Sur les courbes hyperelliptiques cyclotomiques et les équations x p - y p = c z ²

Wilfrid Ivorra

  • 2007

Abstract

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Let p be a prime number ≥ 11 and c be a square-free integer ≥ 3. In this paper, we study the diophantine equation x p - y p = c z ² in the case where p belongs to 11,13,17. More precisely, we prove that for those primes, there is no integer solution (x,y,z) to this equation satisfying gcd(x,y,z) = 1 and xyz ≠ 0 if the integer c has the following property: if ℓ is a prime number dividing c then ℓ ≢ 1 mod p. To obtain this result, we consider the hyperelliptic curves C p : y ² = Φ p ( x ) and D p : p y ² = Φ p ( x ) , where Φ p is the pth cyclotomic polynomial, and we determine the sets C p ( ℚ ) and D p ( ℚ ) . Using the elliptic Chabauty method, we prove that C p ( ℚ ) = ( - 1 , - 1 ) , ( - 1 , 1 ) , ( 0 , - 1 ) , ( 0 , 1 ) and D p ( ℚ ) = ( 1 , - 1 ) , ( 1 , 1 ) for p ∈ 11,13,17.

How to cite

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Wilfrid Ivorra. Sur les courbes hyperelliptiques cyclotomiques et les équations $x^{p} - y^{p} = cz²$. 2007. <http://eudml.org/doc/286030>.

@book{WilfridIvorra2007,
author = {Wilfrid Ivorra},
keywords = {higher degree Diophantine equations; Chabauty's method; -rational points; hyperelliptic curves; cyclotomic polynomial},
language = {fre},
title = {Sur les courbes hyperelliptiques cyclotomiques et les équations $x^\{p\} - y^\{p\} = cz²$},
url = {http://eudml.org/doc/286030},
year = {2007},
}

TY - BOOK
AU - Wilfrid Ivorra
TI - Sur les courbes hyperelliptiques cyclotomiques et les équations $x^{p} - y^{p} = cz²$
PY - 2007
LA - fre
KW - higher degree Diophantine equations; Chabauty's method; -rational points; hyperelliptic curves; cyclotomic polynomial
UR - http://eudml.org/doc/286030
ER -

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