Mohamed Saidi

A Prime-to-p Version of Grothendieck’s Anabelian Conjecture for Hyperbolic Curves over Finite Fields of Characteristic p > 0

In this paper, we prove a prime-to-p version of Grothendiecks anabelian conjecture for hyperbolic curves over finite fields of characteristic p>0, whose original (full profinite) version was proved by Tamagawa in the affine case and by Mochizuki in the proper case.

Non-abelian fundamental groups and Iwasawa theory

List of contributors Preface 1. Lectures on anabelian phenomena in geometry and arithmetic Florian Pop 2. On Galois rigidity of fundamental groups of algebraic curves Hiroaki Nakamura 3. Around the Grothendieck anabelian section conjecture Mohamed Saidi 4. From the classical to the noncommutative Iwasawa theory (for totally real number fields) Mahesh Kakde 5. On the MUH(G)-conjecture J. Coates and R. Sujatha 6. Galois theory and Diophantine geometry Minhyong Kim 7. Potential modularity - a survey Kevin Buzzard 8. Remarks on some locally Qp-analytic representations of GL2(F) in the crystalline case Christophe Breuil 9. Completed cohomology - a survey Frank Calegari and Matthew Emerton 10. Tensor and homotopy criteria for functional equations of l-adic and classical iterated integrals Hiroaki Nakamura and Zdzislaw Wojtkowiak.

On complete families of curves with a given fundamental group in positive characteristic

In this paper we prove that complete families of smooth and projective curves of genus g≥2 in characteristic p>0 with a constant geometric fundamental group are isotrivial.

Non-abelian Fundamental Groups and Iwasawa Theory: Around the Grothendieck anabelian section conjecture

This is the author accepted manuscript. The final version is available from CUP via the DOI in this record

On the Section Conjecture over Function Fields and Finitely Generated Fields

We investigate sections of arithmetic fundamental groups of hyperbolic curves over function fields. As a consequence we prove that the anabelian section conjecture of Grothendieck holds over all finitely generated fields over

On the arithmetic of abelian varieties

\Bbb Q

On the existence of non-geometric sections of arithmetic fundamental groups

if it holds over all number fields, under the condition of finiteness (of the

A local-global principle for torsors under geometric prosolvable fundamental groups

\ell

Non-abelian Fundamental Groups and Iwasawa Theory: List of contributors

-primary parts) of certain Shafarevich-Tate groups. We also prove that if the section conjecture holds over all number fields then it holds over all finitely generated fields for curves which are defined over a number field.

Non-abelian Fundamental Groups and Iwasawa Theory: Contents

Abstract We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects “discrete Selmer groups” and “discrete Shafarevich–Tate groups”, and prove that they are finitely generated ℤ {\mathbb{Z}} -modules. Further, we prove that in the isotrivial case, the discrete Shafarevich–Tate group vanishes and the discrete Selmer group coincides with the Mordell–Weil group. One of the key ingredients to prove these results is a new specialisation theorem for first Galois cohomology groups, which generalises Néron’s specialisation theorem for rational points of abelian varieties.