Details

Autor(en) / Beteiligte

• Klassen, Matthew James

Titel

Algebraic points of low degree on curves of low rank

Ort / Verlag

ProQuest Dissertations Publishing

Erscheinungsjahr

1993

Link zum Volltext

• ProQuest

Quelle

ProQuest Dissertations & Theses A&I

Beschreibungen/Notizen

• This work addresses the problem finiteness of algebraic points of low degree on curves of genus at least 2, with special emphasis on smooth plane curves and Fermat curves. The study of such points is carried out on the symmetric product varieties of the curve, and a Theorem of Faltings is used to determine finiteness. We prove that for the Fermat curve of prime degree N that there are only finitely many points of degree at most $N-2$ over the rational numbers Q. This result is generalized in joint work with Debarre (DK). P-adic techniques are also used in the style of Coleman and Chabauty. In particular, we use the p-adic Abelian integrals of Coleman to show that under certain assumptions on the rank of the Jacobian, all but finitely many of the rational points of the symmetric product are contained in a canonical divisor.