Sur certaines surfaces de Kummer

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Thesis

Reference

Sur certaines surfaces de Kummer

ARGENTIN, Paola

Abstract

Soient C et D des courbes de genre 1 définies sur "Q" [le corps des nombres rationnels]. Soit Y la surface de Kummer de type Kum (C,D). Après avoir déterminé la structure du groupe de Picard de [Y défini sur le complété de "Q"], on calcule le 1er groupe de cohomologie [le premier groupe de cohomologie galoisienne du groupe de Galois de "Q" agissant sur le groupe de Picard de Y défini sur le complété de "Q"] pour tout Y de type Kum (C,D). On démontre le résultat suivant: soit Y = Kum (C,D). Si C et D ne sont pas isogènes sur "Q" (ou si elles sont isogènes sur [le complété de "Q"] mais n'ont pas de multiplication complexe) et [les points rationnels de 2-torsion de la Jacobienne de C sont triviaux] et [les points rationnels de 2-torsion de la Jacobienne de D sont triviaux] alors [le premier groupe de cohomologie galoisienne du groupe de Galois de "Q" agissant sur le groupe de Picard de Y défini sur le complété de "Q"]. Les contre-exemples au principe de Hasse parmi de telles surfaces n'ont pas d'obstruction de Brauer-Manin algébrique. On montre ensuite des contre-exemples au Principe de Hasse où [...]

ARGENTIN, Paola. Sur certaines surfaces de Kummer . Thèse de doctorat : Univ. Genève, 2006, no. Sc. 3759

URN : urn:nbn:ch:unige-4132

DOI : 10.13097/archive-ouverte/unige:413

Available at:

http://archive-ouverte.unige.ch/unige:413

Disclaimer: layout of this document may differ from the published version.

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Sur certaines surfaces de Kummer

Thesis

Reference

Sur certaines surfaces de Kummer

ARGENTIN, Paola

Abstract

ARGENTIN, Paola. Sur certaines surfaces de Kummer . Thèse de doctorat : Univ. Genève, 2006, no. Sc. 3759

URN : urn:nbn:ch:unige-4132

DOI : 10.13097/archive-ouverte/unige:413

Available at:

http://archive-ouverte.unige.ch/unige:413

Disclaimer: layout of this document may differ from the published version.

1 / 1

Sur certaines surfaces de Kummer

o 3759

2006

S 2 (E/K)

S 2 (E/K)

E(K)[2]

Pic( ¯ Y )

Z 18

Pic( ¯ Y )

Z 19

Pic( ¯ Y )

Z 20

F 1

F 2

H 1 (Q,

( ¯ Y ))

S 2,3

V

Q

V /Q

V /Q

R

Q p

Q

p

V

V

Q p

R

Q

W := W n,d,r

P n

r

d

W

k

q

N

W (k)

A(n,d,r)

n,r,d

W n,d,r

k

| N − q r | ≤ δ q r − 1/2 + A(n,d,r) q r − 1

δ = (d − 1)(d − 2)

W

C 1 q r − 1

C 1

X

V

Z

X

p

X ˜ p

p

X ˜ p

R p

p

F p

V

P p

Q p

X ˜ p

R p

F p

X

P p

S ² (E/K)

S ² (E/K)

Z ¹⁸

Z ¹⁹

Z ²⁰

H ¹ (Q,

S _2,3

Q _p

Q _p

W := W _n,d,r

P ⁿ

W _n,d,r

| N − q ^r | ≤ δ q ^r ⁻ ^1/2 + A(n,d,r) q ^r ⁻ ¹

C ₁ q ^r ⁻ ¹

C ₁

X ˜ _p

R _p

F _p

F _p

P _p

Z _p

R _p

Q _p

P ⁿ _Q

ν : Br(k _ν ) → Q/Z

Br(k _ν )

(x _ν )

ν A (x _ν )

X (A _k ) := Y

X (k _ν ).

X (A _k ) ^B := { (x _ν ) ∈ X (A _k ) | X

ν A (x _ν ) = 0

ν A (x _ν )

X (A _k ) ^B

P ³

P ⁴

(A k ) → X × ^k A k