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l-adic,p-adic and geometric invariants in families of varieties.

Abstract : This thesis is divided in 8 chapters. Chapter 1 is of preliminary nature: we recall the tools that we will use in the rest of the thesis and some previously known results. Chapter 2 is devoted to summarize in a uniform way the new results obtained in this thesis. The other six chapters are original. In Chapters 3 and 4, we prove the following: given a smooth proper morphism f:Y→X over a smooth geometrically connected base X over an infinite finitely generated field of positive characteristic, there are lots of closed points x∈|X| such that the rank of the N'eron-Severi group of the geometric fibre of f at x is the same of the rank of the N'eron-Severi group of the geometric generic fibre. To prove this, we first study the specialization of the ℓ-adic lisse sheaf R²f_*ℚℓ(1)(ℓ≠p), then we relate it with the specialization of the F-isocrystal R²f_{*,crys}O_{Y/K}(1) passing trough the category of overconvergent F-isocrystals. Then, the variational Tate conjecture in crystalline cohomology, allows us to deduce the result on the N'eron-Severi groups from the results on R²f_{*,crys}O_{Y/K}(1). These extend to positive characteristic results of Cadoret-Tamagawa and Andr'e in characteristic zero. Chapters 5 and 6 are devoted to the study of the monodromy groups of (over)convergent F-isocrystals. Chapter ref{chaptermarcuzzo} is a joint work with Marco D'Addezio. We study the maximal tori in the monodromy groups of (over)convergent F-isocrystals and using them we prove a special case of a conjecture of Kedlaya on homomorphism of convergent F-isocrystals. Using this special case, we prove that if A is an abelian variety without isotrivial geometric isogeny factors over a function field F over F¯_p, then the group A(F^{perf})_{tors} is finite. This may be regarded as an extension of the Lang--N'eron theorem and answer positively to a question of Esnault. In Chapter 6, we define $overline Q_p-linear category of (over)convergent F-isocrystals and the monodromy groups of their objects. Using the theory of companion for overconvergent F-isocrystals and lisse sheaves, we study the specialization theory of these monodromy groups, transferring the result of Chapter 3 to this setting via the theory of companions. The last two chapters are devoted to complements and refinement of the results in the previous chapters. In Chapter 7, we show that the Tate conjecture for divisors over finitely generated fields of characteristic p above 0 follows from the Tate conjecture for divisors over finite fields of characteristic p above 0. In Chapter 8, we prove uniform boundedness results for the Brauer groups of forms of varieties in positive characteristic, satisfying the ℓ-adic Tate conjecture for divisors. This extends to positive characteristic a result of Orr-Skorobogatov in characteristic zero.
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Submitted on : Monday, September 9, 2019 - 3:31:06 PM
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Emiliano Ambrosi. l-adic,p-adic and geometric invariants in families of varieties.. Algebraic Geometry [math.AG]. Université Paris Saclay (COmUE), 2019. English. ⟨NNT : 2019SACLX019⟩. ⟨tel-02281903⟩



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