Final Report Summary - RATIONAL POINTS (Fundamental groups, etale and motivic, local systems, Hodge theory and rational points)
Over the years we had several breakthroughs in our project which were the core of the ERC Advanced Grant ‘Rational Points’.
We proved the Gieseker conjecture with V. Mehta, which is at the borderline of geometry and arithmetic geometry, the motivation of which was coming from the theorem of Malcev-Grothendieck over the complex numbers: if a complex manifold is simply connected, then complex local systems are trivial. The characteristic p analog which was proposed by Gieseker in 1975 says: a smooth projective simply connected variety in positive characteristic has no non-trivial O-coherent D-module. Our proof is published in: Inventiones math. 181 (2010), 449-465. We use a mix of pure geometry, notably the existence of moduli of semi-stable bundles by Langer, and dynamics over the algebraic closure of a finite field, coded in the theorem of Hrushovsky, to solve the conjecture.
We formulated and proved (under the assumption of the finiteness of the Shafarevich group) an abelian birational version of the section conjecture. It is published in Journal of the American Society 23 (2010), 713—724. It is joint work with Olivier Wittenberg.
We made concrete Deligne’s motivic meta conjecture on the analogy between Hodge level in characteristic 0 and congruence for the number of rational points over a finite field. Our theorem, with P. Berthelot and K. Ruelling, says: the special fibre of a regular model of a smooth projective p-adic variety with Hodge level at least 1 has slopes at least 1. It is published in: Annals of Mathematics 176 (2012), 1-96.
We made a step towards the C1 conjecture after which a (separably) rationally connected over a C1 field admits a rational point. The main problem is for C1 fields such as the maximal unramified extension of the p-adic numbers, for the other ones, it is proven (for example by myself over finite fields). Using new methods in the topic, we prove that a smooth projective rationally connected variety over a C1 field in unequal characteristic, which admits a regular model, has index 1. To appear in: Journal of Algebraic Geometry. This is joint work with Marc Levine and Olivier Wittenberg.
We proved the Gieseker conjecture with V. Mehta, which is at the borderline of geometry and arithmetic geometry, the motivation of which was coming from the theorem of Malcev-Grothendieck over the complex numbers: if a complex manifold is simply connected, then complex local systems are trivial. The characteristic p analog which was proposed by Gieseker in 1975 says: a smooth projective simply connected variety in positive characteristic has no non-trivial O-coherent D-module. Our proof is published in: Inventiones math. 181 (2010), 449-465. We use a mix of pure geometry, notably the existence of moduli of semi-stable bundles by Langer, and dynamics over the algebraic closure of a finite field, coded in the theorem of Hrushovsky, to solve the conjecture.
We formulated and proved (under the assumption of the finiteness of the Shafarevich group) an abelian birational version of the section conjecture. It is published in Journal of the American Society 23 (2010), 713—724. It is joint work with Olivier Wittenberg.
We made concrete Deligne’s motivic meta conjecture on the analogy between Hodge level in characteristic 0 and congruence for the number of rational points over a finite field. Our theorem, with P. Berthelot and K. Ruelling, says: the special fibre of a regular model of a smooth projective p-adic variety with Hodge level at least 1 has slopes at least 1. It is published in: Annals of Mathematics 176 (2012), 1-96.
We made a step towards the C1 conjecture after which a (separably) rationally connected over a C1 field admits a rational point. The main problem is for C1 fields such as the maximal unramified extension of the p-adic numbers, for the other ones, it is proven (for example by myself over finite fields). Using new methods in the topic, we prove that a smooth projective rationally connected variety over a C1 field in unequal characteristic, which admits a regular model, has index 1. To appear in: Journal of Algebraic Geometry. This is joint work with Marc Levine and Olivier Wittenberg.