Conjectures in Arithmetic Algebraic Geometry: A Survey by Wilfried W. J. Hulsbergen

By Wilfried W. J. Hulsbergen

In this expository textual content we cartoon a few interrelations among a number of recognized conjectures in quantity conception and algebraic geometry that experience intrigued math­ ematicians for a protracted time period. ranging from Fermat's final Theorem one is of course ended in introduce L­ services, 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 category of exponents. sooner than Kummer, Dirichlet had already effectively utilized his L-functions to the facts of the theory on mathematics progressions. one other renowned visual appeal of an L-function is Riemann's paper the place the now recognized Riemann speculation used to be acknowledged. briefly, 19th century quantity thought confirmed that a lot, if now not all, of quantity conception is mirrored through houses of L-functions. 20th century quantity concept, classification box concept and algebraic geome­ attempt in simple terms increase the 19th century quantity theorists's view. We simply point out the paintings of E. H~cke, E. Artin, A. Weil and A. Grothendieck along 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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Unfortunately the Hasse principle is not valid for curves of higher genus, in general. E. Selmer (1951) gave one of the first counter-examples to the Hasse principle. He showed that for all primes p the curve r : 3x 3 + 4y 3 + 5z 3 = 0 has a Qp-point and, obviously r has real points, but there exist no Q-rational point on r. There are many more examples. In this same spirit, we can't conclude about Fermat's Conjecture! As always, let E /Q be an elliptic curve over Q. To define the Tate-Shafarevich group, one looks at the set of E-torsors.

In fact, for any elliptic curve E defined over a number field F with complex multiplication by (an order in) the ring of integers OK of an imaginary quadratic number field K, M. Deuring, in a series of papers written in the 1950's, proved that the Lfunction L( E IF, s) can be expressed as a (product of such) Heeke L-function( s ). We indicate briefly what is meant by this observation, cf. (Grol], [Lal] and [Sh]. So let E IF be an elliptic curve over the number field F such that B: K ~ End( E)® Q, where K is an imaginary quadratic field (not necessarily of class number 1).

This (-function is a quasi-periodic function with respect to A: ((u +w) = ((u) +77(w), wE A. 21) The function 77 : A - - t C can be extended to an IR-linear function on the whole complex plane, 77 : C - - t C. This implies in particular that the Klein function satisfies the relation II<(u + w)l = II<(u)l for all wE A, hence the real analyticity of the Green's function G(P, Q) = II<(z- w)l off the diagonal. An algorithm and examples of H(P) and R(E/Q), especially for the BremnerCassels type curves and modular elliptic curves, can be found in [TZ].

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