Section conjecture for proper curves

(1)

Lifting sections along torsors

Niels Borne

(j. w. Michel Emsalem [Lille]

Jakob Stix [Heidelberg])

Laboratoire Paul Painlevé Université Lille 1, France

Scuola Normale Superiore Pisa 10.10.2012

(2)

Outline

Anabelian conjectures Main conjecture Section conjecture

Cuspidalization conjecture Evaluation of units

Lifting Galois sections The torsors

Torsion packets Main result Sketch of a proof

Arithmetic first Chern class Stix’s result

Conclusion

Niels Borne (j. w. Michel Emsalem [Lille] Jakob Stix [Heidelberg]) Lifting sections along torsors

(3)

The Hom conjecture

Letk be a number field, and Gal_k its absolute Galois group. Let X/k be a smooth, geometrically connected curve. Denote byg the genus ofX_k, andnits numbers of "holes". Letx be a geometric point ofX_k. The algebraic fundamental group π1(X_k,x)'Γdg,nfits into a short exact sequence:

1→π₁(X_k,x)→π₁(X,x)→Gal_k →1 Conjecture (Grothendieck 1983, Tamagawa 1997, Mochizuki 1999)

If X,Y are hyperbolic curves the morphism Hom^dom_k (X,Y)→Hom^open_Gal

k(π₁(X), π₁(Y))/∼ is one to one (where∼denotes conjugacy byπ1(Y_k)).

(4)

Section conjecture for proper curves

LetX/k be a geometrically connected scheme of finite type andx :SpecΩ→X a geometric point. The “fundamental exact sequence” is the extensionπ1(X/k):

1→π₁(X_k,x)→π₁(X,x)→Gal_k →1

By functoriality,x ∈X(k)induces a sections_X(x), well defined up to conjugacy by an element ofπ₁(X_k,x). One thus get:

s_X :X(k)→Sπ1(X/k)

Conjecture (Section conjecture, Grothendieck 1983) If X a smooth, proper curve of genus greater than2over a field k of finite type overQ, the application s_X is one to one.

Remark

The Mordell-Weil theorem easily implies that s_X is into.

(5)

Section conjecture and diophantine geometry

The Section Conjecture was initially thought as a strategy to show:

Conjecture (Mordell Conjecture 1922, Faltings 1983) Let k be a number field. If X/k is a smooth proper

geometrically connected curve of genus g≥2, then

#X(k)<∞.

Even today, no examples of suchX are known where 0<#Sπ1(X/k)<∞, and it is difficult to produce interesting examples whereSπ1(X/k)=∅.

However, Grothendieck’s idea to relate diophantine issues to fundamental groups (in a broader sense) has been very

successful lately (Kim’s proof of Siegel theorem, 2005, is based on this principle).

(6)

Packet of cuspidal sections

LetX/k be a smooth, proper, geometrically connected curve, andU =X\Dan affine dense open subset. Forx ∈D(k), denote byXc_x a formal completion ofX atx and byUc_x =Xc_x\x. The morphism of extensions

1 ^//π₁(Ud_k,x)

//π₁(cU_x) ^//

Gal_k ^//

1 1 ^//π₁(U_k) ^//π₁(U) ^//Gal_k ^//1 defines a mapS_π

1(cUx/k)→Sπ1(U/k).

The groupπ₁(Ud_k_,x)'Zb(1) =lim←−nµ_nis the inertia group atx.

One can show that the above sequence splits, so thatS_π

1(Ucx/k)

is a torsor under H¹(Gal_k,bZ(1))'kc^∗ :=lim←−nk^∗/(k^∗)ⁿ.

(7)

Section conjecture for affine curves

U(k)` `

x∈D(k)S_π

1(cUx/k) s_U //

Sπ1(U/k)

X(k) ^s^X ^//Sπ1(X/k)

Conjecture (Section conjecture for affine curves)

If U a smooth, hyperbolic curve (that is2−2g−r <0, where r = #D(k)) over a field k of finite type overQ, the application s_U is one to one.

A consequence of the Section conjecture forX is:

Conjecture (Cuspidalization conjecture) Sπ₁(U/k)→Sπ₁(X/k)is surjective.

(8)

Kummer class of a unit

Mochizuki has observed than one can (almost) reconstruct the valuef(x)of a unitf ∈Γ(U,Gm)at a rational pointx ∈U(k) from the Kummer class off and associated sectionsx. Starting from Kummer sequence

1→µn→Gm

−−→×n Gm →1

one defines evaluation·(s)of a unit on a fixed sectionsas:

Γ(U,Gm) ^//H¹(U,Z(1))b

H¹(π₁(U),Zb(1))

∼=

OO

s^∗ //H¹(Gal_k,Zb(1)) ^∼⁼ ^//ck^∗ It is easy to check thatf(sx)is the image off(x)byk^∗ →kc^∗.

(9)

Cuspidally central fundamental group

Mozichuki defines the cuspidally central fundamental group π^cc₁ (U)as the largest intermediate quotient

π₁(U)π₁^cc(U)π₁(X)

such thatK =ker(π₁^cc(U)→π1(X))is abelian, and the induced action ofπ₁(X_k)onK by conjugacy is trivial.

In Mochizuki’s theory of evaluation of units on sections,π₁^cc(U) is almost as good asπ1(U)since:

Lemma (Mochizuki 2005)

The morphismH¹(π₁^cc(U),Zb(1))→H¹(π₁(U),Zb(1))is an isomorphism.

(10)

Set up

Letk be a field of characteristic 0.

LetX/k be a smooth, proper, geometrically connected curve, E →X a torsor under a torusT, trivialized over an affine open subsetU =X\D, so that the chosen section induces an epimorphismπ1(U)π1(E). This gives rise to:

π₁(U)π₁(E)π₁(X)

This will allow us to give a geometric interpretation of the intermediate quotient constructed by Mochizuki by profinite means, and to give a positive result - in a special situation - for lifting Galois sections alongπ^cc₁ (U)→π1(X).

Our construction of the relevant torsor consist of two steps.

(11)

The torsor E

_D

The generalized Jacobian ofUis constructed from the pushout

diagram D ^//

X

Speck ^//X_D

. This leads to an exact sequence

0→Gm→Gm,D→Pic⁰_X

D/k →Pic⁰_X/k →0. So, denoting byS_D the torusGm,D/Gm, we get a naturalS_D-torsor onX by

pullback: E_D ^//

Pic_X_D_/k

X

O(∆)//Pic_X_/k

(12)

The torsor F

_D

Denote byT_D the torusGm,D, it fits into an exact sequence 0→Gm→T_D →S_D →0. We define a canonical line bundle F_D/E_D corresponding to the top row of:

E_D ^//

Pic_X_D_/k

//Pic_k/k =BGm

X O(∆)//Pic_X_/k

where we have to

consider here Picardstacks. The schemeF_D/X is endowed with a canonical structure of aT_D-torsor, trivialized overU.

Proposition

The morphismπ₁(U)→π₁(F_D)is surjective, and induces an isomorphismπ^cc₁ (U)'π₁(F_D).

(13)

Torsion criterion for E

_D

Definition (Baker-Poonen)

A reduced effective divisor D is a torsion packet if any degree0 divisor on X_k with support in D_k is torsion.

Remarks

1. Any rational point defines a torsion packet.

2. Let X be a modular curve. Manin-Drinfeld’s theorem states that the divisor of cusps D is a torsion packet.

Proposition

Let X/k be a smooth, proper, geometrically connected curve.

1. The torsor E_D →X is torsion if and only if D is a torsion packet.

2. The torsor F_D →X is never torsion.

(14)

Lifting Galois sections to F

_D

over the rationals

Theorem (Emsalem-Stix-B. 2012)

Let X/Qbe a smooth, proper, geometrically connected curve of genus≥2, and let D⊂X be a union of torsion packets. Then every Galois section s:Gal_Q→π1(X)lifts to a section

Gal_Q →π₁(F_D).

Corollary

Let X/Qbe a smooth, proper, geometrically connected of genus≥2, and let U=X \X(Q)be the complement of the set of allQ-rational points. Then every Galois section

s:Gal_Q →π₁(X) lifts to a sectionGal_Q →π^cc₁ (U).

(15)

Cohomological translation

LetE →X a torsor under a torusT. The fiberE_x ofE →X at a geometric pointx ofX identifies withT_k, leading to a sequence

1→π1(T_k)→π1(E)→π1(X)→1

which is exact becauseX is an algebraicK(π,1). The same fact shows that the morphism H²(π₁(X), π₁(T_k))→H²(X,T(T)) is an isomorphism. Moreover the image of the classπ₁(E/X)is the arithmetic first Chern class ofE/X, obtained from the cobord morphismc₁:H¹(X,T)→H²(X,T(T))associated to the Kummer sequences 0→T[n]→T −−→^×n T →0.

To a sections:Galk →π₁(X), one can thus associate a class s^∗(c₁(E/X))∈H²(Speck,T(T))

which cancels exactly whenslifts toπ1(E).

(16)

Over Q , a section kills the relative Brauer group

The image of a sections:Galk →π₁(X)corresponds to a (pro-)coverX_s →X, and it is easy to check that for a line bundleLonX

s^∗(c₁(L)) =0 ⇐⇒ Lis divisible onX_s Moreover from the exact sequence:

0→Pic(Xs)→Pic_X_s_/k(k)→Br(k)→Br(Xs)

one deduces that if Br(X_s/k) =ker(Br(k)→Br(X_s))vanishes, then any line bundleLonX is divisible onXs.

Theorem (Stix 2010)

Let X/Qbe a smooth projective curve of positive genus such thatπ₁(X/Q)admits a section. Then the relative Brauer group Br(X/Q)vanishes.

(17)

Killing the obstruction over Q

H¹(X,Gm) _c

1

//

uukkkkkkk H²(X,Zb(1))

uukkkkkkk

s^∗

H¹(X,T_D) _c

1

//

uullllll H²(X,T(T_D))

s^∗

H¹(X,S_D)

H²(k,Zb(1))

uukkkkkkk

0 ^//H¹(k,T_D) =0 ^//H²(k,T(T_D)) ^//T(H²(k,T_D))

(18)

What is missing

I A version of Stix’s theorem over a number field.

I Canonical liftings for non cuspidal sections:

U(k)` `

x∈D(k)S_π

1(Ucx/k) s_U //

Sπ1(U/k)

//Sπ1(F/k)

yyssssssssss

X(k) ^s^X ^//Sπ1(X/k)

If the section conjecture holds, then a non-cuspidal section inSπ1(X/k)should have acanonicallifting inSπ1(F/k).

I Algebraization of value of sections at units: the valuef(s) of a sections inSπ₁(U/k)at a unitΓ(U,Gm)should be algebraic, i.e. should belong tok^∗ ⊂ck^∗.

(19)

Appendix : a sketch of a proof of Stix result

Proposition

Let X/k be a smooth projective curve of positive genus such thatπ1(X/k)admits a section. Then:

1. if k/Qpis a finite extension, then#Br(X/k)is a power of p, 2. if k =RthenBr(X/R) =0.

Going back toX/Q, one uses Hasse-Brauer-Noether theorem that shows that:

Br(X/Q),→ker M

v

Br(X_Q_v/Qv)

P

vinvv

−−−−−→Q/Z

!