El contraexemple de Selmer al principi de Hasse

Abstract

The aim of our work is to present a counterexample to Hasse principle. We will see that Selmer's equation has local non-zero solutions everywhere but not global ones. Thus, its content is divided in 2 parts: the $p$-adic study of Selmer’s equation, which is done in chapters 1 and 3, and its rational study, which is done in chapters 2 and 4. Let us see its content by chapters.

- In chapter 1 we will build $p$-adic rings by using projective limits. We will also present basic results on $p$-adic fields so that we can achieve a local study of Selmer’s equation. We will state Hensel’l lemma, which is an essential tool in proving the existence of $p$-adic solutions of polynomial equations; particularly, we will characterize the cubs of $\mathbb{Q}^{\ast}_{p}$.

- In chapter 2 some basic facts in the geometry of numbers are summarized.

- In chapter 3 we will prove the existence of non-zero solutions of Selmer’s equation over $\mathbb{Q}_p$ , for each prime $p$, and over $\mathbb{R}$. Particularly, we will find them in $\mathbb{Z}_p$ due to the fact that we are dealing with homogenous equations.

- In chapter 4 we will prove the non existence of non-zero solutions of Selmer’s equation over $\mathbb{Q}$. The main tool for that will be the study of √ the ring of integers of the cubic number field $\mathbb{Q}\sqrt[3]{6}$.

We would like to mention that the courses \textit{Topology, Algebraic equations} and \textit{Algebraic methods in number theory} have provided basic prerequisites to perform this work.

We shall note that we make use of the arithmetic of a cubic number field, not carried out in the \textit{Algebraic methods in number theory} course, where the systematic study of number fields was done in the quadratic and cyclotomic cases only.

We mention the recent preprint by K. Conrad [5] as a basic reference. It contains (in 4 pages) a simplified proof of Selmer’s contraexample. Worth mentioning that this preprint has been modified by his author twice while our work was being developed. Furthermore, we have been in contact with him in order to correct some miscalculations detected in his paper. In a first edition of the preprint, Conrad deduced the existence of local solutions by using Hasse’s inequality on the number of congruence solutions of diagonal cubic equations. Later, Conrad uploaded a version of the preprint that excluded the use of that theorem. We have decided to explain the last proof of Conrad but we have added our previous study of diagonal equations in an annex since it can be useful in a future treatment of more general equations as can be seen, for example, in Selmer’s original paper. Most of the proofs are complete and self-contained, based on known results acquired in the above mentioned courses. In general, by looking at the references, one can see that our work presents changes in proofs, if able, in order to make them more direct and avoid the use of unnecessary sophisticated technics. Our main contribution in this sense is given in the theorems of Section 4.1, in which we avoid the complex study of the general cubic fields by providing \textit{ad hoc} proofs for arithmetic results in the cubic field $\mathbb{Q}\sqrt[3]{6}$, the necessary one for the study of Selmer’s equation.