Weil Conjecture I
نویسنده
چکیده
Solving Diophantine equation is one of the main problem in number theory for a long time. It is very difficult but wonderful. For example, it took over 300 years to see that Xn + Y n = Zn has no nontrivial integers solution when n ≥ 3. We would like to consider an easier problem: solving the Diophantine equation modulo p, where p is a prime number. We expect that this problem is easy enough to handle, but still is not too trivial.
منابع مشابه
Complete characterization of the Mordell-Weil group of some families of elliptic curves
The Mordell-Weil theorem states that the group of rational points on an elliptic curve over the rational numbers is a finitely generated abelian group. In our previous paper, H. Daghigh, and S. Didari, On the elliptic curves of the form $ y^2=x^3-3px$, Bull. Iranian Math. Soc. 40 (2014), no. 5, 1119--1133., using Selmer groups, we have shown that for a prime $p...
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By the Mordell-Weil theorem, the group of rational points on an elliptic curve over a number field is a finitely generated abelian group. There is no known algorithm for finding the rank of this group. This paper computes the rank of the family $ E_p:y^2=x^3-3px $ of elliptic curves, where p is a prime.
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On June 23, 1993, Andrew Wiles unveiled his strategy for proving the Shimura-Taniyama-Weil conjecture for semistable elliptic curves defined over the field Q of rational numbers. Thanks to the work of Gerhard Frey, JeanPierre Serre and Kenneth Ribet, this was known to imply Fermat’s Last Theorem. Six years later, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor have finally ann...
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On June 23, 1993, Andrew Wiles unveiled his strategy for proving the Shimura-Taniyama-Weil conjecture for semistable elliptic curves defined over the field Q of rational numbers. Thanks to the work of Gerhard Frey, Jean-Pierre Serre, and Kenneth Ribet, this was known to imply Fermat’s Last Theorem. Six years later Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor have finally an...
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