visible subgroup

H^{1}(K, A) (A/K)

A J K i : AJ K^{1}

visible subgroup H^{1}(K, A)

i J

Vis_{J}^{(i)}(H^{1}(K, A)) := ker {H^{1}(K, A)H^{1}(K, J)},

H^{1}(K, A)H^{1}(K, J)

Note that J is not necessarily a Jacobian of a curve. The only reason we use the letter J is that in most of the computational examples that we will provide later on, we will be using Jacobians of modular i^{2}

H^{1}(K, A)

ˆ a A J i : AJ

0AJC0,

C J/A

0A(K)J(K)C(K)H^{1}(K, A)H^{1}(K, J) ….

0J(K)/A(K)C(K)Vis_{J}^{(i)}(H^{1}(K, A))0.

c K x ∈ C(K) c x JC J A W C(A/K) c x ∈ C(K)

visualizes c

i : AJ

_{J}^{(i)}(H^{1}(K, A))

H^{1}(K, A) C(K) C(K)Vis_{J}^{(i)}(H^{1}(K, A)) Vis_{J}^{(i)}(H^{1}(K, A))

(A/K)

i : AJ

visible subgroup (A/K) i

Vis_{J}^{(i)}(A) := Vis_{J}^{(i)}(H^{1}(K, A))∩(A/K).

(A/K)

H^{1}(K, A)

J i : A ↪ J

c ∈ H^{1}(K, A)

J i : A ↪ J c ∈

Vis_{J}^{(i)}(H^{1}(K, A))

L/K 3

X′ L X K X

Res_{L/K}(X′) : (Sch/K){Sets}, S ↦ Hom_{L}(S×_{K}L, X).

X′/L X/K

X′ X X L/K(X′)

E : y^{2} = x^{3}+x+1

K = (\sqrt{5})

K E

K

\{1, \sqrt{5}\} x = x_{1}+ \sqrt{5}x_{2} y = y_{1}+ \sqrt{5}y_{2} x_{1}, x_{2}, y_{1}, y_{2} x y

X = ⟨x_{1}^{3}+5x_{1}x_{2}^{2}−y_{1}^{2}−5y_{2}^{2}+x_{1}+1, 3x_{1}^{2}x_{2}+5x_{2}^{3}−2y_{1}y_{2}−x_{2}⟩

K E (x, y)(x_{1}, x_{2}, y_{1}, y_{2})

?

[L :

K] = n X′

X′ = L[y_{1}, y_{2}, …, y_{k}]/I.

I = ⟨f_{1}, f_{2}, …, f_{s}⟩

f_{i} ∈ L[y_{1}, …, y_{k}] {e_{1}, …, e_{n}}

L/K y_{i} =

∑_{j=1}^{n} x_{ij}e_{j} x_{ij}

f_{i} y_{j}

f_{i}

{e_{1}, …, e_{n}} n g_{i1}, g_{i2}, …, g_{in}

f_{i} g_{ij} ∈

K[x_{11}, …, x_{1n}, …, x_{kn}]

J = ⟨g_{ij}⟩_{i=1}^{k}_{j=1}^{n}

X = K[x_{11}, …, x_{kn}]/J.

X L/K(X′)

K A

Hom_{L}(A⊗_{K}L, X′)Hom_{K}(A, X),

curves. In some papers, no superscript (i) is used. The only reason we use it here is because we do not necessarily require that the morphism i be an embedding. The same argument applies for the general case, but since we will

A^{4} X

Hom_{L}(L[y_{1}, y_{2}, …, y_{k}]/I, A×_{K}L)Hom_{K}(K[y_{11}, y_{12}, …, y_{1n}, … y_{kn}]/J, A),

5

c ∈ H^{1}(K, A)

C c K c L/K

C L L/K

L/K(c) =

0 L/K :

H^{1}(K, A)H^{1}(L, A)

C/K A/K

C ∈ W C(A/K) c ∈

H^{1}(K, A) f :

A× CC A C P ∈ C(L)

L φ :

A_{L}C_{L} φ(a) = f(a, P)^{6}

A C φ

ψ = φ^{−1}

A_{L} φ

ψ

ϕ : A_{L}×A_{L}A_{L}

ϕ(a, a′) = ψ(f(a, φ(a′))) ϕ(a, e) =

ψ(f(a, P)) = a ϕ(e, a) =

ψ(f(e, φ(a))) =

ψ(f(a, P)) = a μ_{L} :

A_{L}× A_{L}A_{L} ϕ− μ_{L}

ϕ = μ_{L}

J = Res_{L/K}(A_{L})

ψ : C_{L}A_{L} C(K) ↪ C(L) ≅ A_{L}(L) ≅ J(K)

id : A_{L}A_{L}

A(K) ↪ A_{L}(L) ≅ J(K)

AJ CJ ?

A K J_{L} n A_{L}

J_{L} ≅ ∏_{i=1}^{n} A_{L}.

C ↪ J L C_{L}J_{L}

x ↦ (ψ_{1}(x), ψ_{2}(x), …, ψ_{n}(x)),

ψ_{i} :

C_{L}A_{L} ψ :

C_{L}A_{L} n L ↪

\overline{K} ψ

K ψ_{i}

C_{L}A_{L} C A K

C_{L} J_{L} A_{L} A_{L} ↪ J_{L} ≅

∏_{i=1}^{n} A_{L} C_{L}

A_{L}ϕC_{L} ∏_{i=1}^{n} A_{L},

a ↦ (ψ_{1}(φ(a)), ψ_{2}(φ(a)), … ψ_{n}(φ(a))) a ∈

A_{L}(\overline{K})

b f(b, f(a, x)) =

σ_{i}(x) f f(a+ b, x) = σ_{i}(x)

ϕ a+ b+ ψ(σ_{i}(x)) =

ψ(x)

b = ψ(x)−

ψ(σ_{i}(x))− a C_{L} J_{L} A_{L}

(ϕ(x)− ϕ(σ_{1}(x)), ϕ(x)− ϕ(σ_{2}(x)), …, ϕ(x)− ϕ(σ_{n}(x)))

c J c i : A ↪ J

c ∈ H^{1}(K, A) visibility dimension c J c ∈ Vis_{J}^{(i)}(H^{1}(K, A))

c ∈

H^{1}(K, A) n A

G M G c ∈

H^{1}(G, M)

H ⊆ G c H^{1}(H, M)

[G : H] ≤ |M|

H = ker(f)

f :

GM c f

f(στ) = σf(τ)+f(σ).

c H^{1}(H, M)

G/H ↪

M τH ↦ f(τ)

H f(σ) =

f(τ)

f(τ) = f(σ(σ^{−1}τ)) = σf(σ^{−1}τ)+f(σ).

σf(σ^{−1}τ) =

0 f(σ^{−1}τ) =

0 σ^{−1}τ ∈ H G/H ↪

M

c ∈ H^{1}(K, A)

d· n^{2d} n c H^{1}(K, A)

d A

J [L :

K]·

dim A [L :

K]

H^{1}(K, A[n])H^{1}(K, A)[n] c ∈

H^{1}(K, A)[n] (\overline{K}/K)

L/K c H^{1}(L, A[n])

[L : K] |A[n]| =

n^{2d} L [L :

K]·

dim A ≤ d· n^{2d}

É

f :

AB flat

M ↦ M⊗_{A} B A B flat- ness f

A_{f^{−1}(m)}B_{m}

m ⊂ B

φ :

Y X flat X,φ(y)Y,y

y ∈ Y

φ :

Y X x ∈ X φ^{−1}(x)

Y_{x} := φ^{−1}(x)

dim Y_{x} = dim Y −dim X.

φ : Y X k smooth of relative dimension n φ

Y′ ⊆ Y X′ ⊆ X φ(Y′) ⊆

X′

dim Y′ = dim X′+n;

y ∈ Y

dim_{k(y)}(Ω_{Y/X}⊗k(y)) = n.

é

φ :

Y X étale

R henselian

R k R henselian P ∈ R[T]

k / (\overline{P} ∈) k[T]

R P k R strictly henselian

R R é f :

XR x ∈ X f(x) = s R u : RX S u(s) = x

[n] : G(R)G(R)

A K R x ∈ A(K) x é A A n R x ∈ nA(K)

G S n s ∈ S n n_{G} : GG é

U ⊆ G G U = G

G G×_{R} K G K G K §

G×_{K}

\overline{K} = (U×_{K}

\overline{K})· (V ×_{K} \overline{K})

K U G U U^{c}

x ∈ G K x U U· x v u ∈ U ux = v x = u^{−1}v ∈ U

G R G R

be considering abelian varieties over number fields, we will not worry too much about the general case. Hom_{K} denotes morphisms of schemes over K In this case Hom_{K} denotes homomorphisms of Kalgebras If X is a scheme over K and L is an exn_{G} : G(R)G(R)

n ∈ R^{×}

x ∈ G(R) x x : Spec RG

x[r]^{ψ}[d]Spec R[d]^{x} G[r]^{n_{G}}G

n_{G} : G(R)G(R) RY_{g} RY_{g}G

R G x ∈ G(R) n_{G}

R Y_{g}R

é Y_{g} n_{G} : GG é 8 n_{G} n_{G} n_{G}

X K K X X X×_{K}

\overline{K}

é A(K) ≅

A(R) x ∈ A(K)

A^{0}(R) A^{0}(R)

R A^{0} (R) n n_{G} :

G(R)G(R) z ∈ A(K) nz = x

A B R é A B K R ϕ :

AB ϕ

ϕ :

AB ϕ_{K} :

A(K)B(K)

ϕ_{R} :

A(R)B(R) x ∈ B(R)

RB x[r]^{ψ}[d]Spec R[d]^{x}

A[r]^{ϕ}B ψ : Y_{x}R

A K (A/K)

(A/K) (A/K)

A/K B/K J/K

tension of K, then X_{L} will denote the scheme X×_{K} L We say that a scheme X over a field K is geometrically connected if X×_{K} \overline{K} is connected By abuse of notation we denote by n_{G} both the morphism GG and the homomor-