## UE Arithmetics under the influence of Geometry

Diplômes intégrant cet élément pédagogique :

### Descriptif

Exhibiting strange relations between integers is as old as the notion of numbers. The fermat equation

Xp+Yp=Zp

has fascinated many would-be mathematicians for 300 years. With computers, it is possible to get quite surprising relations like

958004+2175194+4145604=4224814

or

26824404+153656394+187967604=206156734

with which Noam Elkies disproved a long standing conjecture of Euler.

Therefore one would like to have criteria to determine from an explicit system of polynomial equations with integer coefficients:

• Whether the system has solutions with integer coordinates;
• Whether it has infinitely many solutions;
• How many such solution one may find with coordinates smaller than a real number B.

The first question is known as Hilbert’s tenth problem. It was proven to be not solvable by Matiyasevich: no algorithm can determine in finite time whether a system of polynomial equations is solvable over the integers.

On the bright side, it turned out that the set of solutions was under the influence of the geometry of the complex variety defined by the same equations. One of the most famous result in that direction is a theorem of Faltings: A complex curve may be seen as an oriented real surface which is classified up to homeomorphism by a topological invariant: its genus. If the genus of a curve is bigger or equal to two, there exists only a finite numbers of points with rational coordinates on the curve. So the nature of the set of solutions is ruled by purely geometric invariants.

In higher dimensions, the situation is more complicated but similar phenomena are expected to be true.

The objective of these lectures is to present the tools of arithmetic geometry which provide the link between the arithmetic setting and the geometry, including projective varieties, line bundles, heights,\dots We shall illustrate these notions on various examples explaining also techniques to produce solutions and to count them.

As an example, the figure below displays points with rational coordinates on the surface given by the equation

Y2+Z2=X3−X. ### Pré-requis recommandés

Notions of algebra (groups, rings, algebras, polynomials in several variables) topology (topological space, compacity) and differential geometry (partial derivatives, jacobian matrix, tangent space).

### Informations complémentaires

Langue(s) : Anglais