Modularity of rational elliptic curves and Fermat's last theorem

• Main Authors: Dris, Jose Arnaldo Bebita, Te, Dixon C.
• Format: text
• Language: English
• Published: Animo Repository 2000
• Subjects: Curves, Elliptic; Fermat's last theorem; Forms, Modular
• Online Access: https://animorepository.dlsu.edu.ph/etd_bachelors/16996
• Institution: De La Salle University

This thesis is an exposition on the theory of elliptic curves and modular forms as applied to the Fermat problem of showing the nonexistence of positive nontrivial integer solutions to the equation x^n + y^n = z^n, for n >= 3. It was Gerhard Frey who, in 1985, made the remarkable observation that if there is a nontrivial solution (x,y,z,n) in positive integers to the equation above, then the elliptic curve defined by the equation Y^2 = X(X -x^n) (X + y^n) is semistable but not modular. This conjecture was refined and formulated in precise mathematical terms by Jean-Pierre Serre, giving rise to the Epsilon Conjecture. It was this last conjecture that was finally proved by Kenneth Ribet in 1987. Now, it was Yutaka Taniyama who, around 1955, conjectured that all semistable elliptic curves defined over the rationals are modular. This conjecture was expressed in increasingly more general forms since then by Goro Shimura and Andre Weil. It is a special case of what is not known as the Taniyama-Shimura-Weil (TSW) Conjecture. This conjecture (with the technical semistable requirement) was finally established by Andrew Wiles, a Princeton mathematician, in October of 1994, thereby proving Fermat's Last Theorem and some more generalizations as a consequence. Elliptic curves and modular forms are discussed in considerable detail, and the two concepts are combined to give rise to the difficult notion of a modular elliptic curve. Modular elliptic curves are, however, not discussed in great detail.