Jacobiennes de Courbes Algebriques de Genre 2 et 3 de Grand Rang Sur Q

Infinite families of curves are constructed of genus 2 and 3over Q whose jacobians have high rank over Q. More precisely,if E is an elliptic curve with rank at least r over Q, an infinitefamily of curves are constructed of genus 2 whose jacobianshave rank at least r+4 over Q, and, under certain conditions,an infinite family of curves are constructed of genus 3 whosejacobians have rank at least 2r over Q. On specialisation, afamily of curves are obtained of genus 2 whose jacobians haverank at least 27 and a family of curves are obtained of genus3 whose jacobians have rank at least 26; one of these has rankat least 42.