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Faltings heights and Zariski density of CM abelian varieties Shou-Wu Zhang Algebraic Geometry, Salt Lake City July 31, 2015 1 / 27
Transcript of Faltings heights and Zariski density of CM abelian varieties › downloads › utah › Notes ›...
Faltings heights and Zariski densityof
CM abelian varieties
Shou-Wu ZhangAlgebraic Geometry, Salt Lake City
July 31, 2015
1 / 27
References
A Quote
The theory of complex multiplication which forms a powerful linkbetween number theory and analysis, is not only the most beautiful partof mathematics but also of all science.
D. Hilbert, ICM’1932
Two preprints
1. J. Tsimerman A proof of the Andre-Oort conjecture for Ag ,arXiv:1506.01466 [math.NT].
2. X. Yuan and S. Zhang On the Averaged Colmez Conjecture,arXiv:1507.06903 [math.NT].
Two previous lectures
1. S. Zhang, Equidistributions for torsion points and small points,AG’95, Santa Cruz
2. S. Zhang, Heights of Heegner cycles and derivatives of L-series,AG’95, Santa Cruz
2 / 27
References
A QuoteThe theory of complex multiplication which forms a powerful linkbetween number theory and analysis,
is not only the most beautiful partof mathematics but also of all science.
D. Hilbert, ICM’1932
Two preprints
1. J. Tsimerman A proof of the Andre-Oort conjecture for Ag ,arXiv:1506.01466 [math.NT].
2. X. Yuan and S. Zhang On the Averaged Colmez Conjecture,arXiv:1507.06903 [math.NT].
Two previous lectures
1. S. Zhang, Equidistributions for torsion points and small points,AG’95, Santa Cruz
2. S. Zhang, Heights of Heegner cycles and derivatives of L-series,AG’95, Santa Cruz
2 / 27
References
A QuoteThe theory of complex multiplication which forms a powerful linkbetween number theory and analysis, is not only the most beautiful partof mathematics but also of all science.
D. Hilbert, ICM’1932
Two preprints
1. J. Tsimerman A proof of the Andre-Oort conjecture for Ag ,arXiv:1506.01466 [math.NT].
2. X. Yuan and S. Zhang On the Averaged Colmez Conjecture,arXiv:1507.06903 [math.NT].
Two previous lectures
1. S. Zhang, Equidistributions for torsion points and small points,AG’95, Santa Cruz
2. S. Zhang, Heights of Heegner cycles and derivatives of L-series,AG’95, Santa Cruz
2 / 27
References
A QuoteThe theory of complex multiplication which forms a powerful linkbetween number theory and analysis, is not only the most beautiful partof mathematics but also of all science.
D. Hilbert, ICM’1932
Two preprints
1. J. Tsimerman A proof of the Andre-Oort conjecture for Ag ,arXiv:1506.01466 [math.NT].
2. X. Yuan and S. Zhang On the Averaged Colmez Conjecture,arXiv:1507.06903 [math.NT].
Two previous lectures
1. S. Zhang, Equidistributions for torsion points and small points,AG’95, Santa Cruz
2. S. Zhang, Heights of Heegner cycles and derivatives of L-series,AG’95, Santa Cruz
2 / 27
References
A QuoteThe theory of complex multiplication which forms a powerful linkbetween number theory and analysis, is not only the most beautiful partof mathematics but also of all science.
D. Hilbert, ICM’1932
Two preprints
1. J. Tsimerman A proof of the Andre-Oort conjecture for Ag ,arXiv:1506.01466 [math.NT].
2. X. Yuan and S. Zhang On the Averaged Colmez Conjecture,arXiv:1507.06903 [math.NT].
Two previous lectures
1. S. Zhang, Equidistributions for torsion points and small points,AG’95, Santa Cruz
2. S. Zhang, Heights of Heegner cycles and derivatives of L-series,AG’95, Santa Cruz
2 / 27
References
A QuoteThe theory of complex multiplication which forms a powerful linkbetween number theory and analysis, is not only the most beautiful partof mathematics but also of all science.
D. Hilbert, ICM’1932
Two preprints
1. J. Tsimerman A proof of the Andre-Oort conjecture for Ag ,arXiv:1506.01466 [math.NT].
2. X. Yuan and S. Zhang On the Averaged Colmez Conjecture,arXiv:1507.06903 [math.NT].
Two previous lectures
1. S. Zhang, Equidistributions for torsion points and small points,AG’95, Santa Cruz
2. S. Zhang, Heights of Heegner cycles and derivatives of L-series,AG’95, Santa Cruz
2 / 27
References
A QuoteThe theory of complex multiplication which forms a powerful linkbetween number theory and analysis, is not only the most beautiful partof mathematics but also of all science.
D. Hilbert, ICM’1932
Two preprints
1. J. Tsimerman A proof of the Andre-Oort conjecture for Ag ,arXiv:1506.01466 [math.NT].
2. X. Yuan and S. Zhang On the Averaged Colmez Conjecture,arXiv:1507.06903 [math.NT].
Two previous lectures
1. S. Zhang, Equidistributions for torsion points and small points,AG’95, Santa Cruz
2. S. Zhang, Heights of Heegner cycles and derivatives of L-series,AG’95, Santa Cruz
2 / 27
References
A QuoteThe theory of complex multiplication which forms a powerful linkbetween number theory and analysis, is not only the most beautiful partof mathematics but also of all science.
D. Hilbert, ICM’1932
Two preprints
1. J. Tsimerman A proof of the Andre-Oort conjecture for Ag ,arXiv:1506.01466 [math.NT].
2. X. Yuan and S. Zhang On the Averaged Colmez Conjecture,arXiv:1507.06903 [math.NT].
Two previous lectures
1. S. Zhang, Equidistributions for torsion points and small points,AG’95, Santa Cruz
2. S. Zhang, Heights of Heegner cycles and derivatives of L-series,AG’95, Santa Cruz
2 / 27
References
A QuoteThe theory of complex multiplication which forms a powerful linkbetween number theory and analysis, is not only the most beautiful partof mathematics but also of all science.
D. Hilbert, ICM’1932
Two preprints
1. J. Tsimerman A proof of the Andre-Oort conjecture for Ag ,arXiv:1506.01466 [math.NT].
2. X. Yuan and S. Zhang On the Averaged Colmez Conjecture,arXiv:1507.06903 [math.NT].
Two previous lectures
1. S. Zhang, Equidistributions for torsion points and small points,AG’95, Santa Cruz
2. S. Zhang, Heights of Heegner cycles and derivatives of L-series,AG’95, Santa Cruz
2 / 27
References
A QuoteThe theory of complex multiplication which forms a powerful linkbetween number theory and analysis, is not only the most beautiful partof mathematics but also of all science.
D. Hilbert, ICM’1932
Two preprints
1. J. Tsimerman A proof of the Andre-Oort conjecture for Ag ,arXiv:1506.01466 [math.NT].
2. X. Yuan and S. Zhang On the Averaged Colmez Conjecture,arXiv:1507.06903 [math.NT].
Two previous lectures
1. S. Zhang, Equidistributions for torsion points and small points,AG’95, Santa Cruz
2. S. Zhang, Heights of Heegner cycles and derivatives of L-series,AG’95, Santa Cruz
2 / 27
Modular curve
An elliptic curve E can be embedded into P2 by a Weiestrass equation
Ca,b : y2 = x3 + ax + b, (a, b) ∈ W := A2 \ 4a3 + 27b2 = 0.
Over complex numbers, it is a complex torus:
Eτ = C/Zτ + Z, τ ∈ H := τ ∈ C, Imτ > 0.
The moduli S1 of elliptic curves is an orbit fold:
S1 = Gm\Wj' A1, S1(C) = SL2(Z)\H
j' C
j(Ea,b) = 17284a3
4a3 + 27b2, j(Eτ ) = e−2πiτ + 744 + 196884e2πiτ + · · ·
3 / 27
Modular curve
An elliptic curve E can be embedded into P2 by a Weiestrass equation
Ca,b : y2 = x3 + ax + b, (a, b) ∈ W := A2 \ 4a3 + 27b2 = 0.
Over complex numbers, it is a complex torus:
Eτ = C/Zτ + Z, τ ∈ H := τ ∈ C, Imτ > 0.
The moduli S1 of elliptic curves is an orbit fold:
S1 = Gm\Wj' A1, S1(C) = SL2(Z)\H
j' C
j(Ea,b) = 17284a3
4a3 + 27b2, j(Eτ ) = e−2πiτ + 744 + 196884e2πiτ + · · ·
3 / 27
Modular curve
An elliptic curve E can be embedded into P2 by a Weiestrass equation
Ca,b : y2 = x3 + ax + b, (a, b) ∈ W := A2 \ 4a3 + 27b2 = 0.
Over complex numbers, it is a complex torus:
Eτ = C/Zτ + Z, τ ∈ H := τ ∈ C, Imτ > 0.
The moduli S1 of elliptic curves is an orbit fold:
S1 = Gm\Wj' A1, S1(C) = SL2(Z)\H
j' C
j(Ea,b) = 17284a3
4a3 + 27b2, j(Eτ ) = e−2πiτ + 744 + 196884e2πiτ + · · ·
3 / 27
Modular curve
An elliptic curve E can be embedded into P2 by a Weiestrass equation
Ca,b : y2 = x3 + ax + b, (a, b) ∈ W := A2 \ 4a3 + 27b2 = 0.
Over complex numbers, it is a complex torus:
Eτ = C/Zτ + Z, τ ∈ H := τ ∈ C, Imτ > 0.
The moduli S1 of elliptic curves is an orbit fold:
S1 = Gm\Wj' A1, S1(C) = SL2(Z)\H
j' C
j(Ea,b) = 17284a3
4a3 + 27b2, j(Eτ ) = e−2πiτ + 744 + 196884e2πiτ + · · ·
3 / 27
Elliptic curves with complex multiplications.
An elliptic curve has CM if End(E ) 6= Z.
Over C, this is equivalent to E = Eτ with τ ∈ K imaginary quadratic.
Equivalently, τ is fixed by an embedded
K× ⊂ GL2(Q)+.
In this description, End(E ) ⊂ OK is an order, and
CM(R) := [E ] : End(E ) = R
is a PHS of Pic(R) under the action I ∗ C/Λ = C/(I · Λ).
4 / 27
Elliptic curves with complex multiplications.
An elliptic curve has CM if End(E ) 6= Z.
Over C, this is equivalent to E = Eτ with τ ∈ K imaginary quadratic.
Equivalently, τ is fixed by an embedded
K× ⊂ GL2(Q)+.
In this description, End(E ) ⊂ OK is an order, and
CM(R) := [E ] : End(E ) = R
is a PHS of Pic(R) under the action I ∗ C/Λ = C/(I · Λ).
4 / 27
Elliptic curves with complex multiplications.
An elliptic curve has CM if End(E ) 6= Z.
Over C, this is equivalent to E = Eτ with τ ∈ K imaginary quadratic.
Equivalently, τ is fixed by an embedded
K× ⊂ GL2(Q)+.
In this description, End(E ) ⊂ OK is an order, and
CM(R) := [E ] : End(E ) = R
is a PHS of Pic(R) under the action I ∗ C/Λ = C/(I · Λ).
4 / 27
Elliptic curves with complex multiplications.
An elliptic curve has CM if End(E ) 6= Z.
Over C, this is equivalent to E = Eτ with τ ∈ K imaginary quadratic.
Equivalently, τ is fixed by an embedded
K× ⊂ GL2(Q)+.
In this description, End(E ) ⊂ OK is an order, and
CM(R) := [E ] : End(E ) = R
is a PHS of Pic(R) under the action I ∗ C/Λ = C/(I · Λ).
4 / 27
Theory of complex multiplication
Let R an order of an imaginary quadratic K , h(R) = #Pic(R).
Theorem (Weber–Fueter)
1. All j(E ),E ∈ CM(R) are algebraic integers and conjugates to eachother;
2. Gal(K (j(E ))/K ) ' Pic(R) compatible with actions on CM(R).
For example, if R = Z[1 +√−163)/2], then h(R) = 1, and
j(Z[(1 +√−163)/2]) = −6403203.
This gives an approximation:
eπ√
163 = 6403203 + 743.99999999999925007..., (Ramanujan)
5 / 27
Theory of complex multiplication
Let R an order of an imaginary quadratic K , h(R) = #Pic(R).
Theorem (Weber–Fueter)
1. All j(E ),E ∈ CM(R) are algebraic integers and conjugates to eachother;
2. Gal(K (j(E ))/K ) ' Pic(R) compatible with actions on CM(R).
For example, if R = Z[1 +√−163)/2], then h(R) = 1, and
j(Z[(1 +√−163)/2]) = −6403203.
This gives an approximation:
eπ√
163 = 6403203 + 743.99999999999925007..., (Ramanujan)
5 / 27
Theory of complex multiplication
Let R an order of an imaginary quadratic K , h(R) = #Pic(R).
Theorem (Weber–Fueter)
1. All j(E ),E ∈ CM(R) are algebraic integers and conjugates to eachother;
2. Gal(K (j(E ))/K ) ' Pic(R) compatible with actions on CM(R).
For example, if R = Z[1 +√−163)/2], then h(R) = 1, and
j(Z[(1 +√−163)/2]) = −6403203.
This gives an approximation:
eπ√
163 = 6403203 + 743.99999999999925007..., (Ramanujan)
5 / 27
Theory of complex multiplication
Let R an order of an imaginary quadratic K , h(R) = #Pic(R).
Theorem (Weber–Fueter)
1. All j(E ),E ∈ CM(R) are algebraic integers and conjugates to eachother;
2. Gal(K (j(E ))/K ) ' Pic(R) compatible with actions on CM(R).
For example, if R = Z[1 +√−163)/2], then h(R) = 1, and
j(Z[(1 +√−163)/2]) = −6403203.
This gives an approximation:
eπ√
163 = 6403203 + 743.99999999999925007..., (Ramanujan)
5 / 27
Degree and distribution of CM points
The schemes CM(R) has dimension 0. As discR−→∞, the degree h(R)grows as follows:
h(R) = |discR|1/2+o(1), (Brauer–Siegel).
The subschemes CM(R) in S1 are equidistubted with respect to thePoincare measure:
1
h(R)
∑x∈CM(R)
δx =3
π
dxdy
y2+ o(1), (W. Duke).
QuestionFor a prime p, are CM(R)’s equidistributed on the p-adic analytic spaceSan1 with respect to some measure?
6 / 27
Degree and distribution of CM points
The schemes CM(R) has dimension 0. As discR−→∞, the degree h(R)grows as follows:
h(R) = |discR|1/2+o(1), (Brauer–Siegel).
The subschemes CM(R) in S1 are equidistubted with respect to thePoincare measure:
1
h(R)
∑x∈CM(R)
δx =3
π
dxdy
y2+ o(1), (W. Duke).
QuestionFor a prime p, are CM(R)’s equidistributed on the p-adic analytic spaceSan1 with respect to some measure?
6 / 27
Degree and distribution of CM points
The schemes CM(R) has dimension 0. As discR−→∞, the degree h(R)grows as follows:
h(R) = |discR|1/2+o(1), (Brauer–Siegel).
The subschemes CM(R) in S1 are equidistubted with respect to thePoincare measure:
1
h(R)
∑x∈CM(R)
δx =3
π
dxdy
y2+ o(1), (W. Duke).
QuestionFor a prime p, are CM(R)’s equidistributed on the p-adic analytic spaceSan1 with respect to some measure?
6 / 27
Siegel moduli spaces
A principally polarized abelian variety A of dimension g can be embeddedinto P3g−1 with the Hilbert polynomial Pg (x) = 3gxg .
Over C, a principally polarized A is a torus A = Cg/(Zgτ + Zg ) with
τ ∈ Hg :=τ ∈ Mg (C), τ t = τ, Imτ > 0
.
The moduli Sg of PPAV of dim g is a quotient variety over Q:
Sg = PGL(3g )\Wg , Wg ⊂ Hilb(Pg ), (Mumford)
Sg (C) = Sp(2g ,Z)\Hg , (Riemann)
7 / 27
Siegel moduli spaces
A principally polarized abelian variety A of dimension g can be embeddedinto P3g−1 with the Hilbert polynomial Pg (x) = 3gxg .
Over C, a principally polarized A is a torus A = Cg/(Zgτ + Zg ) with
τ ∈ Hg :=τ ∈ Mg (C), τ t = τ, Imτ > 0
.
The moduli Sg of PPAV of dim g is a quotient variety over Q:
Sg = PGL(3g )\Wg , Wg ⊂ Hilb(Pg ), (Mumford)
Sg (C) = Sp(2g ,Z)\Hg , (Riemann)
7 / 27
Siegel moduli spaces
A principally polarized abelian variety A of dimension g can be embeddedinto P3g−1 with the Hilbert polynomial Pg (x) = 3gxg .
Over C, a principally polarized A is a torus A = Cg/(Zgτ + Zg ) with
τ ∈ Hg :=τ ∈ Mg (C), τ t = τ, Imτ > 0
.
The moduli Sg of PPAV of dim g is a quotient variety over Q:
Sg = PGL(3g )\Wg , Wg ⊂ Hilb(Pg ), (Mumford)
Sg (C) = Sp(2g ,Z)\Hg , (Riemann)
7 / 27
Siegel moduli spaces
A principally polarized abelian variety A of dimension g can be embeddedinto P3g−1 with the Hilbert polynomial Pg (x) = 3gxg .
Over C, a principally polarized A is a torus A = Cg/(Zgτ + Zg ) with
τ ∈ Hg :=τ ∈ Mg (C), τ t = τ, Imτ > 0
.
The moduli Sg of PPAV of dim g is a quotient variety over Q:
Sg = PGL(3g )\Wg , Wg ⊂ Hilb(Pg ), (Mumford)
Sg (C) = Sp(2g ,Z)\Hg , (Riemann)
7 / 27
Siegel moduli spaces
A principally polarized abelian variety A of dimension g can be embeddedinto P3g−1 with the Hilbert polynomial Pg (x) = 3gxg .
Over C, a principally polarized A is a torus A = Cg/(Zgτ + Zg ) with
τ ∈ Hg :=τ ∈ Mg (C), τ t = τ, Imτ > 0
.
The moduli Sg of PPAV of dim g is a quotient variety over Q:
Sg = PGL(3g )\Wg , Wg ⊂ Hilb(Pg ), (Mumford)
Sg (C) = Sp(2g ,Z)\Hg , (Riemann)
7 / 27
CM abelian variety
An abelian variety A of dimension g is of CM type if End(A)⊗Qcontains a CM-commutative subalgebra K of degree 2g .
Analytically, A = Cg/Φ(Λ) where Φ : K ⊗Q R ' Cg is an isomorphismcalled a type of K and Λ is a lattice of K .
If A is a PPAV, then A = Cg/Zgτ + Zg with τ ∈ Hg fixed by anembedding K1 ⊂ Sp(2g ,Q), K1 := z ∈ K , zz = 1.
Theorem (Shimura–Taniyama)The [A] ∈ Sg (C) is defined over a number field, more precisely,
[A] ∈ Sg (K (Φ)ab), K (Φ) := Q(trΦ(x), x ∈ K ).
8 / 27
CM abelian variety
An abelian variety A of dimension g is of CM type if End(A)⊗Qcontains a CM-commutative subalgebra K of degree 2g .
Analytically, A = Cg/Φ(Λ) where Φ : K ⊗Q R ' Cg is an isomorphismcalled a type of K and Λ is a lattice of K .
If A is a PPAV, then A = Cg/Zgτ + Zg with τ ∈ Hg fixed by anembedding K1 ⊂ Sp(2g ,Q), K1 := z ∈ K , zz = 1.
Theorem (Shimura–Taniyama)The [A] ∈ Sg (C) is defined over a number field, more precisely,
[A] ∈ Sg (K (Φ)ab), K (Φ) := Q(trΦ(x), x ∈ K ).
8 / 27
CM abelian variety
An abelian variety A of dimension g is of CM type if End(A)⊗Qcontains a CM-commutative subalgebra K of degree 2g .
Analytically, A = Cg/Φ(Λ) where Φ : K ⊗Q R ' Cg is an isomorphismcalled a type of K and Λ is a lattice of K .
If A is a PPAV, then A = Cg/Zgτ + Zg with τ ∈ Hg fixed by anembedding K1 ⊂ Sp(2g ,Q), K1 := z ∈ K , zz = 1.
Theorem (Shimura–Taniyama)The [A] ∈ Sg (C) is defined over a number field, more precisely,
[A] ∈ Sg (K (Φ)ab), K (Φ) := Q(trΦ(x), x ∈ K ).
8 / 27
CM abelian variety
An abelian variety A of dimension g is of CM type if End(A)⊗Qcontains a CM-commutative subalgebra K of degree 2g .
Analytically, A = Cg/Φ(Λ) where Φ : K ⊗Q R ' Cg is an isomorphismcalled a type of K and Λ is a lattice of K .
If A is a PPAV, then A = Cg/Zgτ + Zg with τ ∈ Hg fixed by anembedding K1 ⊂ Sp(2g ,Q), K1 := z ∈ K , zz = 1.
Theorem (Shimura–Taniyama)The [A] ∈ Sg (C) is defined over a number field, more precisely,
[A] ∈ Sg (K (Φ)ab), K (Φ) := Q(trΦ(x), x ∈ K ).
8 / 27
Special subvarieties
The Hg can be identified with the GSp(2g ,R)+- conjugacy class of
hg : C×−→GSp(2g ,R)+ x + yi 7→(
x1g y1g
−y1g x1g
)−1
.
An irreducible subvariety V of Sg is special, if V = Γ\X , induced from areductive subgroup G ⊂ GSp(2g) in the following way:
1. X is the image of a G (R)+-conjugacy class in Hom(C×,G (R)+);
2. Γ = G (Q)+ ∩ Sp(2g ,Z).
CM points are exactly the zero dimensional special subvarieties of Sg .
9 / 27
Special subvarieties
The Hg can be identified with the GSp(2g ,R)+- conjugacy class of
hg : C×−→GSp(2g ,R)+ x + yi 7→(
x1g y1g
−y1g x1g
)−1
.
An irreducible subvariety V of Sg is special, if V = Γ\X , induced from areductive subgroup G ⊂ GSp(2g) in the following way:
1. X is the image of a G (R)+-conjugacy class in Hom(C×,G (R)+);
2. Γ = G (Q)+ ∩ Sp(2g ,Z).
CM points are exactly the zero dimensional special subvarieties of Sg .
9 / 27
Special subvarieties
The Hg can be identified with the GSp(2g ,R)+- conjugacy class of
hg : C×−→GSp(2g ,R)+ x + yi 7→(
x1g y1g
−y1g x1g
)−1
.
An irreducible subvariety V of Sg is special, if V = Γ\X , induced from areductive subgroup G ⊂ GSp(2g) in the following way:
1. X is the image of a G (R)+-conjugacy class in Hom(C×,G (R)+);
2. Γ = G (Q)+ ∩ Sp(2g ,Z).
CM points are exactly the zero dimensional special subvarieties of Sg .
9 / 27
Special subvarieties
The Hg can be identified with the GSp(2g ,R)+- conjugacy class of
hg : C×−→GSp(2g ,R)+ x + yi 7→(
x1g y1g
−y1g x1g
)−1
.
An irreducible subvariety V of Sg is special, if V = Γ\X , induced from areductive subgroup G ⊂ GSp(2g) in the following way:
1. X is the image of a G (R)+-conjugacy class in Hom(C×,G (R)+);
2. Γ = G (Q)+ ∩ Sp(2g ,Z).
CM points are exactly the zero dimensional special subvarieties of Sg .
9 / 27
Special subvarieties
The Hg can be identified with the GSp(2g ,R)+- conjugacy class of
hg : C×−→GSp(2g ,R)+ x + yi 7→(
x1g y1g
−y1g x1g
)−1
.
An irreducible subvariety V of Sg is special, if V = Γ\X , induced from areductive subgroup G ⊂ GSp(2g) in the following way:
1. X is the image of a G (R)+-conjugacy class in Hom(C×,G (R)+);
2. Γ = G (Q)+ ∩ Sp(2g ,Z).
CM points are exactly the zero dimensional special subvarieties of Sg .
9 / 27
Degree and density
Conjecture (Edixhoven)There are positive cg , δg such that for any CM abelian variety A of dimg , with the center Z of End(A),
deg[A] ≥ cg |disc(Z )|δg .
Let CMg be the set of CM-points on Sg .
Conjecture (Andre–Oort for Siegel moduli)Let V be an irreducible subvariety of Sg such that V ∩ CMg is Zariskidense in V . Then V is a special subvariety of Sg .
Equidistribution ProblemThe Galois orbits for a generic sequence of CM points on a special varietyX/L are equidistruted with respect to some v -adic measure for eachplace v of L.
10 / 27
Degree and density
Conjecture (Edixhoven)There are positive cg , δg such that for any CM abelian variety A of dimg , with the center Z of End(A),
deg[A] ≥ cg |disc(Z )|δg .
Let CMg be the set of CM-points on Sg .
Conjecture (Andre–Oort for Siegel moduli)Let V be an irreducible subvariety of Sg such that V ∩ CMg is Zariskidense in V . Then V is a special subvariety of Sg .
Equidistribution ProblemThe Galois orbits for a generic sequence of CM points on a special varietyX/L are equidistruted with respect to some v -adic measure for eachplace v of L.
10 / 27
Degree and density
Conjecture (Edixhoven)There are positive cg , δg such that for any CM abelian variety A of dimg , with the center Z of End(A),
deg[A] ≥ cg |disc(Z )|δg .
Let CMg be the set of CM-points on Sg .
Conjecture (Andre–Oort for Siegel moduli)Let V be an irreducible subvariety of Sg such that V ∩ CMg is Zariskidense in V . Then V is a special subvariety of Sg .
Equidistribution ProblemThe Galois orbits for a generic sequence of CM points on a special varietyX/L are equidistruted with respect to some v -adic measure for eachplace v of L.
10 / 27
Degree and density
Conjecture (Edixhoven)There are positive cg , δg such that for any CM abelian variety A of dimg , with the center Z of End(A),
deg[A] ≥ cg |disc(Z )|δg .
Let CMg be the set of CM-points on Sg .
Conjecture (Andre–Oort for Siegel moduli)Let V be an irreducible subvariety of Sg such that V ∩ CMg is Zariskidense in V . Then V is a special subvariety of Sg .
Equidistribution ProblemThe Galois orbits for a generic sequence of CM points on a special varietyX/L are equidistruted with respect to some v -adic measure for eachplace v of L.
10 / 27
Previous results on AO
Theorem (Pila–Tsimerman)
Edixhoven =⇒ Andre–Oort for Sg .
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven
Other major contributors to AOEdixhoven, Yafaev, Ullmo, Klingler, Zannier, Gao, ....
TheoremThe AO conjecture holds for mixed Shimura varieties of abelian type.
11 / 27
Previous results on AO
Theorem (Pila–Tsimerman)
Edixhoven =⇒ Andre–Oort for Sg .
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven
Other major contributors to AOEdixhoven, Yafaev, Ullmo, Klingler, Zannier, Gao, ....
TheoremThe AO conjecture holds for mixed Shimura varieties of abelian type.
11 / 27
Previous results on AO
Theorem (Pila–Tsimerman)
Edixhoven =⇒ Andre–Oort for Sg .
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven
Other major contributors to AOEdixhoven, Yafaev, Ullmo, Klingler, Zannier, Gao, ....
TheoremThe AO conjecture holds for mixed Shimura varieties of abelian type.
11 / 27
Previous results on AO
Theorem (Pila–Tsimerman)
Edixhoven =⇒ Andre–Oort for Sg .
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven
Other major contributors to AO
Edixhoven, Yafaev, Ullmo, Klingler, Zannier, Gao, ....
TheoremThe AO conjecture holds for mixed Shimura varieties of abelian type.
11 / 27
Previous results on AO
Theorem (Pila–Tsimerman)
Edixhoven =⇒ Andre–Oort for Sg .
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven
Other major contributors to AOEdixhoven,
Yafaev, Ullmo, Klingler, Zannier, Gao, ....
TheoremThe AO conjecture holds for mixed Shimura varieties of abelian type.
11 / 27
Previous results on AO
Theorem (Pila–Tsimerman)
Edixhoven =⇒ Andre–Oort for Sg .
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven
Other major contributors to AOEdixhoven, Yafaev,
Ullmo, Klingler, Zannier, Gao, ....
TheoremThe AO conjecture holds for mixed Shimura varieties of abelian type.
11 / 27
Previous results on AO
Theorem (Pila–Tsimerman)
Edixhoven =⇒ Andre–Oort for Sg .
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven
Other major contributors to AOEdixhoven, Yafaev, Ullmo,
Klingler, Zannier, Gao, ....
TheoremThe AO conjecture holds for mixed Shimura varieties of abelian type.
11 / 27
Previous results on AO
Theorem (Pila–Tsimerman)
Edixhoven =⇒ Andre–Oort for Sg .
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven
Other major contributors to AOEdixhoven, Yafaev, Ullmo, Klingler,
Zannier, Gao, ....
TheoremThe AO conjecture holds for mixed Shimura varieties of abelian type.
11 / 27
Previous results on AO
Theorem (Pila–Tsimerman)
Edixhoven =⇒ Andre–Oort for Sg .
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven
Other major contributors to AOEdixhoven, Yafaev, Ullmo, Klingler, Zannier,
Gao, ....
TheoremThe AO conjecture holds for mixed Shimura varieties of abelian type.
11 / 27
Previous results on AO
Theorem (Pila–Tsimerman)
Edixhoven =⇒ Andre–Oort for Sg .
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven
Other major contributors to AOEdixhoven, Yafaev, Ullmo, Klingler, Zannier, Gao, ....
TheoremThe AO conjecture holds for mixed Shimura varieties of abelian type.
11 / 27
Previous results on AO
Theorem (Pila–Tsimerman)
Edixhoven =⇒ Andre–Oort for Sg .
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven
Other major contributors to AOEdixhoven, Yafaev, Ullmo, Klingler, Zannier, Gao, ....
TheoremThe AO conjecture holds for mixed Shimura varieties of abelian type.
11 / 27
Faltings Heights
A/K : abelian variety defined over a number field of dim g .A/OK : unit connected component of the Neron model of A.Ω(A) := Lie(A)∨, invariant differetnial 1-forms on A/OK .
ω(A) := det Ω(A) with metric for each archimedean place v of K :
‖α‖2v := (2π)−g
∫Av (C)
|α ∧ α|, α ∈ ω(Av ) = Γ(Av ,ΩgAv
).
Faltings height of A = h(A) :=1
[K : Q]degω(A).
Assume A is semiabelian, then height is invariant under base change.
12 / 27
Faltings Heights
A/K : abelian variety defined over a number field of dim g .
A/OK : unit connected component of the Neron model of A.Ω(A) := Lie(A)∨, invariant differetnial 1-forms on A/OK .
ω(A) := det Ω(A) with metric for each archimedean place v of K :
‖α‖2v := (2π)−g
∫Av (C)
|α ∧ α|, α ∈ ω(Av ) = Γ(Av ,ΩgAv
).
Faltings height of A = h(A) :=1
[K : Q]degω(A).
Assume A is semiabelian, then height is invariant under base change.
12 / 27
Faltings Heights
A/K : abelian variety defined over a number field of dim g .A/OK : unit connected component of the Neron model of A.
Ω(A) := Lie(A)∨, invariant differetnial 1-forms on A/OK .
ω(A) := det Ω(A) with metric for each archimedean place v of K :
‖α‖2v := (2π)−g
∫Av (C)
|α ∧ α|, α ∈ ω(Av ) = Γ(Av ,ΩgAv
).
Faltings height of A = h(A) :=1
[K : Q]degω(A).
Assume A is semiabelian, then height is invariant under base change.
12 / 27
Faltings Heights
A/K : abelian variety defined over a number field of dim g .A/OK : unit connected component of the Neron model of A.Ω(A) := Lie(A)∨, invariant differetnial 1-forms on A/OK .
ω(A) := det Ω(A) with metric for each archimedean place v of K :
‖α‖2v := (2π)−g
∫Av (C)
|α ∧ α|, α ∈ ω(Av ) = Γ(Av ,ΩgAv
).
Faltings height of A = h(A) :=1
[K : Q]degω(A).
Assume A is semiabelian, then height is invariant under base change.
12 / 27
Faltings Heights
A/K : abelian variety defined over a number field of dim g .A/OK : unit connected component of the Neron model of A.Ω(A) := Lie(A)∨, invariant differetnial 1-forms on A/OK .
ω(A) := det Ω(A) with metric for each archimedean place v of K :
‖α‖2v := (2π)−g
∫Av (C)
|α ∧ α|, α ∈ ω(Av ) = Γ(Av ,ΩgAv
).
Faltings height of A = h(A) :=1
[K : Q]degω(A).
Assume A is semiabelian, then height is invariant under base change.
12 / 27
Faltings Heights
A/K : abelian variety defined over a number field of dim g .A/OK : unit connected component of the Neron model of A.Ω(A) := Lie(A)∨, invariant differetnial 1-forms on A/OK .
ω(A) := det Ω(A) with metric for each archimedean place v of K :
‖α‖2v := (2π)−g
∫Av (C)
|α ∧ α|, α ∈ ω(Av ) = Γ(Av ,ΩgAv
).
Faltings height of A = h(A) :=1
[K : Q]degω(A).
Assume A is semiabelian, then height is invariant under base change.
12 / 27
Faltings Heights
A/K : abelian variety defined over a number field of dim g .A/OK : unit connected component of the Neron model of A.Ω(A) := Lie(A)∨, invariant differetnial 1-forms on A/OK .
ω(A) := det Ω(A) with metric for each archimedean place v of K :
‖α‖2v := (2π)−g
∫Av (C)
|α ∧ α|, α ∈ ω(Av ) = Γ(Av ,ΩgAv
).
Faltings height of A = h(A) :=1
[K : Q]degω(A).
Assume A is semiabelian, then height is invariant under base change.
12 / 27
Faltings Heights
A/K : abelian variety defined over a number field of dim g .A/OK : unit connected component of the Neron model of A.Ω(A) := Lie(A)∨, invariant differetnial 1-forms on A/OK .
ω(A) := det Ω(A) with metric for each archimedean place v of K :
‖α‖2v := (2π)−g
∫Av (C)
|α ∧ α|, α ∈ ω(Av ) = Γ(Av ,ΩgAv
).
Faltings height of A = h(A) :=1
[K : Q]degω(A).
Assume A is semiabelian, then height is invariant under base change.
12 / 27
Colmez conjecture
E : CM field with totally real subfield F , [F : Q] = g .
Φ : E ⊗ R ' Cg an isomorphism of R-algbras.A: a CM abelian variety of type (OE ,Φ).
CM theory: A defined over a # field K with a projective A/OK
Colmez: h(A) depends only on (E ,Φ); denote h(AΦ) = h(Φ)
Colmez conjectureThe h(Φ) is a precise linear combination of logarithmic derivatives ofArtin L-functions at 0.
13 / 27
Colmez conjecture
E : CM field with totally real subfield F , [F : Q] = g .Φ : E ⊗ R ' Cg an isomorphism of R-algbras.
A: a CM abelian variety of type (OE ,Φ).
CM theory: A defined over a # field K with a projective A/OK
Colmez: h(A) depends only on (E ,Φ); denote h(AΦ) = h(Φ)
Colmez conjectureThe h(Φ) is a precise linear combination of logarithmic derivatives ofArtin L-functions at 0.
13 / 27
Colmez conjecture
E : CM field with totally real subfield F , [F : Q] = g .Φ : E ⊗ R ' Cg an isomorphism of R-algbras.A: a CM abelian variety of type (OE ,Φ).
CM theory: A defined over a # field K with a projective A/OK
Colmez: h(A) depends only on (E ,Φ); denote h(AΦ) = h(Φ)
Colmez conjectureThe h(Φ) is a precise linear combination of logarithmic derivatives ofArtin L-functions at 0.
13 / 27
Colmez conjecture
E : CM field with totally real subfield F , [F : Q] = g .Φ : E ⊗ R ' Cg an isomorphism of R-algbras.A: a CM abelian variety of type (OE ,Φ).
CM theory: A defined over a # field K with a projective A/OK
Colmez: h(A) depends only on (E ,Φ); denote h(AΦ) = h(Φ)
Colmez conjectureThe h(Φ) is a precise linear combination of logarithmic derivatives ofArtin L-functions at 0.
13 / 27
Colmez conjecture
E : CM field with totally real subfield F , [F : Q] = g .Φ : E ⊗ R ' Cg an isomorphism of R-algbras.A: a CM abelian variety of type (OE ,Φ).
CM theory: A defined over a # field K with a projective A/OK
Colmez: h(A) depends only on (E ,Φ); denote h(AΦ) = h(Φ)
Colmez conjectureThe h(Φ) is a precise linear combination of logarithmic derivatives ofArtin L-functions at 0.
13 / 27
Colmez conjecture
E : CM field with totally real subfield F , [F : Q] = g .Φ : E ⊗ R ' Cg an isomorphism of R-algbras.A: a CM abelian variety of type (OE ,Φ).
CM theory: A defined over a # field K with a projective A/OK
Colmez: h(A) depends only on (E ,Φ); denote h(AΦ) = h(Φ)
Colmez conjectureThe h(Φ) is a precise linear combination of logarithmic derivatives ofArtin L-functions at 0.
13 / 27
Previous results
The conjecture was proved in the following caes:
1. A is an elliptic curve, Chowla–Selberg formula;
2. E/Q is abelian by Colmez and completed by Obus;
3. [E : Q] = 4 by Tonghai Yang.
The conjecture can be considered as a product formula for the norms ofadelic comparisons between de Rham cohomology and adelic (Betti andetele) cohomologies.
14 / 27
Previous results
The conjecture was proved in the following caes:
1. A is an elliptic curve, Chowla–Selberg formula;
2. E/Q is abelian by Colmez and completed by Obus;
3. [E : Q] = 4 by Tonghai Yang.
The conjecture can be considered as a product formula for the norms ofadelic comparisons between de Rham cohomology and adelic (Betti andetele) cohomologies.
14 / 27
Previous results
The conjecture was proved in the following caes:
1. A is an elliptic curve, Chowla–Selberg formula;
2. E/Q is abelian by Colmez and completed by Obus;
3. [E : Q] = 4 by Tonghai Yang.
The conjecture can be considered as a product formula for the norms ofadelic comparisons between de Rham cohomology and adelic (Betti andetele) cohomologies.
14 / 27
Previous results
The conjecture was proved in the following caes:
1. A is an elliptic curve, Chowla–Selberg formula;
2. E/Q is abelian by Colmez and completed by Obus;
3. [E : Q] = 4 by Tonghai Yang.
The conjecture can be considered as a product formula for the norms ofadelic comparisons between de Rham cohomology and adelic (Betti andetele) cohomologies.
14 / 27
Previous results
The conjecture was proved in the following caes:
1. A is an elliptic curve, Chowla–Selberg formula;
2. E/Q is abelian by Colmez and completed by Obus;
3. [E : Q] = 4 by Tonghai Yang.
The conjecture can be considered as a product formula for the norms ofadelic comparisons between de Rham cohomology and adelic (Betti andetele) cohomologies.
14 / 27
Previous results
The conjecture was proved in the following caes:
1. A is an elliptic curve, Chowla–Selberg formula;
2. E/Q is abelian by Colmez and completed by Obus;
3. [E : Q] = 4 by Tonghai Yang.
The conjecture can be considered as a product formula for the norms ofadelic comparisons between de Rham cohomology and adelic (Betti andetele) cohomologies.
14 / 27
A brief history before the conjecture
I the Chowla-Selberg formula (1967);
I Deligne–Gross’ conjecture for the periods of motives with CM by anabelian field (1978);
I Anderson reformulation of DG in terms of the logarithmic derivativesof odd Dirichlet L-functions at s = 0, all up to algebraic numbers(1982);
I Colmez used the Faltings height (or adelic periods) instead of justthe Archimedean periods, to make the conjectures precise (1993).
15 / 27
A brief history before the conjectureI the Chowla-Selberg formula (1967);
I Deligne–Gross’ conjecture for the periods of motives with CM by anabelian field (1978);
I Anderson reformulation of DG in terms of the logarithmic derivativesof odd Dirichlet L-functions at s = 0, all up to algebraic numbers(1982);
I Colmez used the Faltings height (or adelic periods) instead of justthe Archimedean periods, to make the conjectures precise (1993).
15 / 27
A brief history before the conjectureI the Chowla-Selberg formula (1967);
I Deligne–Gross’ conjecture for the periods of motives with CM by anabelian field (1978);
I Anderson reformulation of DG in terms of the logarithmic derivativesof odd Dirichlet L-functions at s = 0, all up to algebraic numbers(1982);
I Colmez used the Faltings height (or adelic periods) instead of justthe Archimedean periods, to make the conjectures precise (1993).
15 / 27
A brief history before the conjectureI the Chowla-Selberg formula (1967);
I Deligne–Gross’ conjecture for the periods of motives with CM by anabelian field (1978);
I Anderson reformulation of DG in terms of the logarithmic derivativesof odd Dirichlet L-functions at s = 0, all up to algebraic numbers(1982);
I Colmez used the Faltings height (or adelic periods) instead of justthe Archimedean periods, to make the conjectures precise (1993).
15 / 27
A brief history before the conjectureI the Chowla-Selberg formula (1967);
I Deligne–Gross’ conjecture for the periods of motives with CM by anabelian field (1978);
I Anderson reformulation of DG in terms of the logarithmic derivativesof odd Dirichlet L-functions at s = 0, all up to algebraic numbers(1982);
I Colmez used the Faltings height (or adelic periods) instead of justthe Archimedean periods, to make the conjectures precise (1993).
15 / 27
Averaged Colmez conjecture
dF : the absolute discriminant of F
dE/F := dE/d2F the norm of the relative discriminant of E/F .
ηE/F : the corresponding quadratic character of A×F .Lf (s, η): the finite part of the completed L-function L(s, η).
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
16 / 27
Averaged Colmez conjecture
dF : the absolute discriminant of FdE/F := dE/d
2F the norm of the relative discriminant of E/F .
ηE/F : the corresponding quadratic character of A×F .Lf (s, η): the finite part of the completed L-function L(s, η).
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
16 / 27
Averaged Colmez conjecture
dF : the absolute discriminant of FdE/F := dE/d
2F the norm of the relative discriminant of E/F .
ηE/F : the corresponding quadratic character of A×F .
Lf (s, η): the finite part of the completed L-function L(s, η).
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
16 / 27
Averaged Colmez conjecture
dF : the absolute discriminant of FdE/F := dE/d
2F the norm of the relative discriminant of E/F .
ηE/F : the corresponding quadratic character of A×F .Lf (s, η): the finite part of the completed L-function L(s, η).
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
16 / 27
Averaged Colmez conjecture
dF : the absolute discriminant of FdE/F := dE/d
2F the norm of the relative discriminant of E/F .
ηE/F : the corresponding quadratic character of A×F .Lf (s, η): the finite part of the completed L-function L(s, η).
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
16 / 27
RemarkThe identity mod Og (log 2) has been announced by Andreatta, Howard,Goren, and Madapusi Pera with a different method of proof:
1. Orthorgonal Shimura varieties and their integral models by Kisin;
2. Borcherds liftings.
RemarkThe averaged Colmez can be considered as an extension to L′(0, η) of thethe class number formula:
L(0, η) =RE/F
wE/F· hE/F
QuestionIs there a arithmetic/geometric formula for every derivative L(n)(0, η)?
17 / 27
RemarkThe identity mod Og (log 2) has been announced by Andreatta, Howard,Goren, and Madapusi Pera with a different method of proof:
1. Orthorgonal Shimura varieties and their integral models by Kisin;
2. Borcherds liftings.
RemarkThe averaged Colmez can be considered as an extension to L′(0, η) of thethe class number formula:
L(0, η) =RE/F
wE/F· hE/F
QuestionIs there a arithmetic/geometric formula for every derivative L(n)(0, η)?
17 / 27
RemarkThe identity mod Og (log 2) has been announced by Andreatta, Howard,Goren, and Madapusi Pera with a different method of proof:
1. Orthorgonal Shimura varieties and their integral models by Kisin;
2. Borcherds liftings.
RemarkThe averaged Colmez can be considered as an extension to L′(0, η) of thethe class number formula:
L(0, η) =RE/F
wE/F· hE/F
QuestionIs there a arithmetic/geometric formula for every derivative L(n)(0, η)?
17 / 27
RemarkThe identity mod Og (log 2) has been announced by Andreatta, Howard,Goren, and Madapusi Pera with a different method of proof:
1. Orthorgonal Shimura varieties and their integral models by Kisin;
2. Borcherds liftings.
RemarkThe averaged Colmez can be considered as an extension to L′(0, η) of thethe class number formula:
L(0, η) =RE/F
wE/F· hE/F
QuestionIs there a arithmetic/geometric formula for every derivative L(n)(0, η)?
17 / 27
RemarkThe identity mod Og (log 2) has been announced by Andreatta, Howard,Goren, and Madapusi Pera with a different method of proof:
1. Orthorgonal Shimura varieties and their integral models by Kisin;
2. Borcherds liftings.
RemarkThe averaged Colmez can be considered as an extension to L′(0, η) of thethe class number formula:
L(0, η) =RE/F
wE/F· hE/F
QuestionIs there a arithmetic/geometric formula for every derivative L(n)(0, η)?
17 / 27
RemarkThe identity mod Og (log 2) has been announced by Andreatta, Howard,Goren, and Madapusi Pera with a different method of proof:
1. Orthorgonal Shimura varieties and their integral models by Kisin;
2. Borcherds liftings.
RemarkThe averaged Colmez can be considered as an extension to L′(0, η) of thethe class number formula:
L(0, η) =RE/F
wE/F· hE/F
QuestionIs there a arithmetic/geometric formula for every derivative L(n)(0, η)?
17 / 27
Proof of Tsimerman’s theorem
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven.
Sketch of proof.For any two elements A,B by OE of type Φ., let deg(A,B) denote theminimal degree of isogenies between A and B.
1. Brauer–Siegel: there are A,B such thatdeg(A,B) >> |disc(E )|1/4+og (1);
2. Masser–Wustholz: there is a positive cg such that for any A,B,max(h(A), deg([A])) >> deg(A,B)cg ;
3. Averaged Colmez: h(A) < |discE |og (1).
Combine to obtain deg[A] >> |discE |cg/4+og (1).
18 / 27
Proof of Tsimerman’s theorem
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven.
Sketch of proof.For any two elements A,B by OE of type Φ., let deg(A,B) denote theminimal degree of isogenies between A and B.
1. Brauer–Siegel: there are A,B such thatdeg(A,B) >> |disc(E )|1/4+og (1);
2. Masser–Wustholz: there is a positive cg such that for any A,B,max(h(A), deg([A])) >> deg(A,B)cg ;
3. Averaged Colmez: h(A) < |discE |og (1).
Combine to obtain deg[A] >> |discE |cg/4+og (1).
18 / 27
Proof of Tsimerman’s theorem
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven.
Sketch of proof.For any two elements A,B by OE of type Φ., let deg(A,B) denote theminimal degree of isogenies between A and B.
1. Brauer–Siegel: there are A,B such thatdeg(A,B) >> |disc(E )|1/4+og (1);
2. Masser–Wustholz: there is a positive cg such that for any A,B,max(h(A), deg([A])) >> deg(A,B)cg ;
3. Averaged Colmez: h(A) < |discE |og (1).
Combine to obtain deg[A] >> |discE |cg/4+og (1).
18 / 27
Proof of Tsimerman’s theorem
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven.
Sketch of proof.For any two elements A,B by OE of type Φ., let deg(A,B) denote theminimal degree of isogenies between A and B.
1. Brauer–Siegel: there are A,B such thatdeg(A,B) >> |disc(E )|1/4+og (1);
2. Masser–Wustholz: there is a positive cg such that for any A,B,max(h(A), deg([A])) >> deg(A,B)cg ;
3. Averaged Colmez: h(A) < |discE |og (1).
Combine to obtain deg[A] >> |discE |cg/4+og (1).
18 / 27
Proof of Tsimerman’s theorem
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven.
Sketch of proof.For any two elements A,B by OE of type Φ., let deg(A,B) denote theminimal degree of isogenies between A and B.
1. Brauer–Siegel: there are A,B such thatdeg(A,B) >> |disc(E )|1/4+og (1);
2. Masser–Wustholz: there is a positive cg such that for any A,B,max(h(A), deg([A])) >> deg(A,B)cg ;
3. Averaged Colmez: h(A) < |discE |og (1).
Combine to obtain deg[A] >> |discE |cg/4+og (1).
18 / 27
Proof of Tsimerman’s theorem
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven.
Sketch of proof.For any two elements A,B by OE of type Φ., let deg(A,B) denote theminimal degree of isogenies between A and B.
1. Brauer–Siegel: there are A,B such thatdeg(A,B) >> |disc(E )|1/4+og (1);
2. Masser–Wustholz: there is a positive cg such that for any A,B,max(h(A), deg([A])) >> deg(A,B)cg ;
3. Averaged Colmez: h(A) < |discE |og (1).
Combine to obtain deg[A] >> |discE |cg/4+og (1).
18 / 27
Proof of Tsimerman’s theorem
Theorem (Tsimerman)
Averaged Colmez =⇒ Edixhoven.
Sketch of proof.For any two elements A,B by OE of type Φ., let deg(A,B) denote theminimal degree of isogenies between A and B.
1. Brauer–Siegel: there are A,B such thatdeg(A,B) >> |disc(E )|1/4+og (1);
2. Masser–Wustholz: there is a positive cg such that for any A,B,max(h(A), deg([A])) >> deg(A,B)cg ;
3. Averaged Colmez: h(A) < |discE |og (1).
Combine to obtain deg[A] >> |discE |cg/4+og (1).
18 / 27
Proof of Colmez conjecture: g = 1
If g = 1, then ω(A) has a (Q-section) given by modular form ` of weight1 with q-expansion at the Tate curve Gm/q
Z:
` = η(q)2 du
u, η(q) = q1/24
∏n
(1− qn).
h(A) =1
12[K : Q]
(log |disc(A)| −
∑σ:K→C
log∣∣η(qσ)24(4πImτσ)6
∣∣) .When A has CM, disc(A) = 1; for the second term, apply eitherKronecker–Limit or Chowla–Selberg formula.
19 / 27
Proof of Colmez conjecture: g = 1
If g = 1, then ω(A) has a (Q-section) given by modular form ` of weight1 with q-expansion at the Tate curve Gm/q
Z:
` = η(q)2 du
u, η(q) = q1/24
∏n
(1− qn).
h(A) =1
12[K : Q]
(log |disc(A)| −
∑σ:K→C
log∣∣η(qσ)24(4πImτσ)6
∣∣) .When A has CM, disc(A) = 1; for the second term, apply eitherKronecker–Limit or Chowla–Selberg formula.
19 / 27
Proof of Colmez conjecture: g = 1
If g = 1, then ω(A) has a (Q-section) given by modular form ` of weight1 with q-expansion at the Tate curve Gm/q
Z:
` = η(q)2 du
u, η(q) = q1/24
∏n
(1− qn).
h(A) =1
12[K : Q]
(log |disc(A)| −
∑σ:K→C
log∣∣η(qσ)24(4πImτσ)6
∣∣) .When A has CM, disc(A) = 1; for the second term, apply eitherKronecker–Limit or Chowla–Selberg formula.
19 / 27
Proof of Colmez conjecture: g = 1
If g = 1, then ω(A) has a (Q-section) given by modular form ` of weight1 with q-expansion at the Tate curve Gm/q
Z:
` = η(q)2 du
u, η(q) = q1/24
∏n
(1− qn).
h(A) =1
12[K : Q]
(log |disc(A)| −
∑σ:K→C
log∣∣η(qσ)24(4πImτσ)6
∣∣) .
When A has CM, disc(A) = 1; for the second term, apply eitherKronecker–Limit or Chowla–Selberg formula.
19 / 27
Proof of Colmez conjecture: g = 1
If g = 1, then ω(A) has a (Q-section) given by modular form ` of weight1 with q-expansion at the Tate curve Gm/q
Z:
` = η(q)2 du
u, η(q) = q1/24
∏n
(1− qn).
h(A) =1
12[K : Q]
(log |disc(A)| −
∑σ:K→C
log∣∣η(qσ)24(4πImτσ)6
∣∣) .When A has CM, disc(A) = 1; for the second term, apply eitherKronecker–Limit or Chowla–Selberg formula.
19 / 27
Proof of Colmez conjecture: g > 1
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
Sketch of proof.If g > 1, there is no natural Q-sections for ω(A). We prove the theoremindirecly in the following three steps:
1. RHS ∼ 12h(P) for a CM point on the quaternionic Shimura curve X ;
this is due to Xinyi Yuan.
2. LHS ∼ 12h(P ′) for a CM point on a unitary Shimura curve X ′;
3. h(P) ∼ h(P ′).
20 / 27
Proof of Colmez conjecture: g > 1
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
Sketch of proof.If g > 1, there is no natural Q-sections for ω(A). We prove the theoremindirecly in the following three steps:
1. RHS ∼ 12h(P) for a CM point on the quaternionic Shimura curve X ;
this is due to Xinyi Yuan.
2. LHS ∼ 12h(P ′) for a CM point on a unitary Shimura curve X ′;
3. h(P) ∼ h(P ′).
20 / 27
Proof of Colmez conjecture: g > 1
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
Sketch of proof.
If g > 1, there is no natural Q-sections for ω(A). We prove the theoremindirecly in the following three steps:
1. RHS ∼ 12h(P) for a CM point on the quaternionic Shimura curve X ;
this is due to Xinyi Yuan.
2. LHS ∼ 12h(P ′) for a CM point on a unitary Shimura curve X ′;
3. h(P) ∼ h(P ′).
20 / 27
Proof of Colmez conjecture: g > 1
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
Sketch of proof.If g > 1, there is no natural Q-sections for ω(A).
We prove the theoremindirecly in the following three steps:
1. RHS ∼ 12h(P) for a CM point on the quaternionic Shimura curve X ;
this is due to Xinyi Yuan.
2. LHS ∼ 12h(P ′) for a CM point on a unitary Shimura curve X ′;
3. h(P) ∼ h(P ′).
20 / 27
Proof of Colmez conjecture: g > 1
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
Sketch of proof.If g > 1, there is no natural Q-sections for ω(A). We prove the theoremindirecly in the following three steps:
1. RHS ∼ 12h(P) for a CM point on the quaternionic Shimura curve X ;
this is due to Xinyi Yuan.
2. LHS ∼ 12h(P ′) for a CM point on a unitary Shimura curve X ′;
3. h(P) ∼ h(P ′).
20 / 27
Proof of Colmez conjecture: g > 1
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
Sketch of proof.If g > 1, there is no natural Q-sections for ω(A). We prove the theoremindirecly in the following three steps:
1. RHS ∼ 12h(P) for a CM point on the quaternionic Shimura curve X ;
this is due to Xinyi Yuan.
2. LHS ∼ 12h(P ′) for a CM point on a unitary Shimura curve X ′;
3. h(P) ∼ h(P ′).
20 / 27
Proof of Colmez conjecture: g > 1
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
Sketch of proof.If g > 1, there is no natural Q-sections for ω(A). We prove the theoremindirecly in the following three steps:
1. RHS ∼ 12h(P) for a CM point on the quaternionic Shimura curve X ;
this is due to Xinyi Yuan.
2. LHS ∼ 12h(P ′) for a CM point on a unitary Shimura curve X ′;
3. h(P) ∼ h(P ′).
20 / 27
Proof of Colmez conjecture: g > 1
Theorem (Averged Colmez conjecture, Xinyi Yuan –)
1
2g
∑Φ
h(Φ) = −1
2
L′f (ηE/F , 0)
Lf (ηE/F , 0)− 1
4log(dE/FdF ).
where Φ runs through the set of CM types of E .
Sketch of proof.If g > 1, there is no natural Q-sections for ω(A). We prove the theoremindirecly in the following three steps:
1. RHS ∼ 12h(P) for a CM point on the quaternionic Shimura curve X ;
this is due to Xinyi Yuan.
2. LHS ∼ 12h(P ′) for a CM point on a unitary Shimura curve X ′;
3. h(P) ∼ h(P ′).
20 / 27
Heights on quaternionic Shimura curve
B/F : a quaternion algebra split at exactly one archimedean place τ of FE → B: an F -embeddingOB : maximal order of of B which contains of OE
X/F : the Shimura curve defined by (G ,X ) = (B×,H±) with level O×B(abelian type)X/OF : a canonical integral model which is Q-factorial.L: the arithmetic Hodge Q-bundle of X , with Hermitian metrics
‖dz‖v = 2 Im(z), v | ∞
P ∈ X (E ab): a CM point by OE with the height defined by
hL(P) =1
[F (P) : F ]deg(L|P).
21 / 27
Heights on quaternionic Shimura curve
B/F : a quaternion algebra split at exactly one archimedean place τ of F
E → B: an F -embeddingOB : maximal order of of B which contains of OE
X/F : the Shimura curve defined by (G ,X ) = (B×,H±) with level O×B(abelian type)X/OF : a canonical integral model which is Q-factorial.L: the arithmetic Hodge Q-bundle of X , with Hermitian metrics
‖dz‖v = 2 Im(z), v | ∞
P ∈ X (E ab): a CM point by OE with the height defined by
hL(P) =1
[F (P) : F ]deg(L|P).
21 / 27
Heights on quaternionic Shimura curve
B/F : a quaternion algebra split at exactly one archimedean place τ of FE → B: an F -embedding
OB : maximal order of of B which contains of OE
X/F : the Shimura curve defined by (G ,X ) = (B×,H±) with level O×B(abelian type)X/OF : a canonical integral model which is Q-factorial.L: the arithmetic Hodge Q-bundle of X , with Hermitian metrics
‖dz‖v = 2 Im(z), v | ∞
P ∈ X (E ab): a CM point by OE with the height defined by
hL(P) =1
[F (P) : F ]deg(L|P).
21 / 27
Heights on quaternionic Shimura curve
B/F : a quaternion algebra split at exactly one archimedean place τ of FE → B: an F -embeddingOB : maximal order of of B which contains of OE
X/F : the Shimura curve defined by (G ,X ) = (B×,H±) with level O×B(abelian type)X/OF : a canonical integral model which is Q-factorial.L: the arithmetic Hodge Q-bundle of X , with Hermitian metrics
‖dz‖v = 2 Im(z), v | ∞
P ∈ X (E ab): a CM point by OE with the height defined by
hL(P) =1
[F (P) : F ]deg(L|P).
21 / 27
Heights on quaternionic Shimura curve
B/F : a quaternion algebra split at exactly one archimedean place τ of FE → B: an F -embeddingOB : maximal order of of B which contains of OE
X/F : the Shimura curve defined by (G ,X ) = (B×,H±) with level O×B(abelian type)
X/OF : a canonical integral model which is Q-factorial.L: the arithmetic Hodge Q-bundle of X , with Hermitian metrics
‖dz‖v = 2 Im(z), v | ∞
P ∈ X (E ab): a CM point by OE with the height defined by
hL(P) =1
[F (P) : F ]deg(L|P).
21 / 27
Heights on quaternionic Shimura curve
B/F : a quaternion algebra split at exactly one archimedean place τ of FE → B: an F -embeddingOB : maximal order of of B which contains of OE
X/F : the Shimura curve defined by (G ,X ) = (B×,H±) with level O×B(abelian type)X/OF : a canonical integral model which is Q-factorial.
L: the arithmetic Hodge Q-bundle of X , with Hermitian metrics
‖dz‖v = 2 Im(z), v | ∞
P ∈ X (E ab): a CM point by OE with the height defined by
hL(P) =1
[F (P) : F ]deg(L|P).
21 / 27
Heights on quaternionic Shimura curve
B/F : a quaternion algebra split at exactly one archimedean place τ of FE → B: an F -embeddingOB : maximal order of of B which contains of OE
X/F : the Shimura curve defined by (G ,X ) = (B×,H±) with level O×B(abelian type)X/OF : a canonical integral model which is Q-factorial.L: the arithmetic Hodge Q-bundle of X , with Hermitian metrics
‖dz‖v = 2 Im(z), v | ∞
P ∈ X (E ab): a CM point by OE with the height defined by
hL(P) =1
[F (P) : F ]deg(L|P).
21 / 27
Heights on quaternionic Shimura curve
B/F : a quaternion algebra split at exactly one archimedean place τ of FE → B: an F -embeddingOB : maximal order of of B which contains of OE
X/F : the Shimura curve defined by (G ,X ) = (B×,H±) with level O×B(abelian type)X/OF : a canonical integral model which is Q-factorial.L: the arithmetic Hodge Q-bundle of X , with Hermitian metrics
‖dz‖v = 2 Im(z), v | ∞
P ∈ X (E ab): a CM point by OE with the height defined by
hL(P) =1
[F (P) : F ]deg(L|P).
21 / 27
Yuan’s theorem
Theorem (Xinyi Yuan)
hL(P) =L′f (ηE/F , 0)
Lf (ηE/F , 0)+
1
2log
dBdE/F
.
Proof.Refining the proof of the Gross–Zagier formula on Shimura curves in AnnMath Studies vol 184. The main ideal is to use Kudla’s generating seriesT (q) of arithmetic Hecke divisors on the product X × X . In the case ofmodular curve X (1) = P1
Z, such a series takes form:
T (q) = T 0
(1− 3
πy
)+∑
T nqn, T 0 = −π∗1ω − π∗2ω.
This series is proportional to the Eisenstein series of weight 2.
22 / 27
Yuan’s theorem
Theorem (Xinyi Yuan)
hL(P) =L′f (ηE/F , 0)
Lf (ηE/F , 0)+
1
2log
dBdE/F
.
Proof.Refining the proof of the Gross–Zagier formula on Shimura curves in AnnMath Studies vol 184. The main ideal is to use Kudla’s generating seriesT (q) of arithmetic Hecke divisors on the product X × X . In the case ofmodular curve X (1) = P1
Z, such a series takes form:
T (q) = T 0
(1− 3
πy
)+∑
T nqn, T 0 = −π∗1ω − π∗2ω.
This series is proportional to the Eisenstein series of weight 2.
22 / 27
Yuan’s theorem
Theorem (Xinyi Yuan)
hL(P) =L′f (ηE/F , 0)
Lf (ηE/F , 0)+
1
2log
dBdE/F
.
Proof.Refining the proof of the Gross–Zagier formula on Shimura curves in AnnMath Studies vol 184.
The main ideal is to use Kudla’s generating seriesT (q) of arithmetic Hecke divisors on the product X × X . In the case ofmodular curve X (1) = P1
Z, such a series takes form:
T (q) = T 0
(1− 3
πy
)+∑
T nqn, T 0 = −π∗1ω − π∗2ω.
This series is proportional to the Eisenstein series of weight 2.
22 / 27
Yuan’s theorem
Theorem (Xinyi Yuan)
hL(P) =L′f (ηE/F , 0)
Lf (ηE/F , 0)+
1
2log
dBdE/F
.
Proof.Refining the proof of the Gross–Zagier formula on Shimura curves in AnnMath Studies vol 184. The main ideal is to use Kudla’s generating seriesT (q) of arithmetic Hecke divisors on the product X × X .
In the case ofmodular curve X (1) = P1
Z, such a series takes form:
T (q) = T 0
(1− 3
πy
)+∑
T nqn, T 0 = −π∗1ω − π∗2ω.
This series is proportional to the Eisenstein series of weight 2.
22 / 27
Yuan’s theorem
Theorem (Xinyi Yuan)
hL(P) =L′f (ηE/F , 0)
Lf (ηE/F , 0)+
1
2log
dBdE/F
.
Proof.Refining the proof of the Gross–Zagier formula on Shimura curves in AnnMath Studies vol 184. The main ideal is to use Kudla’s generating seriesT (q) of arithmetic Hecke divisors on the product X × X . In the case ofmodular curve X (1) = P1
Z, such a series takes form:
T (q) = T 0
(1− 3
πy
)+∑
T nqn, T 0 = −π∗1ω − π∗2ω.
This series is proportional to the Eisenstein series of weight 2.
22 / 27
Yuan’s theorem
Theorem (Xinyi Yuan)
hL(P) =L′f (ηE/F , 0)
Lf (ηE/F , 0)+
1
2log
dBdE/F
.
Proof.Refining the proof of the Gross–Zagier formula on Shimura curves in AnnMath Studies vol 184. The main ideal is to use Kudla’s generating seriesT (q) of arithmetic Hecke divisors on the product X × X . In the case ofmodular curve X (1) = P1
Z, such a series takes form:
T (q) = T 0
(1− 3
πy
)+∑
T nqn, T 0 = −π∗1ω − π∗2ω.
This series is proportional to the Eisenstein series of weight 2.
22 / 27
Decomposition of bundles
K ⊂ C: a number field containing all Galois conjugates of E .
A/OK : a CM abelian variety of type (OE ,Φ).
W (A, τ) := Ω(A)⊗OK⊗OE ,τ OK , ∀τ ∈ Φ.
ω(A)−→⊗τ∈Φ
W (A, τ).
But there is no natural metrics defined on the individual Ω(A)τ .To solve this problem, we bring the dual At into the picture to define:
N (A, τ) := W (A, τ)⊗W (At , τc).
PropositionThere are natural metrics to make hermitian line bundles N (A, τ).
23 / 27
Decomposition of bundles
K ⊂ C: a number field containing all Galois conjugates of E .A/OK : a CM abelian variety of type (OE ,Φ).
W (A, τ) := Ω(A)⊗OK⊗OE ,τ OK , ∀τ ∈ Φ.
ω(A)−→⊗τ∈Φ
W (A, τ).
But there is no natural metrics defined on the individual Ω(A)τ .To solve this problem, we bring the dual At into the picture to define:
N (A, τ) := W (A, τ)⊗W (At , τc).
PropositionThere are natural metrics to make hermitian line bundles N (A, τ).
23 / 27
Decomposition of bundles
K ⊂ C: a number field containing all Galois conjugates of E .A/OK : a CM abelian variety of type (OE ,Φ).
W (A, τ) := Ω(A)⊗OK⊗OE ,τ OK , ∀τ ∈ Φ.
ω(A)−→⊗τ∈Φ
W (A, τ).
But there is no natural metrics defined on the individual Ω(A)τ .To solve this problem, we bring the dual At into the picture to define:
N (A, τ) := W (A, τ)⊗W (At , τc).
PropositionThere are natural metrics to make hermitian line bundles N (A, τ).
23 / 27
Decomposition of bundles
K ⊂ C: a number field containing all Galois conjugates of E .A/OK : a CM abelian variety of type (OE ,Φ).
W (A, τ) := Ω(A)⊗OK⊗OE ,τ OK , ∀τ ∈ Φ.
ω(A)−→⊗τ∈Φ
W (A, τ).
But there is no natural metrics defined on the individual Ω(A)τ .To solve this problem, we bring the dual At into the picture to define:
N (A, τ) := W (A, τ)⊗W (At , τc).
PropositionThere are natural metrics to make hermitian line bundles N (A, τ).
23 / 27
Decomposition of bundles
K ⊂ C: a number field containing all Galois conjugates of E .A/OK : a CM abelian variety of type (OE ,Φ).
W (A, τ) := Ω(A)⊗OK⊗OE ,τ OK , ∀τ ∈ Φ.
ω(A)−→⊗τ∈Φ
W (A, τ).
But there is no natural metrics defined on the individual Ω(A)τ .
To solve this problem, we bring the dual At into the picture to define:
N (A, τ) := W (A, τ)⊗W (At , τc).
PropositionThere are natural metrics to make hermitian line bundles N (A, τ).
23 / 27
Decomposition of bundles
K ⊂ C: a number field containing all Galois conjugates of E .A/OK : a CM abelian variety of type (OE ,Φ).
W (A, τ) := Ω(A)⊗OK⊗OE ,τ OK , ∀τ ∈ Φ.
ω(A)−→⊗τ∈Φ
W (A, τ).
But there is no natural metrics defined on the individual Ω(A)τ .To solve this problem, we bring the dual At into the picture to define:
N (A, τ) := W (A, τ)⊗W (At , τc).
PropositionThere are natural metrics to make hermitian line bundles N (A, τ).
23 / 27
Decomposition of bundles
K ⊂ C: a number field containing all Galois conjugates of E .A/OK : a CM abelian variety of type (OE ,Φ).
W (A, τ) := Ω(A)⊗OK⊗OE ,τ OK , ∀τ ∈ Φ.
ω(A)−→⊗τ∈Φ
W (A, τ).
But there is no natural metrics defined on the individual Ω(A)τ .To solve this problem, we bring the dual At into the picture to define:
N (A, τ) := W (A, τ)⊗W (At , τc).
PropositionThere are natural metrics to make hermitian line bundles N (A, τ).
23 / 27
Decompositions of heights
Definition
h(A, τ) :=1
2deg(N (A, τ)).
The h(A, τ) depends only on the pair (Φ, τ); denote it as h(Φ, τ).EΦ: is the reflex field of (E ,Φ).dΦ, dΦc : absolute discriminants of Φ,Φc .
Theorem
h(Φ)−∑τ∈Φ
h(Φ, τ) =1
4[EΦ : Q]log(dΦdΦc ).
QuestionIs there a Colmez conjecture for h(Φ, τ)?
24 / 27
Decompositions of heights
Definition
h(A, τ) :=1
2deg(N (A, τ)).
The h(A, τ) depends only on the pair (Φ, τ); denote it as h(Φ, τ).EΦ: is the reflex field of (E ,Φ).dΦ, dΦc : absolute discriminants of Φ,Φc .
Theorem
h(Φ)−∑τ∈Φ
h(Φ, τ) =1
4[EΦ : Q]log(dΦdΦc ).
QuestionIs there a Colmez conjecture for h(Φ, τ)?
24 / 27
Decompositions of heights
Definition
h(A, τ) :=1
2deg(N (A, τ)).
The h(A, τ) depends only on the pair (Φ, τ); denote it as h(Φ, τ).
EΦ: is the reflex field of (E ,Φ).dΦ, dΦc : absolute discriminants of Φ,Φc .
Theorem
h(Φ)−∑τ∈Φ
h(Φ, τ) =1
4[EΦ : Q]log(dΦdΦc ).
QuestionIs there a Colmez conjecture for h(Φ, τ)?
24 / 27
Decompositions of heights
Definition
h(A, τ) :=1
2deg(N (A, τ)).
The h(A, τ) depends only on the pair (Φ, τ); denote it as h(Φ, τ).EΦ: is the reflex field of (E ,Φ).
dΦ, dΦc : absolute discriminants of Φ,Φc .
Theorem
h(Φ)−∑τ∈Φ
h(Φ, τ) =1
4[EΦ : Q]log(dΦdΦc ).
QuestionIs there a Colmez conjecture for h(Φ, τ)?
24 / 27
Decompositions of heights
Definition
h(A, τ) :=1
2deg(N (A, τ)).
The h(A, τ) depends only on the pair (Φ, τ); denote it as h(Φ, τ).EΦ: is the reflex field of (E ,Φ).dΦ, dΦc : absolute discriminants of Φ,Φc .
Theorem
h(Φ)−∑τ∈Φ
h(Φ, τ) =1
4[EΦ : Q]log(dΦdΦc ).
QuestionIs there a Colmez conjecture for h(Φ, τ)?
24 / 27
Decompositions of heights
Definition
h(A, τ) :=1
2deg(N (A, τ)).
The h(A, τ) depends only on the pair (Φ, τ); denote it as h(Φ, τ).EΦ: is the reflex field of (E ,Φ).dΦ, dΦc : absolute discriminants of Φ,Φc .
Theorem
h(Φ)−∑τ∈Φ
h(Φ, τ) =1
4[EΦ : Q]log(dΦdΦc ).
QuestionIs there a Colmez conjecture for h(Φ, τ)?
24 / 27
Decompositions of heights
Definition
h(A, τ) :=1
2deg(N (A, τ)).
The h(A, τ) depends only on the pair (Φ, τ); denote it as h(Φ, τ).EΦ: is the reflex field of (E ,Φ).dΦ, dΦc : absolute discriminants of Φ,Φc .
Theorem
h(Φ)−∑τ∈Φ
h(Φ, τ) =1
4[EΦ : Q]log(dΦdΦc ).
QuestionIs there a Colmez conjecture for h(Φ, τ)?
24 / 27
Heights of CM points on unitary Shimura curves
(Φ1,Φ2): a nearby pair of CM types: |Φ1 ∩ Φ2| = g − 1.τi : the complement of Φ1 ∩ Φ2 in Φi .
h(Φ1,Φ2) :=1
2(h(Φ1, τ1) + h(Φ2, τ2))
PropositionLet A0 be an abelian variety with action by OE and isogenous toAΦ1 + AΦ2 .Then
h(Φ1,Φ2) =1
2h(A0, τ).
Now we construct an unitary Shimura curve X ′ (PEL type) Such an A0
corresponds to a CM-point P ′ on X ′.
Theorem
1
2g
∑Φ
h(Φ) ∼ h(Φ1,Φ2) ∼ 1
2h(P ′).
25 / 27
Heights of CM points on unitary Shimura curves
(Φ1,Φ2): a nearby pair of CM types: |Φ1 ∩ Φ2| = g − 1.
τi : the complement of Φ1 ∩ Φ2 in Φi .
h(Φ1,Φ2) :=1
2(h(Φ1, τ1) + h(Φ2, τ2))
PropositionLet A0 be an abelian variety with action by OE and isogenous toAΦ1 + AΦ2 .Then
h(Φ1,Φ2) =1
2h(A0, τ).
Now we construct an unitary Shimura curve X ′ (PEL type) Such an A0
corresponds to a CM-point P ′ on X ′.
Theorem
1
2g
∑Φ
h(Φ) ∼ h(Φ1,Φ2) ∼ 1
2h(P ′).
25 / 27
Heights of CM points on unitary Shimura curves
(Φ1,Φ2): a nearby pair of CM types: |Φ1 ∩ Φ2| = g − 1.τi : the complement of Φ1 ∩ Φ2 in Φi .
h(Φ1,Φ2) :=1
2(h(Φ1, τ1) + h(Φ2, τ2))
PropositionLet A0 be an abelian variety with action by OE and isogenous toAΦ1 + AΦ2 .Then
h(Φ1,Φ2) =1
2h(A0, τ).
Now we construct an unitary Shimura curve X ′ (PEL type) Such an A0
corresponds to a CM-point P ′ on X ′.
Theorem
1
2g
∑Φ
h(Φ) ∼ h(Φ1,Φ2) ∼ 1
2h(P ′).
25 / 27
Heights of CM points on unitary Shimura curves
(Φ1,Φ2): a nearby pair of CM types: |Φ1 ∩ Φ2| = g − 1.τi : the complement of Φ1 ∩ Φ2 in Φi .
h(Φ1,Φ2) :=1
2(h(Φ1, τ1) + h(Φ2, τ2))
PropositionLet A0 be an abelian variety with action by OE and isogenous toAΦ1 + AΦ2 .Then
h(Φ1,Φ2) =1
2h(A0, τ).
Now we construct an unitary Shimura curve X ′ (PEL type) Such an A0
corresponds to a CM-point P ′ on X ′.
Theorem
1
2g
∑Φ
h(Φ) ∼ h(Φ1,Φ2) ∼ 1
2h(P ′).
25 / 27
Heights of CM points on unitary Shimura curves
(Φ1,Φ2): a nearby pair of CM types: |Φ1 ∩ Φ2| = g − 1.τi : the complement of Φ1 ∩ Φ2 in Φi .
h(Φ1,Φ2) :=1
2(h(Φ1, τ1) + h(Φ2, τ2))
Proposition
Let A0 be an abelian variety with action by OE and isogenous toAΦ1 + AΦ2 .Then
h(Φ1,Φ2) =1
2h(A0, τ).
Now we construct an unitary Shimura curve X ′ (PEL type) Such an A0
corresponds to a CM-point P ′ on X ′.
Theorem
1
2g
∑Φ
h(Φ) ∼ h(Φ1,Φ2) ∼ 1
2h(P ′).
25 / 27
Heights of CM points on unitary Shimura curves
(Φ1,Φ2): a nearby pair of CM types: |Φ1 ∩ Φ2| = g − 1.τi : the complement of Φ1 ∩ Φ2 in Φi .
h(Φ1,Φ2) :=1
2(h(Φ1, τ1) + h(Φ2, τ2))
PropositionLet A0 be an abelian variety with action by OE and isogenous toAΦ1 + AΦ2 .
Then
h(Φ1,Φ2) =1
2h(A0, τ).
Now we construct an unitary Shimura curve X ′ (PEL type) Such an A0
corresponds to a CM-point P ′ on X ′.
Theorem
1
2g
∑Φ
h(Φ) ∼ h(Φ1,Φ2) ∼ 1
2h(P ′).
25 / 27
Heights of CM points on unitary Shimura curves
(Φ1,Φ2): a nearby pair of CM types: |Φ1 ∩ Φ2| = g − 1.τi : the complement of Φ1 ∩ Φ2 in Φi .
h(Φ1,Φ2) :=1
2(h(Φ1, τ1) + h(Φ2, τ2))
PropositionLet A0 be an abelian variety with action by OE and isogenous toAΦ1 + AΦ2 .Then
h(Φ1,Φ2) =1
2h(A0, τ).
Now we construct an unitary Shimura curve X ′ (PEL type) Such an A0
corresponds to a CM-point P ′ on X ′.
Theorem
1
2g
∑Φ
h(Φ) ∼ h(Φ1,Φ2) ∼ 1
2h(P ′).
25 / 27
Heights of CM points on unitary Shimura curves
(Φ1,Φ2): a nearby pair of CM types: |Φ1 ∩ Φ2| = g − 1.τi : the complement of Φ1 ∩ Φ2 in Φi .
h(Φ1,Φ2) :=1
2(h(Φ1, τ1) + h(Φ2, τ2))
PropositionLet A0 be an abelian variety with action by OE and isogenous toAΦ1 + AΦ2 .Then
h(Φ1,Φ2) =1
2h(A0, τ).
Now we construct an unitary Shimura curve X ′ (PEL type) Such an A0
corresponds to a CM-point P ′ on X ′.
Theorem
1
2g
∑Φ
h(Φ) ∼ h(Φ1,Φ2) ∼ 1
2h(P ′).
25 / 27
Heights of CM points on unitary Shimura curves
(Φ1,Φ2): a nearby pair of CM types: |Φ1 ∩ Φ2| = g − 1.τi : the complement of Φ1 ∩ Φ2 in Φi .
h(Φ1,Φ2) :=1
2(h(Φ1, τ1) + h(Φ2, τ2))
PropositionLet A0 be an abelian variety with action by OE and isogenous toAΦ1 + AΦ2 .Then
h(Φ1,Φ2) =1
2h(A0, τ).
Now we construct an unitary Shimura curve X ′ (PEL type) Such an A0
corresponds to a CM-point P ′ on X ′.
Theorem
1
2g
∑Φ
h(Φ) ∼ h(Φ1,Φ2) ∼ 1
2h(P ′).
25 / 27
From X to X ′
Theorem
h(P) ∼ h(P ′).
Proof.The curve X and X ′ has the same geometric connected component.The heights of CM points on X and X ′ can be compared using work ofDeligne, Carayol, Cerednik–Drinfeld, and Breuil–Kisin modules forp-divisible groups.
26 / 27
From X to X ′
Theorem
h(P) ∼ h(P ′).
Proof.The curve X and X ′ has the same geometric connected component.
The heights of CM points on X and X ′ can be compared using work ofDeligne, Carayol, Cerednik–Drinfeld, and Breuil–Kisin modules forp-divisible groups.
26 / 27
From X to X ′
Theorem
h(P) ∼ h(P ′).
Proof.The curve X and X ′ has the same geometric connected component.The heights of CM points on X and X ′ can be compared using work of
Deligne, Carayol, Cerednik–Drinfeld, and Breuil–Kisin modules forp-divisible groups.
26 / 27
From X to X ′
Theorem
h(P) ∼ h(P ′).
Proof.The curve X and X ′ has the same geometric connected component.The heights of CM points on X and X ′ can be compared using work ofDeligne, Carayol, Cerednik–Drinfeld, and Breuil–Kisin modules forp-divisible groups.
26 / 27
Questions
1. How to prove Colmez conjecture (without average)?
2. How to generalize Colmez conjecture to CM motives, e.g. oneconsidered by Deligne–Gross?
3. How about high derivatives of L-series?
4. How to prove Andre–Oort conjecture for Shimura varieties ofnon-abelian type?
5. How about the unlikely intersection conjectures of Zilber–Pink?
27 / 27
Questions
1. How to prove Colmez conjecture (without average)?
2. How to generalize Colmez conjecture to CM motives, e.g. oneconsidered by Deligne–Gross?
3. How about high derivatives of L-series?
4. How to prove Andre–Oort conjecture for Shimura varieties ofnon-abelian type?
5. How about the unlikely intersection conjectures of Zilber–Pink?
27 / 27
Questions
1. How to prove Colmez conjecture (without average)?
2. How to generalize Colmez conjecture to CM motives, e.g. oneconsidered by Deligne–Gross?
3. How about high derivatives of L-series?
4. How to prove Andre–Oort conjecture for Shimura varieties ofnon-abelian type?
5. How about the unlikely intersection conjectures of Zilber–Pink?
27 / 27
Questions
1. How to prove Colmez conjecture (without average)?
2. How to generalize Colmez conjecture to CM motives, e.g. oneconsidered by Deligne–Gross?
3. How about high derivatives of L-series?
4. How to prove Andre–Oort conjecture for Shimura varieties ofnon-abelian type?
5. How about the unlikely intersection conjectures of Zilber–Pink?
27 / 27
Questions
1. How to prove Colmez conjecture (without average)?
2. How to generalize Colmez conjecture to CM motives, e.g. oneconsidered by Deligne–Gross?
3. How about high derivatives of L-series?
4. How to prove Andre–Oort conjecture for Shimura varieties ofnon-abelian type?
5. How about the unlikely intersection conjectures of Zilber–Pink?
27 / 27
Questions
1. How to prove Colmez conjecture (without average)?
2. How to generalize Colmez conjecture to CM motives, e.g. oneconsidered by Deligne–Gross?
3. How about high derivatives of L-series?
4. How to prove Andre–Oort conjecture for Shimura varieties ofnon-abelian type?
5. How about the unlikely intersection conjectures of Zilber–Pink?
27 / 27
