We consider Thue equations of the form , and assuming the truth of the abc-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations of degree at least six.
This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009.We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve over . The focus is on practical aspects of this problem in the case that the genus of is at least , and therefore the set of rational points is finite.
Článek představuje zjednodušené základy teorie kvadratických zbytků a algebraické teorie čísel a jejich užití při řešení diofantických rovnic. Obsahuje i několik příkladů pro čtenáře.