| Abstract |
One of the main problems in Number Theory is the solution of Diophantine equa¬tions, which means finding all the rational points of algebraic curves and more generally, of algebraic varieties. Given an affine curve C defined over a field K, the problem is finding all of its K-rational points. The field K can be the field of rational numbers Q, an algebraic number field (i.e. a finite extension of Q), the field of p-adic numbers Qp for some p G P or a local field (i.e. a finite extension of some Qp, p G P), or even an algebraic function field.
Curves are divided in three main categories, depending on their genus being g = 0, g = 1 or g > 2. A different approach is needed for each category. In 1983, Faltings proved that if K is an algebraic number field and C is a curve defined over K, then the set of its rational points, C(K), is finite. However, the proof is not effective, which means that it does not provide an algorithm for finding C(K).
In this thesis, different methods for finding the set of rational points of a curve are studied. Basically, we want to find the set of rational points (x,y) G Q x Q of the quadratic equation Y2 = f (X) where f (X) G Z[X] is an irreducible polynomial over Q and of degree deg(f) > 5.
In the first chapter, we develop a number of facts from Algebraic Geometry (Theory of Algebraic Curves), usefull in the next chapters.
In the second chapter, we develop some basic facts about p-adic fileds, three different versions of Hensel's Lemma and the local-global principle.
The third, fourth and fifth chapters are the main part of the thesis. They are based on notes from the lectures of Michael Stoll at the University of Bayreuth in 2014
and 2019.
In the third chapter, we apply the theorems of the first chapter in the case of hy-perelliptic curves. The K-rational points of the hyperelliptic curve C are embedded in the K-rational points of the Jacobian, J(K). Thus, our problem becomes find¬ing the rational points of the Jacobian. The Jacobian J(K) is a finitely generated abelian group.
In the fourth chapter, we calculate an upper rank of J(K) by using the 2-Selmer
group. This group is defined here in a way that favors calculations. Generally, the 2-Selmer group is defined by using Galois Cohomology but this approach makes calculations harder to achieve.
Finally, in the fifth chapter, we describe Chabauty's Method, which became effec¬tive by Coleman. The latter, defined a p-adic theory of integration, which is used to provide an upper bound for |C(Q)| in the case that the Mordell-Weil rank of J(Q) of the curve is strictly less than its genus.
By combining all the above, it is possible to solve a given diophantine equation in many cases.
Finally, it is worth mentioning that hyperelliptic curves of small genus (2 or 3) are used in cryptography.
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