S-Integral Points on Hyperelliptic Curves

Abstract

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.

Trinomials ax8+bx+c with Galois groups of order 1344

“…We recall some notation and results from [11,13] related to S-integral points on hyperelliptic curves that will be used later on. Consider the hyperelliptic curve (2.1) C :…”

“…[4,10,18,22]). In [11] an improved completely explicit upper bound for integral points were proved combining ideas from [10,12,[15][16][17]22] and in [13,14] for S-integral points, the main results stated in Section 2. Let α be a root of F. We have that…”

“…According to the Remark at page 42 in [13] we only need to compute bounds for some of these possible values. In our case only 4 values remain…”

“…We prove the following statement. The proof is based on techniques developed in [11] for integral points on hyperelliptic curves and [13,14] for S-integral points.…”

The Diophantine equation x² + 3ᵃ · 5ᵇ · 11ᶜ · 19ᵈ = 4yⁿ

“…Case 3: = 55. Then ( 1 , 2 , 3 , 4 ) = (0, 1, 5, 4), (4, 1, 3, 4), (4, 5, 3, 4), (4, 5, 5, 2), (4,5,5,4). Equation (3.6) reduces to )︀ = −1.…”