Author(s):  
Yann Bugeaud ◽  
Maurice Mignotte ◽  
Samir Siksek ◽  
Michael Stoll ◽  
Szabolcs Tengely
Keyword(s):  
Hyperelliptic Curves ◽  
Integral Points

scholarly journals AND THE ARITHMETIC OF A FAMILY OF NON-HYPERELLIPTIC CURVES OF GENUS 3

2015 ◽  
Vol 3 ◽  
Author(s):  
JACK A. THORNE
Keyword(s):  
Rational Numbers ◽  
Hyperelliptic Curves ◽  
Simple Singularity ◽  
Integral Points ◽  
Average Size

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.

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

2016 ◽  
Vol 2016 (720) ◽  
Author(s):  
Jennifer S. Balakrishnan ◽  
Amnon Besser ◽  
J. Steffen Müller
Keyword(s):  
Hyperelliptic Curves ◽  
Integral Points

AbstractWe give a formula for the component at

S-INTEGRAL POINTS ON HYPERELLIPTIC CURVES

2011 ◽  
Vol 07 (03) ◽  
pp. 803-824 ◽  
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.

scholarly journals Coleman integration for even-degree models of hyperelliptic curves

2015 ◽  
Vol 18 (1) ◽  
pp. 258-265 ◽  
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.

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

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 .

The surfaces whose prime-sections contain a

1934 ◽  
Vol 30 (2) ◽  
pp. 170-177 ◽  
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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