By Wilfried W. J. Hulsbergen

In this expository textual content we caricature a few interrelations among a number of recognized conjectures in quantity thought and algebraic geometry that experience intrigued math ematicians for an extended time period. ranging from Fermat's final Theorem one is of course resulted in introduce L capabilities, the most, motivation being the calculation of sophistication numbers. In partic ular, Kummer confirmed that the category numbers of cyclotomic fields play a decisive position within the corroboration of Fermat's final Theorem for a wide classification of exponents. sooner than Kummer, Dirichlet had already effectively utilized his L-functions to the evidence of the theory on mathematics progressions. one other well known visual appeal of an L-function is Riemann's paper the place the now recognized Riemann speculation was once said. briefly, 19th century quantity idea confirmed that a lot, if now not all, of quantity thought is mirrored via houses of L-functions. 20th century quantity conception, category box conception and algebraic geome try out in basic terms develop the 19th century quantity theorists's view. We simply point out the paintings of E. H~cke, E. Artin, A. Weil and A. Grothendieck together with his collaborators. Heeke generalized Dirichlet's L-functions to procure effects at the distribution of primes in quantity fields. Artin brought his L-functions as a non-abelian generalization of Dirichlet's L-functions with a generalization of sophistication box concept to non-abelian Galois extensions of quantity fields in mind.

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Mazur, K. Ribet and J-P. Serre (1987) ( cf. 3 (Frey-Mazur-Ribet-Serre) The truth of the Taniyama- Weil Conjecture implies Fermat's Last Theorem. This theorem is closely related to a beautiful conjecture of Serre, dating back to 1973 and sharpened in [Se3], on 'modular' representations of Gal(Q/Q). Serre's work would imply "Taniyama-Weil+c: => Fermat", and the c; was eliminated by Ribet. 22) where a and b belong to the triple (a, b, c) of relatively prime integers, satisfying at + bi + ci = a= 0, ordered in such a way that b is even and 3 (mod 4).

These are the only ones with conductor smaller than 1000. 4 ( cf. (Cr]) The curves = EI/Q: E2/Q: E3/Q: have rank zero and # (III(E;)) x3 x3 x3 - x2 - - x2 - + x2 - 929x- 10595, 900x- 10098, 20x - 42 and = 4, i = 1, 2, 3. 5 ( cf. (Cr]) The curve E/Q: y 2 + xy = x 3 + x 2 - 1154x- 15345 has rank zero and# (III( E))= 9. In 1988 V. Kolyvagin (d. (Koll), (Ru3]) proved the finiteness of E(Q) and III(E) for any modular elliptic curve E which admits a Heegner point 2 of infinite order in E(K), where K is an imaginary quadratic number field in which the primes dividing the conductor of E split.

N, Varieties over Finite Fields 25 The corresponding projective formulation of the global minimal WeierstraB equation for E fQ is evident. In the sequel we will always assume that E fQ is given by its minimal WeierstraB equation, thus Li(E) = D(E). This global equation may be reduced modulo arbitrary primes. Denote the reduction modp by Ep· In this way we get, for all primes not dividing the discriminant Li(E), an elliptic curve EP over the finite field lFP. For the primes that divide Li(E) we get a singular curve with a node or a cusp.