=. Gl, V ) be the Zariski closure of ?. Then: 1. ? n : B n ? B n?1 is a (possibly ramied) Galois cover of smooth proper curves with group 1. The morphism C ? B is separable by Lemma A. Then the generator of the Galois group of C ? B is an hyperelliptic involution and so we can apply

. =-c/-<-i, > where i an hyperelliptic involution. Then the proof goes exactly as in

, every U ? C j,n (?) contains ?(n) (resp. ?(?(n + 1))), hence C j

, ? 1)) and use

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, ? Yvan André and Mauritz Kerz for accepting refereeing my manuscript

C. ?-françois, Hélène Esnault and Javier Fresán for accepting to participate to the jury in the day of the defence

?. Vincent, for having the courage to read some of my drafts, for translating into french Chapter

T. ?-akio, for hosting me at RIMS for two summers and for enlightening discussions

A. ?-tomoyuki, for his innite kindness, for the time he spent in listening and helping me, for all the subtleties of the p-adic machinery that he taught me

S. ?-atsushi, for his innite patience in kindly answering to all my trivial questions

A. Skorobogatov,

?. Marco and D. Addezio, for being almost a mathematical girlfriend, for helping me decoding cryptic papers on overconvergent F-isocrystals and independence, for going crazy on epimorphic subgroup with me for a couple of weeks working on, Chapter

M. ?-matilde, for being almost a non-mathematical girlfriend, for our travels on foot, by car, train, plane, for almost dying in a beach to see wonderful Greek ruins and in general for being always with me in the last 2 years

G. Baldi, for lending me his shirt the day of my thesis defence, for preparing an amazing breakfast with avocado and eggs, for a number of mathematical (crazy) discussion and for teaching me how to learn without learning

, ? Various people that have participated and/or organized various working groups (even if I don't remember all of them, let me mention some

F. Battistoni, for some funny discussion on Galois theory of number elds and because some of the ideas in Chapter 5 arised to me when we were eating an "Hetero classic

P. Fuseau, because french bureaucracy can be crazy, but at the Ecole polytechnique everything its easier thanks to her

, Ludovica, for our beers at 18.00, the two British house-mates for teaching me that it's necessary to rinse the dishes, Tommaso and the German girl, for helping me to survive the st couples of months in Paris with a crazy house owner and various problem, ? The two Romans (Elena and Nicola)

?. Various, . Milan-(lilo, . Zava, . Valeria, . Gabri et al., for what we have done (and we will do) together

?. Hugo, for often asking (coherent) mathematical questions, and Matthew, for coming all days in our oce at 11.15 to ask for having lunch

, ? My family (my mother, my father, my sister and my nephew) for their support in all the years of my life