![Page 1: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/1.jpg)
Vojta’s conjecture and level structures on abelianvarieties
Dan Abramovich, Brown UniversityJoint work with Anthony Várilly-Alvarado and Keerthi Padapusi Pera
ICERM workshop on Birational Geometry and ArithmeticMay 17, 2018
Dan Abramovich Vojta and levels May 17, 2018 1 / 24
![Page 2: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/2.jpg)
Torsion on elliptic curves
Following [Mazur 1977]. . .
Theorem (Merel, 1996)Fix d ∈Z>0. There is an integer c = c(d) such that:For all number fields k with [k :Q]= d and all elliptic curves E/k ,
#E (k)tors < c .
Mazur: d = 1.What about higher dimension?
(Jump to theorem)
Dan Abramovich Vojta and levels May 17, 2018 2 / 24
![Page 3: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/3.jpg)
Torsion on elliptic curves
Following [Mazur 1977]. . .
Theorem (Merel, 1996)Fix d ∈Z>0. There is an integer c = c(d) such that:For all number fields k with [k :Q]= d and all elliptic curves E/k ,
#E (k)tors < c .
Mazur: d = 1.What about higher dimension?
(Jump to theorem)
Dan Abramovich Vojta and levels May 17, 2018 2 / 24
![Page 4: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/4.jpg)
Torsion on elliptic curves
Following [Mazur 1977]. . .
Theorem (Merel, 1996)Fix d ∈Z>0. There is an integer c = c(d) such that:For all number fields k with [k :Q]= d and all elliptic curves E/k ,
#E (k)tors < c .
Mazur: d = 1.What about higher dimension?
(Jump to theorem)
Dan Abramovich Vojta and levels May 17, 2018 2 / 24
![Page 5: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/5.jpg)
Torsion on abelian varieties
Theorem (Cadoret, Tamagawa 2012)Let k be a field, finitely generated over Q; let p be a prime.
Let A→ S be an abelian scheme over a k-curve S .
There is an integer c = c(A,S ,k ,p) such that
#As(k)[p∞]≤ c
for all s ∈ S(k).
What about all torsion?What about all abelian varieties of fixed dimension together?
Dan Abramovich Vojta and levels May 17, 2018 3 / 24
![Page 6: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/6.jpg)
Torsion on abelian varieties
Theorem (Cadoret, Tamagawa 2012)Let k be a field, finitely generated over Q; let p be a prime.
Let A→ S be an abelian scheme over a k-curve S .
There is an integer c = c(A,S ,k ,p) such that
#As(k)[p∞]≤ c
for all s ∈ S(k).
What about all torsion?What about all abelian varieties of fixed dimension together?
Dan Abramovich Vojta and levels May 17, 2018 3 / 24
![Page 7: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/7.jpg)
Torsion on abelian varieties
Theorem (Cadoret, Tamagawa 2012)Let k be a field, finitely generated over Q; let p be a prime.
Let A→ S be an abelian scheme over a k-curve S .
There is an integer c = c(A,S ,k ,p) such that
#As(k)[p∞]≤ c
for all s ∈ S(k).
What about all torsion?What about all abelian varieties of fixed dimension together?
Dan Abramovich Vojta and levels May 17, 2018 3 / 24
![Page 8: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/8.jpg)
Main Theorem
Let A be a g -dimensional abelian variety over a number field k .
A full-level m structure on A is an isomorphism of k-group schemes
A[m]∼−→ (Z/mZ)g × (µm)
g
Theorem (ℵ, V.-A., M. P. 2017)
Assume Vojta’s conjecture.Fix g ∈Z>0 and a number field k .There is an integer m0 =m0(k ,g) such that:For any m>m0 there is no principally polarized abelian variety A/k ofdimension g with full-level m structure.
Why not torsion?What’s with Vojta?
Dan Abramovich Vojta and levels May 17, 2018 4 / 24
![Page 9: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/9.jpg)
Main Theorem
Let A be a g -dimensional abelian variety over a number field k .
A full-level m structure on A is an isomorphism of k-group schemes
A[m]∼−→ (Z/mZ)g × (µm)
g
Theorem (ℵ, V.-A., M. P. 2017)
Assume Vojta’s conjecture.Fix g ∈Z>0 and a number field k .There is an integer m0 =m0(k ,g) such that:For any m>m0 there is no principally polarized abelian variety A/k ofdimension g with full-level m structure.
Why not torsion?What’s with Vojta?
Dan Abramovich Vojta and levels May 17, 2018 4 / 24
![Page 10: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/10.jpg)
Main Theorem
Let A be a g -dimensional abelian variety over a number field k .
A full-level m structure on A is an isomorphism of k-group schemes
A[m]∼−→ (Z/mZ)g × (µm)
g
Theorem (ℵ, V.-A., M. P. 2017)
Assume Vojta’s conjecture.Fix g ∈Z>0 and a number field k .There is an integer m0 =m0(k ,g) such that:For any m>m0 there is no principally polarized abelian variety A/k ofdimension g with full-level m structure.
Why not torsion?What’s with Vojta?
Dan Abramovich Vojta and levels May 17, 2018 4 / 24
![Page 11: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/11.jpg)
Main Theorem
Let A be a g -dimensional abelian variety over a number field k .
A full-level m structure on A is an isomorphism of k-group schemes
A[m]∼−→ (Z/mZ)g × (µm)
g
Theorem (ℵ, V.-A., M. P. 2017)
Assume Vojta’s conjecture.Fix g ∈Z>0 and a number field k .There is an integer m0 =m0(k ,g) such that:For any m>m0 there is no principally polarized abelian variety A/k ofdimension g with full-level m structure.
Why not torsion?What’s with Vojta?
Dan Abramovich Vojta and levels May 17, 2018 4 / 24
![Page 12: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/12.jpg)
Main Theorem
Let A be a g -dimensional abelian variety over a number field k .
A full-level m structure on A is an isomorphism of k-group schemes
A[m]∼−→ (Z/mZ)g × (µm)
g
Theorem (ℵ, V.-A., M. P. 2017)
Assume Vojta’s conjecture.Fix g ∈Z>0 and a number field k .There is an integer m0 =m0(k ,g) such that:For any m>m0 there is no principally polarized abelian variety A/k ofdimension g with full-level m structure.
Why not torsion?What’s with Vojta?
Dan Abramovich Vojta and levels May 17, 2018 4 / 24
![Page 13: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/13.jpg)
Mazur’s theorem revisited
Consider the curves πm :X1(m)→X (1).
X1(m) parametrizes elliptic curves with m-torsion.
Observation: g(X1(m)) m→∞ //∞ (quadratically)
Faltings (1983) =⇒X1(m)(Q) finite for large m.Manin (1969!):1 =⇒X1(p
k)(Q) finite for some k ,and by Mordell–Weil X1(p
k)(Q)=; for large k .
But there are infinitely many primes >m0 !
(Jump to Flexor–Oesterlé)
1DemjanjenkoDan Abramovich Vojta and levels May 17, 2018 5 / 24
![Page 14: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/14.jpg)
Mazur’s theorem revisited
Consider the curves πm :X1(m)→X (1).
X1(m) parametrizes elliptic curves with m-torsion.
Observation: g(X1(m)) m→∞ //∞ (quadratically)
Faltings (1983) =⇒X1(m)(Q) finite for large m.Manin (1969!):1 =⇒X1(p
k)(Q) finite for some k ,and by Mordell–Weil X1(p
k)(Q)=; for large k .
But there are infinitely many primes >m0 !
(Jump to Flexor–Oesterlé)
1DemjanjenkoDan Abramovich Vojta and levels May 17, 2018 5 / 24
![Page 15: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/15.jpg)
Mazur’s theorem revisited
Consider the curves πm :X1(m)→X (1).
X1(m) parametrizes elliptic curves with m-torsion.
Observation: g(X1(m)) m→∞ //∞ (quadratically)
Faltings (1983) =⇒X1(m)(Q) finite for large m.Manin (1969!):1 =⇒X1(p
k)(Q) finite for some k ,and by Mordell–Weil X1(p
k)(Q)=; for large k .
But there are infinitely many primes >m0 !
(Jump to Flexor–Oesterlé)
1DemjanjenkoDan Abramovich Vojta and levels May 17, 2018 5 / 24
![Page 16: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/16.jpg)
Mazur’s theorem revisited
Consider the curves πm :X1(m)→X (1).
X1(m) parametrizes elliptic curves with m-torsion.
Observation: g(X1(m)) m→∞ //∞ (quadratically)
Faltings (1983) =⇒X1(m)(Q) finite for large m.Manin (1969!):1 =⇒X1(p
k)(Q) finite for some k ,and by Mordell–Weil X1(p
k)(Q)=; for large k .
But there are infinitely many primes >m0 !
(Jump to Flexor–Oesterlé)
1DemjanjenkoDan Abramovich Vojta and levels May 17, 2018 5 / 24
![Page 17: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/17.jpg)
Mazur’s theorem revisited
Consider the curves πm :X1(m)→X (1).
X1(m) parametrizes elliptic curves with m-torsion.
Observation: g(X1(m)) m→∞ //∞ (quadratically)
Faltings (1983) =⇒X1(m)(Q) finite for large m.Manin (1969!):1 =⇒X1(p
k)(Q) finite for some k ,and by Mordell–Weil X1(p
k)(Q)=; for large k .
But there are infinitely many primes >m0 !
(Jump to Flexor–Oesterlé)
1DemjanjenkoDan Abramovich Vojta and levels May 17, 2018 5 / 24
![Page 18: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/18.jpg)
Mazur’s theorem revisited
Consider the curves πm :X1(m)→X (1).
X1(m) parametrizes elliptic curves with m-torsion.
Observation: g(X1(m)) m→∞ //∞ (quadratically)
Faltings (1983) =⇒X1(m)(Q) finite for large m.Manin (1969!):1 =⇒X1(p
k)(Q) finite for some k ,and by Mordell–Weil X1(p
k)(Q)=; for large k .
But there are infinitely many primes >m0 !
(Jump to Flexor–Oesterlé)
1DemjanjenkoDan Abramovich Vojta and levels May 17, 2018 5 / 24
![Page 19: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/19.jpg)
Mazur’s theorem revisited
Consider the curves πm :X1(m)→X (1).
X1(m) parametrizes elliptic curves with m-torsion.
Observation: g(X1(m)) m→∞ //∞ (quadratically)
Faltings (1983) =⇒X1(m)(Q) finite for large m.Manin (1969!):1 =⇒X1(p
k)(Q) finite for some k ,and by Mordell–Weil X1(p
k)(Q)=; for large k .
But there are infinitely many primes >m0 !
(Jump to Flexor–Oesterlé)
1DemjanjenkoDan Abramovich Vojta and levels May 17, 2018 5 / 24
![Page 20: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/20.jpg)
Aside: Cadoret-Tamagawa
Cadoret-Tamagawa consider similarly S1(m)→ S , with componentsS1(m)j .
They show g(S j1(p
k)) //∞ ,. . .
unless they correspond to torsion on an isotrivial factor of A/S .Again this suffices by Faltings and Mordell–Weil for their pk theorem.
Is there an analogue for higher dimensional base?
Dan Abramovich Vojta and levels May 17, 2018 6 / 24
![Page 21: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/21.jpg)
Aside: Cadoret-Tamagawa
Cadoret-Tamagawa consider similarly S1(m)→ S , with componentsS1(m)j .
They show g(S j1(p
k)) //∞ ,. . .
unless they correspond to torsion on an isotrivial factor of A/S .Again this suffices by Faltings and Mordell–Weil for their pk theorem.
Is there an analogue for higher dimensional base?
Dan Abramovich Vojta and levels May 17, 2018 6 / 24
![Page 22: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/22.jpg)
Aside: Cadoret-Tamagawa
Cadoret-Tamagawa consider similarly S1(m)→ S , with componentsS1(m)j .
They show g(S j1(p
k)) //∞ ,. . .
unless they correspond to torsion on an isotrivial factor of A/S .Again this suffices by Faltings and Mordell–Weil for their pk theorem.
Is there an analogue for higher dimensional base?
Dan Abramovich Vojta and levels May 17, 2018 6 / 24
![Page 23: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/23.jpg)
Aside: Cadoret-Tamagawa
Cadoret-Tamagawa consider similarly S1(m)→ S , with componentsS1(m)j .
They show g(S j1(p
k)) //∞ ,. . .
unless they correspond to torsion on an isotrivial factor of A/S .Again this suffices by Faltings and Mordell–Weil for their pk theorem.
Is there an analogue for higher dimensional base?
Dan Abramovich Vojta and levels May 17, 2018 6 / 24
![Page 24: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/24.jpg)
Mazur’s theorem revisited: Flexor–Oesterlé, SilverbergProposition (Flexor–Oesterlé 1988, Silverberg 1992)
There is an integer M =M(g) so that: Suppose A(Q)[p] 6= 0, suppose q isa prime, and suppose p > (1+p
qM)2g . Then the reduction of A at q is“not even potentially good”.
p torsion reduced injectively moduo q.The reduction is not good because of Lang-Weil: there are just toomany points!For potentially good reduction, there is good reduction after anextension of degree <M, so that follows too.
Remark:Flexor and Oesterlé proceed to show that ABC implies uniformboundedness for elliptic curves.This is what we follow: Vojta gives a higher dimensional ABC.Mazur proceeds in another wayDan Abramovich Vojta and levels May 17, 2018 7 / 24
![Page 25: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/25.jpg)
Mazur’s theorem revisited: Flexor–Oesterlé, SilverbergProposition (Flexor–Oesterlé 1988, Silverberg 1992)
There is an integer M =M(g) so that: Suppose A(Q)[p] 6= 0, suppose q isa prime, and suppose p > (1+p
qM)2g . Then the reduction of A at q is“not even potentially good”.
p torsion reduced injectively moduo q.The reduction is not good because of Lang-Weil: there are just toomany points!For potentially good reduction, there is good reduction after anextension of degree <M, so that follows too.
Remark:Flexor and Oesterlé proceed to show that ABC implies uniformboundedness for elliptic curves.This is what we follow: Vojta gives a higher dimensional ABC.Mazur proceeds in another wayDan Abramovich Vojta and levels May 17, 2018 7 / 24
![Page 26: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/26.jpg)
Mazur’s theorem revisited: Flexor–Oesterlé, SilverbergProposition (Flexor–Oesterlé 1988, Silverberg 1992)
There is an integer M =M(g) so that: Suppose A(Q)[p] 6= 0, suppose q isa prime, and suppose p > (1+p
qM)2g . Then the reduction of A at q is“not even potentially good”.
p torsion reduced injectively moduo q.The reduction is not good because of Lang-Weil: there are just toomany points!For potentially good reduction, there is good reduction after anextension of degree <M, so that follows too.
Remark:Flexor and Oesterlé proceed to show that ABC implies uniformboundedness for elliptic curves.This is what we follow: Vojta gives a higher dimensional ABC.Mazur proceeds in another wayDan Abramovich Vojta and levels May 17, 2018 7 / 24
![Page 27: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/27.jpg)
Mazur’s theorem revisited: Flexor–Oesterlé, SilverbergProposition (Flexor–Oesterlé 1988, Silverberg 1992)
There is an integer M =M(g) so that: Suppose A(Q)[p] 6= 0, suppose q isa prime, and suppose p > (1+p
qM)2g . Then the reduction of A at q is“not even potentially good”.
p torsion reduced injectively moduo q.The reduction is not good because of Lang-Weil: there are just toomany points!For potentially good reduction, there is good reduction after anextension of degree <M, so that follows too.
Remark:Flexor and Oesterlé proceed to show that ABC implies uniformboundedness for elliptic curves.This is what we follow: Vojta gives a higher dimensional ABC.Mazur proceeds in another wayDan Abramovich Vojta and levels May 17, 2018 7 / 24
![Page 28: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/28.jpg)
Mazur’s theorem revisited: Flexor–Oesterlé, SilverbergProposition (Flexor–Oesterlé 1988, Silverberg 1992)
There is an integer M =M(g) so that: Suppose A(Q)[p] 6= 0, suppose q isa prime, and suppose p > (1+p
qM)2g . Then the reduction of A at q is“not even potentially good”.
p torsion reduced injectively moduo q.The reduction is not good because of Lang-Weil: there are just toomany points!For potentially good reduction, there is good reduction after anextension of degree <M, so that follows too.
Remark:Flexor and Oesterlé proceed to show that ABC implies uniformboundedness for elliptic curves.This is what we follow: Vojta gives a higher dimensional ABC.Mazur proceeds in another wayDan Abramovich Vojta and levels May 17, 2018 7 / 24
![Page 29: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/29.jpg)
Mazur’s theorem revisited: Flexor–Oesterlé, SilverbergProposition (Flexor–Oesterlé 1988, Silverberg 1992)
There is an integer M =M(g) so that: Suppose A(Q)[p] 6= 0, suppose q isa prime, and suppose p > (1+p
qM)2g . Then the reduction of A at q is“not even potentially good”.
p torsion reduced injectively moduo q.The reduction is not good because of Lang-Weil: there are just toomany points!For potentially good reduction, there is good reduction after anextension of degree <M, so that follows too.
Remark:Flexor and Oesterlé proceed to show that ABC implies uniformboundedness for elliptic curves.This is what we follow: Vojta gives a higher dimensional ABC.Mazur proceeds in another wayDan Abramovich Vojta and levels May 17, 2018 7 / 24
![Page 30: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/30.jpg)
Mazur’s theorem revisited: Flexor–Oesterlé, SilverbergProposition (Flexor–Oesterlé 1988, Silverberg 1992)
There is an integer M =M(g) so that: Suppose A(Q)[p] 6= 0, suppose q isa prime, and suppose p > (1+p
qM)2g . Then the reduction of A at q is“not even potentially good”.
p torsion reduced injectively moduo q.The reduction is not good because of Lang-Weil: there are just toomany points!For potentially good reduction, there is good reduction after anextension of degree <M, so that follows too.
Remark:Flexor and Oesterlé proceed to show that ABC implies uniformboundedness for elliptic curves.This is what we follow: Vojta gives a higher dimensional ABC.Mazur proceeds in another wayDan Abramovich Vojta and levels May 17, 2018 7 / 24
![Page 31: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/31.jpg)
Mazur’s theorem revisited after Merel: Kolyvagin-Logachev,Bump–Friedberg–Hoffstein, Kamienny
The following suffices for Mazur’s theorem:
TheoremFor all large p, X1(p)(Q) consists of cusps.
[Merel] There are many weight-2 cusp forms f on Γ0(p) with analyticrank ords=1L(f ,s)= 0.[KL, BFH 1990] The corresponding factor J0(p)f has rank 0.[Mazur, Kamienny 1982] The composite map X1(p)→ J0(p)f sendingcusp to 0 is immersive at the cusp, even modulo small q.But reduction of torsion of J0(p)f modulo q is injective.Combining with Flexor–Oesterlé we get the result.
Is there a replacement for g > 1???????
Dan Abramovich Vojta and levels May 17, 2018 8 / 24
![Page 32: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/32.jpg)
Mazur’s theorem revisited after Merel: Kolyvagin-Logachev,Bump–Friedberg–Hoffstein, Kamienny
The following suffices for Mazur’s theorem:
TheoremFor all large p, X1(p)(Q) consists of cusps.
[Merel] There are many weight-2 cusp forms f on Γ0(p) with analyticrank ords=1L(f ,s)= 0.[KL, BFH 1990] The corresponding factor J0(p)f has rank 0.[Mazur, Kamienny 1982] The composite map X1(p)→ J0(p)f sendingcusp to 0 is immersive at the cusp, even modulo small q.But reduction of torsion of J0(p)f modulo q is injective.Combining with Flexor–Oesterlé we get the result.
Is there a replacement for g > 1???????
Dan Abramovich Vojta and levels May 17, 2018 8 / 24
![Page 33: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/33.jpg)
Mazur’s theorem revisited after Merel: Kolyvagin-Logachev,Bump–Friedberg–Hoffstein, Kamienny
The following suffices for Mazur’s theorem:
TheoremFor all large p, X1(p)(Q) consists of cusps.
[Merel] There are many weight-2 cusp forms f on Γ0(p) with analyticrank ords=1L(f ,s)= 0.[KL, BFH 1990] The corresponding factor J0(p)f has rank 0.[Mazur, Kamienny 1982] The composite map X1(p)→ J0(p)f sendingcusp to 0 is immersive at the cusp, even modulo small q.But reduction of torsion of J0(p)f modulo q is injective.Combining with Flexor–Oesterlé we get the result.
Is there a replacement for g > 1???????
Dan Abramovich Vojta and levels May 17, 2018 8 / 24
![Page 34: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/34.jpg)
Mazur’s theorem revisited after Merel: Kolyvagin-Logachev,Bump–Friedberg–Hoffstein, Kamienny
The following suffices for Mazur’s theorem:
TheoremFor all large p, X1(p)(Q) consists of cusps.
[Merel] There are many weight-2 cusp forms f on Γ0(p) with analyticrank ords=1L(f ,s)= 0.[KL, BFH 1990] The corresponding factor J0(p)f has rank 0.[Mazur, Kamienny 1982] The composite map X1(p)→ J0(p)f sendingcusp to 0 is immersive at the cusp, even modulo small q.But reduction of torsion of J0(p)f modulo q is injective.Combining with Flexor–Oesterlé we get the result.
Is there a replacement for g > 1???????
Dan Abramovich Vojta and levels May 17, 2018 8 / 24
![Page 35: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/35.jpg)
Mazur’s theorem revisited after Merel: Kolyvagin-Logachev,Bump–Friedberg–Hoffstein, Kamienny
The following suffices for Mazur’s theorem:
TheoremFor all large p, X1(p)(Q) consists of cusps.
[Merel] There are many weight-2 cusp forms f on Γ0(p) with analyticrank ords=1L(f ,s)= 0.[KL, BFH 1990] The corresponding factor J0(p)f has rank 0.[Mazur, Kamienny 1982] The composite map X1(p)→ J0(p)f sendingcusp to 0 is immersive at the cusp, even modulo small q.But reduction of torsion of J0(p)f modulo q is injective.Combining with Flexor–Oesterlé we get the result.
Is there a replacement for g > 1???????
Dan Abramovich Vojta and levels May 17, 2018 8 / 24
![Page 36: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/36.jpg)
Mazur’s theorem revisited after Merel: Kolyvagin-Logachev,Bump–Friedberg–Hoffstein, Kamienny
The following suffices for Mazur’s theorem:
TheoremFor all large p, X1(p)(Q) consists of cusps.
[Merel] There are many weight-2 cusp forms f on Γ0(p) with analyticrank ords=1L(f ,s)= 0.[KL, BFH 1990] The corresponding factor J0(p)f has rank 0.[Mazur, Kamienny 1982] The composite map X1(p)→ J0(p)f sendingcusp to 0 is immersive at the cusp, even modulo small q.But reduction of torsion of J0(p)f modulo q is injective.Combining with Flexor–Oesterlé we get the result.
Is there a replacement for g > 1???????
Dan Abramovich Vojta and levels May 17, 2018 8 / 24
![Page 37: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/37.jpg)
Main Theorem
Let A be a g -dimensional abelian variety over a number field k .
A full-level m structure on A is an isomorphism of k-group schemes
A[m]∼−→ (Z/mZ)g × (µm)
g
Theorem (ℵ, V.-A., M. P. 2017)Assume Vojta’s conjecture.
Fix g ∈Z>0 and a number field k .
There is an integer m0 =m0(k ,g) such that:
For any prime p >m0 there is no (pp) abelian variety A/k of dimension gwith full-level p structure.
Dan Abramovich Vojta and levels May 17, 2018 9 / 24
![Page 38: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/38.jpg)
Strategy
Ag → SpecZ := moduli stack of ppav’s of dimension g .Ag (k)[m] := k-rational points of Ag corresponding to ppav’s A/kadmitting a full-level m structure.
Ag (k)[m] =πm(Ag[m]
(k)),
where Ag[m]
is the space of ppav with full level.
Wi :=⋃p≥i
Ag (k)[p]
Wi is closed in Ag and Wi ⊇Wi+1.Ag is Noetherian, so Wn =Wn+1 = ·· · for some n> 0.Vojta for stacks ⇒ Wn has dimension ≤ 0.
(Jump to Vojta)
Dan Abramovich Vojta and levels May 17, 2018 10 / 24
![Page 39: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/39.jpg)
Strategy
Ag → SpecZ := moduli stack of ppav’s of dimension g .Ag (k)[m] := k-rational points of Ag corresponding to ppav’s A/kadmitting a full-level m structure.
Ag (k)[m] =πm(Ag[m]
(k)),
where Ag[m]
is the space of ppav with full level.
Wi :=⋃p≥i
Ag (k)[p]
Wi is closed in Ag and Wi ⊇Wi+1.Ag is Noetherian, so Wn =Wn+1 = ·· · for some n> 0.Vojta for stacks ⇒ Wn has dimension ≤ 0.
(Jump to Vojta)
Dan Abramovich Vojta and levels May 17, 2018 10 / 24
![Page 40: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/40.jpg)
Strategy
Ag → SpecZ := moduli stack of ppav’s of dimension g .Ag (k)[m] := k-rational points of Ag corresponding to ppav’s A/kadmitting a full-level m structure.
Ag (k)[m] =πm(Ag[m]
(k)),
where Ag[m]
is the space of ppav with full level.
Wi :=⋃p≥i
Ag (k)[p]
Wi is closed in Ag and Wi ⊇Wi+1.Ag is Noetherian, so Wn =Wn+1 = ·· · for some n> 0.Vojta for stacks ⇒ Wn has dimension ≤ 0.
(Jump to Vojta)
Dan Abramovich Vojta and levels May 17, 2018 10 / 24
![Page 41: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/41.jpg)
Strategy
Ag → SpecZ := moduli stack of ppav’s of dimension g .Ag (k)[m] := k-rational points of Ag corresponding to ppav’s A/kadmitting a full-level m structure.
Ag (k)[m] =πm(Ag[m]
(k)),
where Ag[m]
is the space of ppav with full level.
Wi :=⋃p≥i
Ag (k)[p]
Wi is closed in Ag and Wi ⊇Wi+1.Ag is Noetherian, so Wn =Wn+1 = ·· · for some n> 0.Vojta for stacks ⇒ Wn has dimension ≤ 0.
(Jump to Vojta)
Dan Abramovich Vojta and levels May 17, 2018 10 / 24
![Page 42: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/42.jpg)
Strategy
Ag → SpecZ := moduli stack of ppav’s of dimension g .Ag (k)[m] := k-rational points of Ag corresponding to ppav’s A/kadmitting a full-level m structure.
Ag (k)[m] =πm(Ag[m]
(k)),
where Ag[m]
is the space of ppav with full level.
Wi :=⋃p≥i
Ag (k)[p]
Wi is closed in Ag and Wi ⊇Wi+1.Ag is Noetherian, so Wn =Wn+1 = ·· · for some n> 0.Vojta for stacks ⇒ Wn has dimension ≤ 0.
(Jump to Vojta)
Dan Abramovich Vojta and levels May 17, 2018 10 / 24
![Page 43: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/43.jpg)
Strategy
Ag → SpecZ := moduli stack of ppav’s of dimension g .Ag (k)[m] := k-rational points of Ag corresponding to ppav’s A/kadmitting a full-level m structure.
Ag (k)[m] =πm(Ag[m]
(k)),
where Ag[m]
is the space of ppav with full level.
Wi :=⋃p≥i
Ag (k)[p]
Wi is closed in Ag and Wi ⊇Wi+1.Ag is Noetherian, so Wn =Wn+1 = ·· · for some n> 0.Vojta for stacks ⇒ Wn has dimension ≤ 0.
(Jump to Vojta)
Dan Abramovich Vojta and levels May 17, 2018 10 / 24
![Page 44: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/44.jpg)
Dimension 0 case (with Flexor–Oesterlé)
Suppose that Wn =⋃p≥n
Ag (k)[p] has dimension 0.
representing finitely many geometric isomorphism classes of ppav’s.Fix a point in Wn that comes from some A/k .Pick a prime q ∈ SpecOk of potentially good reduction for A.Twists of A with full-level p structure (p > 2; q - p) have goodreduction at q.p-torsion injects modulo q =⇒ p ≤ (1+Nq1/2)2.
There are other approaches!
Dan Abramovich Vojta and levels May 17, 2018 11 / 24
![Page 45: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/45.jpg)
Dimension 0 case (with Flexor–Oesterlé)
Suppose that Wn =⋃p≥n
Ag (k)[p] has dimension 0.
representing finitely many geometric isomorphism classes of ppav’s.Fix a point in Wn that comes from some A/k .Pick a prime q ∈ SpecOk of potentially good reduction for A.Twists of A with full-level p structure (p > 2; q - p) have goodreduction at q.p-torsion injects modulo q =⇒ p ≤ (1+Nq1/2)2.
There are other approaches!
Dan Abramovich Vojta and levels May 17, 2018 11 / 24
![Page 46: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/46.jpg)
Dimension 0 case (with Flexor–Oesterlé)
Suppose that Wn =⋃p≥n
Ag (k)[p] has dimension 0.
representing finitely many geometric isomorphism classes of ppav’s.Fix a point in Wn that comes from some A/k .Pick a prime q ∈ SpecOk of potentially good reduction for A.Twists of A with full-level p structure (p > 2; q - p) have goodreduction at q.p-torsion injects modulo q =⇒ p ≤ (1+Nq1/2)2.
There are other approaches!
Dan Abramovich Vojta and levels May 17, 2018 11 / 24
![Page 47: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/47.jpg)
Dimension 0 case (with Flexor–Oesterlé)
Suppose that Wn =⋃p≥n
Ag (k)[p] has dimension 0.
representing finitely many geometric isomorphism classes of ppav’s.Fix a point in Wn that comes from some A/k .Pick a prime q ∈ SpecOk of potentially good reduction for A.Twists of A with full-level p structure (p > 2; q - p) have goodreduction at q.p-torsion injects modulo q =⇒ p ≤ (1+Nq1/2)2.
There are other approaches!
Dan Abramovich Vojta and levels May 17, 2018 11 / 24
![Page 48: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/48.jpg)
Towards Vojta’s conjecture
k a number field; S a finite set of places containing infinite places.(X ,D) a pair with:
Ï X → SpecOk ,S a smooth proper morphism of schemes;Ï D a fiber-wise normal crossings divisor on X .
(X ,D) := the generic fiber of (X ,D); D =∑i Di .
We view x ∈X (k) as a point of X (Ok(x)),or a scheme Tx := SpecOk(x) →X .
Dan Abramovich Vojta and levels May 17, 2018 12 / 24
![Page 49: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/49.jpg)
Towards Vojta’s conjecture
k a number field; S a finite set of places containing infinite places.(X ,D) a pair with:
Ï X → SpecOk ,S a smooth proper morphism of schemes;Ï D a fiber-wise normal crossings divisor on X .
(X ,D) := the generic fiber of (X ,D); D =∑i Di .
We view x ∈X (k) as a point of X (Ok(x)),or a scheme Tx := SpecOk(x) →X .
Dan Abramovich Vojta and levels May 17, 2018 12 / 24
![Page 50: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/50.jpg)
Towards Vojta’s conjecture
k a number field; S a finite set of places containing infinite places.(X ,D) a pair with:
Ï X → SpecOk ,S a smooth proper morphism of schemes;Ï D a fiber-wise normal crossings divisor on X .
(X ,D) := the generic fiber of (X ,D); D =∑i Di .
We view x ∈X (k) as a point of X (Ok(x)),or a scheme Tx := SpecOk(x) →X .
Dan Abramovich Vojta and levels May 17, 2018 12 / 24
![Page 51: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/51.jpg)
Towards Vojta’s conjecture
k a number field; S a finite set of places containing infinite places.(X ,D) a pair with:
Ï X → SpecOk ,S a smooth proper morphism of schemes;Ï D a fiber-wise normal crossings divisor on X .
(X ,D) := the generic fiber of (X ,D); D =∑i Di .
We view x ∈X (k) as a point of X (Ok(x)),or a scheme Tx := SpecOk(x) →X .
Dan Abramovich Vojta and levels May 17, 2018 12 / 24
![Page 52: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/52.jpg)
Towards Vojta: counting functions and discriminants
Definition
For x ∈X (k) with residue field k(x) define the truncated counting function
N(1)k
(D ,x)= 1[k(x) : k]
∑q∈SpecOk ,S
(D |Tx )q 6=;
log |κ(q)|︸ ︷︷ ︸size of
residue field
.
and the relative logarithmic discriminant
dk(k(x))=1
[k(x) : k]log |DiscOk(x)|− log |DiscOk |
= 1[k(x) : k]
degΩOk(x)/Ok.
Dan Abramovich Vojta and levels May 17, 2018 13 / 24
![Page 53: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/53.jpg)
Towards Vojta: counting functions and discriminants
Definition
For x ∈X (k) with residue field k(x) define the truncated counting function
N(1)k
(D ,x)= 1[k(x) : k]
∑q∈SpecOk ,S
(D |Tx )q 6=;
log |κ(q)|︸ ︷︷ ︸size of
residue field
.
and the relative logarithmic discriminant
dk(k(x))=1
[k(x) : k]log |DiscOk(x)|− log |DiscOk |
= 1[k(x) : k]
degΩOk(x)/Ok.
Dan Abramovich Vojta and levels May 17, 2018 13 / 24
![Page 54: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/54.jpg)
Vojta’s conjecture
Conjecture (Vojta c. 1984; 1998)X a smooth projective variety over a number field k .D a normal crossings divisor on X ; H a big line bundle on X .Fix a positive integer r and δ> 0.There is a proper Zariski closed Z ⊂X containing D such that
N(1)X
(D ,x)+dk(k(x))≥ hKX+D(x)−δhH(x)−Or (1)
for all x ∈X (k)àZ (k) with [k(x) : k]≤ r .
dk(k(x)) measure failure of being in X (k)
N(1)X
(D ,x) measure failure of being in X 0(Ok)= (X \D)(Ok)
Dan Abramovich Vojta and levels May 17, 2018 14 / 24
![Page 55: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/55.jpg)
Vojta’s conjecture
Conjecture (Vojta c. 1984; 1998)X a smooth projective variety over a number field k .D a normal crossings divisor on X ; H a big line bundle on X .Fix a positive integer r and δ> 0.There is a proper Zariski closed Z ⊂X containing D such that
N(1)X
(D ,x)+dk(k(x))≥ hKX+D(x)−δhH(x)−Or (1)
for all x ∈X (k)àZ (k) with [k(x) : k]≤ r .
dk(k(x)) measure failure of being in X (k)
N(1)X
(D ,x) measure failure of being in X 0(Ok)= (X \D)(Ok)
Dan Abramovich Vojta and levels May 17, 2018 14 / 24
![Page 56: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/56.jpg)
Vojta’s conjecture
Conjecture (Vojta c. 1984; 1998)X a smooth projective variety over a number field k .D a normal crossings divisor on X ; H a big line bundle on X .Fix a positive integer r and δ> 0.There is a proper Zariski closed Z ⊂X containing D such that
N(1)X
(D ,x)+dk(k(x))≥ hKX+D(x)−δhH(x)−Or (1)
for all x ∈X (k)àZ (k) with [k(x) : k]≤ r .
dk(k(x)) measure failure of being in X (k)
N(1)X
(D ,x) measure failure of being in X 0(Ok)= (X \D)(Ok)
Dan Abramovich Vojta and levels May 17, 2018 14 / 24
![Page 57: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/57.jpg)
Vojta’s conjecture
Conjecture (Vojta c. 1984; 1998)X a smooth projective variety over a number field k .D a normal crossings divisor on X ; H a big line bundle on X .Fix a positive integer r and δ> 0.There is a proper Zariski closed Z ⊂X containing D such that
N(1)X
(D ,x)+dk(k(x))≥ hKX+D(x)−δhH(x)−Or (1)
for all x ∈X (k)àZ (k) with [k(x) : k]≤ r .
dk(k(x)) measure failure of being in X (k)
N(1)X
(D ,x) measure failure of being in X 0(Ok)= (X \D)(Ok)
Dan Abramovich Vojta and levels May 17, 2018 14 / 24
![Page 58: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/58.jpg)
Vojta’s conjecture: special cases
D =;; H =KX ; r = 1; X of general type:Lang’s conjecture: X (k) not Zariski dense.H =KX (D); r = 1; S a finite set of places ; (X ,D) of log general type:Lang–Vojta conjecture: X 0(Ok ,S) not Zariski dense.X =P1;r = 1;D = 0,1,∞: Masser–Oesterlé’s ABC conjecture.
Dan Abramovich Vojta and levels May 17, 2018 15 / 24
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Vojta’s conjecture: special cases
D =;; H =KX ; r = 1; X of general type:Lang’s conjecture: X (k) not Zariski dense.H =KX (D); r = 1; S a finite set of places ; (X ,D) of log general type:Lang–Vojta conjecture: X 0(Ok ,S) not Zariski dense.X =P1;r = 1;D = 0,1,∞: Masser–Oesterlé’s ABC conjecture.
Dan Abramovich Vojta and levels May 17, 2018 15 / 24
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Vojta’s conjecture: special cases
D =;; H =KX ; r = 1; X of general type:Lang’s conjecture: X (k) not Zariski dense.H =KX (D); r = 1; S a finite set of places ; (X ,D) of log general type:Lang–Vojta conjecture: X 0(Ok ,S) not Zariski dense.X =P1;r = 1;D = 0,1,∞: Masser–Oesterlé’s ABC conjecture.
Dan Abramovich Vojta and levels May 17, 2018 15 / 24
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Vojta’s conjecture: special cases
D =;; H =KX ; r = 1; X of general type:Lang’s conjecture: X (k) not Zariski dense.H =KX (D); r = 1; S a finite set of places ; (X ,D) of log general type:Lang–Vojta conjecture: X 0(Ok ,S) not Zariski dense.X =P1;r = 1;D = 0,1,∞: Masser–Oesterlé’s ABC conjecture.
Dan Abramovich Vojta and levels May 17, 2018 15 / 24
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Extending Vojta to DM stacks
Recall: Vojta ⇒ Lang.Example: X =P2(pC ), where C a smooth curve of degree > 6.Then KX ∼O(d/2−3) is big, so X of general type, but X (k) is dense.The point is that a rational point might still fail to be integral: it mayhave “potentially good reduction” but not “good reduction”!The correct form of Lang’s conjecture is: if X is of general type thenX (Ok ,S) is not Zariski-dense.
What about a quantitative version? We need to account that even rationalpoints may be ramified.
Heights and intersection numbers are defined as usual.We must define the discriminant of a point x ∈X (k).
Dan Abramovich Vojta and levels May 17, 2018 16 / 24
![Page 63: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/63.jpg)
Extending Vojta to DM stacks
Recall: Vojta ⇒ Lang.Example: X =P2(pC ), where C a smooth curve of degree > 6.Then KX ∼O(d/2−3) is big, so X of general type, but X (k) is dense.The point is that a rational point might still fail to be integral: it mayhave “potentially good reduction” but not “good reduction”!The correct form of Lang’s conjecture is: if X is of general type thenX (Ok ,S) is not Zariski-dense.
What about a quantitative version? We need to account that even rationalpoints may be ramified.
Heights and intersection numbers are defined as usual.We must define the discriminant of a point x ∈X (k).
Dan Abramovich Vojta and levels May 17, 2018 16 / 24
![Page 64: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/64.jpg)
Extending Vojta to DM stacks
Recall: Vojta ⇒ Lang.Example: X =P2(pC ), where C a smooth curve of degree > 6.Then KX ∼O(d/2−3) is big, so X of general type, but X (k) is dense.The point is that a rational point might still fail to be integral: it mayhave “potentially good reduction” but not “good reduction”!The correct form of Lang’s conjecture is: if X is of general type thenX (Ok ,S) is not Zariski-dense.
What about a quantitative version? We need to account that even rationalpoints may be ramified.
Heights and intersection numbers are defined as usual.We must define the discriminant of a point x ∈X (k).
Dan Abramovich Vojta and levels May 17, 2018 16 / 24
![Page 65: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/65.jpg)
Extending Vojta to DM stacks
Recall: Vojta ⇒ Lang.Example: X =P2(pC ), where C a smooth curve of degree > 6.Then KX ∼O(d/2−3) is big, so X of general type, but X (k) is dense.The point is that a rational point might still fail to be integral: it mayhave “potentially good reduction” but not “good reduction”!The correct form of Lang’s conjecture is: if X is of general type thenX (Ok ,S) is not Zariski-dense.
What about a quantitative version? We need to account that even rationalpoints may be ramified.
Heights and intersection numbers are defined as usual.We must define the discriminant of a point x ∈X (k).
Dan Abramovich Vojta and levels May 17, 2018 16 / 24
![Page 66: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/66.jpg)
Extending Vojta to DM stacks
Recall: Vojta ⇒ Lang.Example: X =P2(pC ), where C a smooth curve of degree > 6.Then KX ∼O(d/2−3) is big, so X of general type, but X (k) is dense.The point is that a rational point might still fail to be integral: it mayhave “potentially good reduction” but not “good reduction”!The correct form of Lang’s conjecture is: if X is of general type thenX (Ok ,S) is not Zariski-dense.
What about a quantitative version? We need to account that even rationalpoints may be ramified.
Heights and intersection numbers are defined as usual.We must define the discriminant of a point x ∈X (k).
Dan Abramovich Vojta and levels May 17, 2018 16 / 24
![Page 67: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/67.jpg)
Discriminant of a rational point
X → SpecOk ,S smooth proper, X a DM stack.For x ∈X (k) with residue field k(x), take Zariski closure andnormalization of its image.Get a morphism Tx →X , with Tx a normal stack with coarse modulischeme SpecOk(x),S .The relative logarithmic discriminant is
dk(Tx)= 1degTx/Ok
degΩTx/Ok.
Dan Abramovich Vojta and levels May 17, 2018 17 / 24
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Discriminant of a rational point
X → SpecOk ,S smooth proper, X a DM stack.For x ∈X (k) with residue field k(x), take Zariski closure andnormalization of its image.Get a morphism Tx →X , with Tx a normal stack with coarse modulischeme SpecOk(x),S .The relative logarithmic discriminant is
dk(Tx)= 1degTx/Ok
degΩTx/Ok.
Dan Abramovich Vojta and levels May 17, 2018 17 / 24
![Page 69: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/69.jpg)
Discriminant of a rational point
X → SpecOk ,S smooth proper, X a DM stack.For x ∈X (k) with residue field k(x), take Zariski closure andnormalization of its image.Get a morphism Tx →X , with Tx a normal stack with coarse modulischeme SpecOk(x),S .The relative logarithmic discriminant is
dk(Tx)= 1degTx/Ok
degΩTx/Ok.
Dan Abramovich Vojta and levels May 17, 2018 17 / 24
![Page 70: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/70.jpg)
Vojta’s conjecture for stacks
Conjecturek number field; S a finite set of places (including infinite ones).X → SpecOk ,S a smooth proper DM stack.X =Xk generic fiber (assume irreducible)X coarse moduli of X ; assume projective with big line bundle H.D ⊆X NC divisor with generic fiber D.Fix a positive integer r and δ> 0.
There is a proper Zariski closed Z ⊂X containing D such that
N(1)X
(D ,x)+dk(Tx)≥ hKX+D(x)−δhH(x)−O(1)
for all x ∈X (k)àZ (k) with [k(x) : k]≤ r .
Dan Abramovich Vojta and levels May 17, 2018 18 / 24
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Vojta’s conjecture for stacks
Conjecturek number field; S a finite set of places (including infinite ones).X → SpecOk ,S a smooth proper DM stack.X =Xk generic fiber (assume irreducible)X coarse moduli of X ; assume projective with big line bundle H.D ⊆X NC divisor with generic fiber D.Fix a positive integer r and δ> 0.
There is a proper Zariski closed Z ⊂X containing D such that
N(1)X
(D ,x)+dk(Tx)≥ hKX+D(x)−δhH(x)−O(1)
for all x ∈X (k)àZ (k) with [k(x) : k]≤ r .
Dan Abramovich Vojta and levels May 17, 2018 18 / 24
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Vojta is flexible
Proposition (ℵ, V.-A. 2017)Vojta for DM stacks follows from Vojta for schemes.
Key: Vojta showed that Vojta’s conjecture is compatible with takingbranched covers.
Proposition (Kresch–Vistoli)There is a finite flat surjective morphism π : Y →X
with Y a smooth projective irreducible schemeand DY :=π∗D a NC divisor.
Vojta for Y =⇒ Vojta for X .
Dan Abramovich Vojta and levels May 17, 2018 19 / 24
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Vojta is flexible
Proposition (ℵ, V.-A. 2017)Vojta for DM stacks follows from Vojta for schemes.
Key: Vojta showed that Vojta’s conjecture is compatible with takingbranched covers.
Proposition (Kresch–Vistoli)There is a finite flat surjective morphism π : Y →X
with Y a smooth projective irreducible schemeand DY :=π∗D a NC divisor.
Vojta for Y =⇒ Vojta for X .
Dan Abramovich Vojta and levels May 17, 2018 19 / 24
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Completing the proof of the Main Theorem
Recall:
Wi :=⋃p≥i
Ag (k)[p]
and Wn =Wn+1 = ·· · for some n> 0.Want to show: dimWn ≤ 0. Proceed by contradiction.Let X is an irreducible positive dimensional component of Wn.X ′ →X a resolution of singularities.
X ′ ⊆X′smooth compactification with D :=X
′àX NC divisor.
Pick model (X ,D) of (X′,D) over SpecOk ,S (Olsson)
Dan Abramovich Vojta and levels May 17, 2018 20 / 24
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Completing the proof of the Main Theorem
Recall:
Wi :=⋃p≥i
Ag (k)[p]
and Wn =Wn+1 = ·· · for some n> 0.Want to show: dimWn ≤ 0. Proceed by contradiction.Let X is an irreducible positive dimensional component of Wn.X ′ →X a resolution of singularities.
X ′ ⊆X′smooth compactification with D :=X
′àX NC divisor.
Pick model (X ,D) of (X′,D) over SpecOk ,S (Olsson)
Dan Abramovich Vojta and levels May 17, 2018 20 / 24
![Page 76: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/76.jpg)
Completing the proof of the Main Theorem
Recall:
Wi :=⋃p≥i
Ag (k)[p]
and Wn =Wn+1 = ·· · for some n> 0.Want to show: dimWn ≤ 0. Proceed by contradiction.Let X is an irreducible positive dimensional component of Wn.X ′ →X a resolution of singularities.
X ′ ⊆X′smooth compactification with D :=X
′àX NC divisor.
Pick model (X ,D) of (X′,D) over SpecOk ,S (Olsson)
Dan Abramovich Vojta and levels May 17, 2018 20 / 24
![Page 77: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/77.jpg)
Completing the proof of the Main Theorem
Recall:
Wi :=⋃p≥i
Ag (k)[p]
and Wn =Wn+1 = ·· · for some n> 0.Want to show: dimWn ≤ 0. Proceed by contradiction.Let X is an irreducible positive dimensional component of Wn.X ′ →X a resolution of singularities.
X ′ ⊆X′smooth compactification with D :=X
′àX NC divisor.
Pick model (X ,D) of (X′,D) over SpecOk ,S (Olsson)
Dan Abramovich Vojta and levels May 17, 2018 20 / 24
![Page 78: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/78.jpg)
Birational geometry
[Zuo 2000] KX
′ +D is big.Remark [Brunebarbe 2017]:As soon as m> 12g , every subvariety of A
[m]g is of general type.
Uses the fact that A[m]g →Ag is highly ramified along the boundary.
Implies a Manin-type result for full [pr ]-levels.
Can one prove a result for torsion rather than full level?
Taking H =KX
′ +D get by Northcott an observation on the right handside of Vojta’s conjecture
N(1)X
(D ,x)+dk(Tx)≥ hKX′+D(x)−δhH(x)︸ ︷︷ ︸
large for small δ away from some Z
−O(1)
Dan Abramovich Vojta and levels May 17, 2018 21 / 24
![Page 79: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/79.jpg)
Birational geometry
[Zuo 2000] KX
′ +D is big.Remark [Brunebarbe 2017]:As soon as m> 12g , every subvariety of A
[m]g is of general type.
Uses the fact that A[m]g →Ag is highly ramified along the boundary.
Implies a Manin-type result for full [pr ]-levels.
Can one prove a result for torsion rather than full level?
Taking H =KX
′ +D get by Northcott an observation on the right handside of Vojta’s conjecture
N(1)X
(D ,x)+dk(Tx)≥ hKX′+D(x)−δhH(x)︸ ︷︷ ︸
large for small δ away from some Z
−O(1)
Dan Abramovich Vojta and levels May 17, 2018 21 / 24
![Page 80: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/80.jpg)
Birational geometry
[Zuo 2000] KX
′ +D is big.Remark [Brunebarbe 2017]:As soon as m> 12g , every subvariety of A
[m]g is of general type.
Uses the fact that A[m]g →Ag is highly ramified along the boundary.
Implies a Manin-type result for full [pr ]-levels.
Can one prove a result for torsion rather than full level?
Taking H =KX
′ +D get by Northcott an observation on the right handside of Vojta’s conjecture
N(1)X
(D ,x)+dk(Tx)≥ hKX′+D(x)−δhH(x)︸ ︷︷ ︸
large for small δ away from some Z
−O(1)
Dan Abramovich Vojta and levels May 17, 2018 21 / 24
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Key Lemma
X (k)[p] = k-rational points of X corresponding to ppav’s A/k admitting afull-level p structure.
LemmaFix ε1,ε2 > 0. For all pÀ 0 and x ∈X (k)[p], we have(1)
N(1)X
(D ,x)≤ ε1hD(x)+O(1)
and(1)
dk(Tx)≤ ε2hD(x)+O(1).
Note: hD ¿ hH outside some Z .Vojta gives, outside some Z ,hH(x)¿N
(1)X
(D ,x)+dk(Tx)¿ εhH(x),giving finiteness outside this Z by Northcott.
Dan Abramovich Vojta and levels May 17, 2018 22 / 24
![Page 82: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/82.jpg)
Key Lemma
X (k)[p] = k-rational points of X corresponding to ppav’s A/k admitting afull-level p structure.
LemmaFix ε1,ε2 > 0. For all pÀ 0 and x ∈X (k)[p], we have(1)
N(1)X
(D ,x)≤ ε1hD(x)+O(1)
and(1)
dk(Tx)≤ ε2hD(x)+O(1).
Note: hD ¿ hH outside some Z .Vojta gives, outside some Z ,hH(x)¿N
(1)X
(D ,x)+dk(Tx)¿ εhH(x),giving finiteness outside this Z by Northcott.
Dan Abramovich Vojta and levels May 17, 2018 22 / 24
![Page 83: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/83.jpg)
Key Lemma
X (k)[p] = k-rational points of X corresponding to ppav’s A/k admitting afull-level p structure.
LemmaFix ε1,ε2 > 0. For all pÀ 0 and x ∈X (k)[p], we have(1)
N(1)X
(D ,x)≤ ε1hD(x)+O(1)
and(1)
dk(Tx)≤ ε2hD(x)+O(1).
Note: hD ¿ hH outside some Z .Vojta gives, outside some Z ,hH(x)¿N
(1)X
(D ,x)+dk(Tx)¿ εhH(x),giving finiteness outside this Z by Northcott.
Dan Abramovich Vojta and levels May 17, 2018 22 / 24
![Page 84: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/84.jpg)
Key Lemma
X (k)[p] = k-rational points of X corresponding to ppav’s A/k admitting afull-level p structure.
LemmaFix ε1,ε2 > 0. For all pÀ 0 and x ∈X (k)[p], we have(1)
N(1)X
(D ,x)≤ ε1hD(x)+O(1)
and(1)
dk(Tx)≤ ε2hD(x)+O(1).
Note: hD ¿ hH outside some Z .Vojta gives, outside some Z ,hH(x)¿N
(1)X
(D ,x)+dk(Tx)¿ εhH(x),giving finiteness outside this Z by Northcott.
Dan Abramovich Vojta and levels May 17, 2018 22 / 24
![Page 85: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/85.jpg)
(1) N (1)(D ,x)¿ ε1hD(x)
x is the image of a rational point on A[p]g
πp :A[p]g →Ag is highly ramified along D (Mumford / Madapusi
Pera).So whenever (D|Tx )q 6= ; its multiplicity is À p.
so N(1)(D ,x)¿ hD(x)1p︸︷︷︸∼ε1
.
Dan Abramovich Vojta and levels May 17, 2018 23 / 24
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(1) N (1)(D ,x)¿ ε1hD(x)
x is the image of a rational point on A[p]g
πp :A[p]g →Ag is highly ramified along D (Mumford / Madapusi
Pera).So whenever (D|Tx )q 6= ; its multiplicity is À p.
so N(1)(D ,x)¿ hD(x)1p︸︷︷︸∼ε1
.
Dan Abramovich Vojta and levels May 17, 2018 23 / 24
![Page 87: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/87.jpg)
(1) N (1)(D ,x)¿ ε1hD(x)
x is the image of a rational point on A[p]g
πp :A[p]g →Ag is highly ramified along D (Mumford / Madapusi
Pera).So whenever (D|Tx )q 6= ; its multiplicity is À p.
so N(1)(D ,x)¿ hD(x)1p︸︷︷︸∼ε1
.
Dan Abramovich Vojta and levels May 17, 2018 23 / 24
![Page 88: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/88.jpg)
(1) N (1)(D ,x)¿ ε1hD(x)
x is the image of a rational point on A[p]g
πp :A[p]g →Ag is highly ramified along D (Mumford / Madapusi
Pera).So whenever (D|Tx )q 6= ; its multiplicity is À p.
so N(1)(D ,x)¿ hD(x)1p︸︷︷︸∼ε1
.
Dan Abramovich Vojta and levels May 17, 2018 23 / 24
![Page 89: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/89.jpg)
(2) dk(Tx)≤ ε2hD(x)
x corresponds to an abelian variety with many p-torsion points.Flexor–Oesterlé at any small prime ⇒ hD(x)À ps .
x has semistable reduction outside p ⇒ dk(Tx)¿ logp
so dk(Tx)¿ hD(x)logpps︸ ︷︷ ︸∼ε2
.
Dan Abramovich Vojta and levels May 17, 2018 24 / 24
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(2) dk(Tx)≤ ε2hD(x)
x corresponds to an abelian variety with many p-torsion points.Flexor–Oesterlé at any small prime ⇒ hD(x)À ps .
x has semistable reduction outside p ⇒ dk(Tx)¿ logp
so dk(Tx)¿ hD(x)logpps︸ ︷︷ ︸∼ε2
.
Dan Abramovich Vojta and levels May 17, 2018 24 / 24
![Page 91: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/91.jpg)
(2) dk(Tx)≤ ε2hD(x)
x corresponds to an abelian variety with many p-torsion points.Flexor–Oesterlé at any small prime ⇒ hD(x)À ps .
x has semistable reduction outside p ⇒ dk(Tx)¿ logp
so dk(Tx)¿ hD(x)logpps︸ ︷︷ ︸∼ε2
.
Dan Abramovich Vojta and levels May 17, 2018 24 / 24
![Page 92: Vojta’sconjectureandlevelstructuresonabelian varietiesabrmovic/PAPERS/ICERM2018.pdf · Vojta’sconjectureandlevelstructuresonabelian varieties DanAbramovich,BrownUniversity JointworkwithAnthonyVárilly-AlvaradoandKeerthiPadapusiPera](https://reader031.vdocument.in/reader031/viewer/2022021713/5bb0994a09d3f272478bb9ac/html5/thumbnails/92.jpg)
(2) dk(Tx)≤ ε2hD(x)
x corresponds to an abelian variety with many p-torsion points.Flexor–Oesterlé at any small prime ⇒ hD(x)À ps .
x has semistable reduction outside p ⇒ dk(Tx)¿ logp
so dk(Tx)¿ hD(x)logpps︸ ︷︷ ︸∼ε2
.
Dan Abramovich Vojta and levels May 17, 2018 24 / 24