By Anthony W. Knapp
An elliptic curve is a selected type of cubic equation in variables whose projective strategies shape a gaggle. Modular kinds are analytic features within the top part airplane with definite transformation legislation and progress houses. the 2 subjects--elliptic curves and modular forms--come jointly in Eichler-Shimura conception, which constructs elliptic curves out of modular sorts of a unique variety. The communicate, that every one rational elliptic curves come up this fashion, is named the Taniyama-Weil Conjecture and is understood to suggest Fermat's final Theorem.
Elliptic curves and the modeular kinds within the Eichler- Shimura conception either have linked L features, and it's a final result of the idea that the 2 forms of L features fit. the speculation coated through Anthony Knapp during this e-book is, for this reason, a window right into a extensive expanse of mathematics--including category box conception, mathematics algebraic geometry, and team representations--in which the concidence of L features relates research and algebra within the such a lot basic ways.
Developing, with many examples, the effortless idea of elliptic curves, the publication is going directly to the topic of modular kinds and the 1st connections with elliptic curves. The final chapters problem Eichler-Shimura conception, which establishes a miles deeper courting among the 2 matters. No different booklet in print treats the elemental idea of elliptic curves with purely undergraduate arithmetic, and no different explains Eichler-Shimura concept in such an obtainable manner.
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