Unlikely intersections in a family of semi-abelian surfaces.
D. Bertrand (IMJ-PRG)
CIRM, Géométrie diophantienne, Mai 2018

(Joint work with H. Schmidt)(Joint work with H. Schmidt)
D. Bertrand (IMJ-PRG) (CIRM, Géométrie
Unlikely
diophantienne,intersectionsMaiin2018)a family of semi-abelian surfaces.

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Abstract[Joint with Harry Schmidt, see ArXiV:1803.04835v1).]
Let E0 be an elliptic curve with CM, and let G /S be a one-dimensionalfamily of extensions of E0 by Gm . Then, G contains a dense set C of
special curves, known as Ribet curves. This set C contains all the torsion
curves of G .
Let W be an irreducible curve in G . With everything over Q, we show that
W meets C Zariski densely if and only if it is a translate of a Ribet curve bya multiplicative section, or it lies in a fiber of the family.
The proof involves the Pila-Zannier/Masser-Zannier strategy, a new type ofbounded height, and the Mordell-Lang theorem.
See also D.B.-B. Edixhoven (ArXiv:1104.5178v1), and current work of Bas.

(Joint work with H. Schmidt)
D. Bertrand (IMJ-PRG) (CIRM, Géométrie
Unlikely
diophantienne,intersections
Maiin2018)a family of semi-abelian surfaces.

2/8Motivation
Pink’s conjecture for an irreducible curve Z /Q in a mixed Shimuravariety S says : let S[2] = union of all the special subvarieties ofcodimension ≥ 2. Assume that Z ∩ S[2] is infinite. Then, Z is contained ina proper special subvariety S0 of S.
Special points are usually easy to describe, higher dim’l ones less so.
Examples with dim S = 2 (⇒ Z is a special curve) :• pure S = Y (2) × Y (2) (André);• mixed S = the Legendre family E/Y (2) (André, Pila).
Examples with dim S = 3 (⇒ Z is special or lies in a special surface) :• mixed S = E ×Y (2) E. Two types of special curves (see Gao) :- torsion curves (dominant over Y (2)) : see Masser-Zannier;- torsion translates of elliptic curves in a CM fiber E0 × E0 : see Barroero.
• “doubly mixed" S = P0 , the Poincaré Gm -torsor over E0 × Ê0 .
(Joint work with H. Schmidt)
D. Bertrand (IMJ-PRG) (CIRM, Géométrie
Unlikely
diophantienne,intersections
Maiin2018)a family of semi-abelian surfaces.

3/8Relative Manin-Mumford
A reduction in mixed cases : instead of Z ⊂ S, study a section s of agroup scheme G /S over a curve S/Q, with s(S) = Z 0 ⊂ G .

Theorem[Masser-Zannier] Let A = E ×S E , with E /S non isoconstant and lets = (p, q) ∈ A(S). The set
Sstor := {λ ∈ S(Q), s(λ) ∈ Atorλ }is infinite if and only if one of the following conditions holds :a) s is a torsion section;b) there exists an elliptic subscheme E 0 /S of A/S such that a multiple of sfactors through E 0 [and is not a constant section if E 0 /S is isoconstant].
(b) = relative dimension 1" : if s is a section of E 0 /S which is not constant(and not torsion), it meets E 0 tor densely (but with bounded height).
In relative dimension 2, same for G /S ∈ ExtS (E , Ga ) (H. Schmidt), but(Joint work with H. Schmidt)
D. Bertrand (IMJ-PRG) (CIRM, Géométrie
Unlikelydiophantienne,intersections
Maiin2018)a family of semi-abelian surfaces.

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for G /S ∈ ExtS (E0 , Gm ) ' Ê0 (S) ' E0 (S),Theorem[B-Masser-Pillay-Zannier] Let S/Q, G /S, parametrized by a non constantq ∈ E0 (S), and let s ∈ G (S) be a section, with projection p ∈ E0 (S).
Then, the set
Sstor := {λ ∈ S(Q), s(λ) ∈ Gλtor }is infinite if and only if one of the following conditions is satisfied :a) s is a torsion section, or more generally :a’) s is a Ribet section.
b) a multiple of s factors though Gm/S and is not constant.
At the generic fiber of G , Ribet sections form a group of rank 1. They
satisfy p(λ) ∈ Eλtor ⇒ s(λ) ∈ Gλtor , so their Sstor is infinite. Their image(Ribet curves) form a set of special curves of P0 , strictly containing thetorsion curves.
Ribet points at any fiber Gλ are defined in the same way. If q(λ) ∈/ (Ê0 )tor ,they form a group of rank 1. Every Ribet point lies on a Ribet curve.
(Joint work with H. Schmidt)
D. Bertrand (IMJ-PRG) (CIRM, Géométrie
Unlikelydiophantienne,intersections
Maiin2018)a family of semi-abelian surfaces.

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Ribet curves
Let S = Pic 0 (E0 ) = Ê0 be the moduli space of extensions of E0 by Gm ,and let π : G → S be the universal semi-abelian scheme over S : dimension
3, relative dim. 2.
This G can also be viewed as the Poincaré biextension P0 of E0 × Ê0 by Gm .
Let B be an elliptic curve in E0 × Ê0 ' E0 × E0 , so there is an isogeny :(a, b) : E0 → B, with a, b ∈ End (E0 ). Then, the Gm -torsor P0|B , or atleast (P0|B )⊗2 , is trivial iff N(a − b) = N(a) + N(b), i.e. iff α := a/b (or(b/a)) is purely imaginary. If so, the “isotropic"B ⊂ {(p, q), bp − aq = 0}
has a canonical lift to P0 , which we call a basic Ribet curve of P0 . Moregenerally, any curve C in G such that [n]G .C is a basic Ribet curve forsome n ∈ Z>0 is called a Ribet curve (in the sense of G). So, these include
all the (local) torsion curves of G.
Duality

Ribet curves in the sense of G 0 ∈ ExtE0 (Ê0 , Gm ).

(Joint work with H. Schmidt)
D. Bertrand (IMJ-PRG) (CIRM, Géométrie
Unlikely
diophantienne,intersections
Maiin2018)a family of semi-abelian surfaces.

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“Relative Mordell-Lang"
Let G ∈ ExtS (E0 , Gm ) as above, so G = Gq for a non constant q ∈ E0 (S).

Theorem[D.B.-H. Schmidt] Let s ∈ G (S). Assume that the set
SsRib = {λ ∈ S(Q), s(λ) is a Ribet point of its fiber Gλ .}is infinite. Then, s = s 0 + s”, where s 0 is a Ribet section, and s” factors
through Gm . (So s factors trough Gm × some isotropic B.)
List of special curves of S = P0 : if Z is special, then :- if Z → Ê0 is dominant, Z is a Ribet curves in the sense of G;- if Z → E0 is dominant, Z is a Ribet curves in the sense of G 0 ;- if Z ∼ Gm projects to a point (p0 , q0 ) ∈ E0 × Ê0 , then (p0 , q0 ) istorsion and Z is the fibre of P0 above (p0 , q0 ).
[WiP] : check that the list is complete, and conclude the proof that Pink’s
conjecture holds [for a curve] over Q in the MSV P0 .
(Joint work with H. Schmidt)
D. Bertrand (IMJ-PRG) (CIRM, Géométrie
Unlikely
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Maiin2018)a family of semi-abelian surfaces.

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Sketch of proof
As a first case, assume that p and q are linearly independent over End (E0 ),modulo constants. Then,- λ ∈ SsRib ⇒ bounded height , since bλ p(λ) − aλ q(λ) = 0.
- height of relations controled by degrees (Masser-type estimate forns(λ) = msR (λ))- o-minimal count on an incidence variety (Habegger-Pila) ;- logarithmic Ax
no algebraic subset under the “non-weakly"
assumption. NB : the Betti coordinates of a Ribet section sR generate thesame field as the torsion sections.
2nd case: same argument if p = rq + p0 with r ∈ Z, reducing finally to
3rd case : p = p0 . Using duality, transfer the problem into a special case of
Mordell-Lang for the semi-abelian variety Gp0 0 ∈ Ext(Ê0 , Gm ), and the rank
1 subgroup of Gp0 0 (Q) formed by its Ribet points.
(Joint work with H. Schmidt)
D. Bertrand (IMJ-PRG) (CIRM, Géométrie
Unlikely
diophantienne,intersections
Maiin2018)a family of semi-abelian surfaces.

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