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Rational points on curves and tropical geometry.
David Zureick-Brown (Emory University)Eric Katz (Waterloo University)
Slides available at http://www.mathcs.emory.edu/~dzb/slides/
Specialization of Linear Series for Algebraic and Tropical CurvesBIRS
April 3, 2014
http://www.mathcs.emory.edu/~dzb/slides/
Faltings’ theorem
Theorem (Faltings)
Let X be a smooth curve over Q with genus at least 2. Then X (Q) isfinite.
Example
For g ≥ 2, y 2 = x2g+1 + 1 has only finitely many solutions with x , y ∈ Q.
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 2 / 38
Uniformity
Problem1 Given X , compute X (Q) exactly.2 Compute bounds on #X (Q).
Conjecture (Uniformity)
There exists a constant N(g) such that every smooth curve of genus gover Q has at most N(g) rational points.
Theorem (Caporaso, Harris, Mazur)
Lang’s conjecture ⇒ uniformity.
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 3 / 38
Coleman’s bound
Theorem (Coleman)
Let X be a curve of genus g and let r = rankZ JacX (Q). Suppose p > 2gis a prime of good reduction. Suppose r < g. Then
#X (Q) ≤ #X (Fp) + 2g − 2.
Remark1 A modified statement holds for p ≤ 2g or for K 6= Q.2 Note: this does not prove uniformity (since the first good p might be
large).
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 4 / 38
Stoll’s bound
Theorem (Stoll)
Let X be a curve of genus g and let r = rankZ JacX (Q). Suppose p > 2gis a prime of good reduction. Suppose r < g. Then
#X (Q) ≤ #X (Fp) + 2r .
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 5 / 38
Bad reduction bound
Theorem (Lorenzini-Tucker, McCallum-Poonen)
Let X be a curve of genus g and let r = rankZ JacX (Q). Suppose p > 2gis a prime. Suppose r < g.
Let X be a regular proper model of X . Then
#X (Q) ≤ #X sm(Fp) + 2g − 2.
Remark
A recent improvement due to Stoll gives a uniform bound if r ≤ g − 3 andX is hyperelliptic.
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 6 / 38
Main Theorem
Theorem (Katz-ZB)
Let X be a curve of genus g and let r = rankZ JacX (Q). Suppose p > 2gis a prime. Let X be a regular proper model of X . Suppose r < g. Then
#X (Q) ≤ #X sm(Fp) + 2r .
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 7 / 38
Example (hyperelliptic curve with cuspidal reduction)
−2 · 11 · 19 · 173 · y 2 = (x − 50)(x − 9)(x − 3)(x + 13)(x3 + 2x2 + 3x + 4)
= x(x + 1)(x + 2)(x + 3)(x + 4)3 mod 5.
Analysis
1 X (Q) contains
{∞, (50, 0), (9, 0), (3, 0), (−13, 0), (25, 20247920), (25,−20247920)}
2 #X sm5 (F5) = 53 7 ≤ #X (Q) ≤ #X sm5 (F5) + 2 · 1 = 7
This determines X (Q).
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 8 / 38
Non-example
y 2 = x6 + 5
= x6 mod 5.
Analysis
1 X (Q) ⊃ {∞+,∞−}2 X sm(F5) = {∞+,∞−,±(1,±1),±(2,±23),±(3,±33),±(4,±43)}3 2 ≤ #X (Q) ≤ #X sm5 (F5) + 2 · 1 = 20
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 9 / 38
Models (X /Zp)
y 2 = x6 + 5
= x6 mod 5.
Note: no Zp-point can reduce to (0, 0).
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 10 / 38
Models – not regular
y 2 = x6 + 52
= x6 mod 5
Now: (0, 5) reduces to (0, 0).
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 11 / 38
Models – not regular (blow up)
y 2 = x6 + 52
= x6 mod 5
Blow up.
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 12 / 38
Models – semistable example
y 2 = (x(x − 1)(x − 2))3 + 5
= x6 mod 5.
Note: no point can reduce to (0, 0). Local equation looks like xy = 5
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 13 / 38
Models – semistable example (not regular)
y 2 = (x(x − 1)(x − 2))3 + 54
= x6 mod 5
Now: (0, 52) reduces to (0, 0). Local equation looks like xy = 54
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 14 / 38
Models – semistable example
y 2 = (x(x − 1)(x − 2))3 + 54
= x6 mod 5
Blow up. Local equation looks like xy = 53
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 15 / 38
Models – semistable example (regular at (0,0))
y 2 = (x(x − 1)(x − 2))3 + 54
= x6 mod 5
Blow up. Local equation looks like xy = 5
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 16 / 38
Main Theorem
Theorem (Katz-ZB)
Let X be a curve of genus g and let r = rankZ JacX (Q). Suppose p > 2gis a prime. Let X be a regular proper model of X . Suppose r < g. Then
#X (Q) ≤ #X sm(Fp) + 2r .
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 17 / 38
Chabauty’s method
(p-adic integration) There exists V ⊂ H0(XQp ,Ω1X ) withdimQp V ≥ g − r such that,∫ Q
Pω = 0 ∀P,Q ∈ X (Q), ω ∈ V
(Coleman, via Newton Polygons) Number of zeroes in a residuedisc DP is ≤ 1 + nP , where nP = # (divω ∩ DP)
(Riemann-Roch)∑
nP = 2g − 2.(Coleman’s bound)
∑P∈X (Fp)(1 + nP) = #X (Fp) + 2g − 2.
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 18 / 38
Example (from McCallum-Poonen’s survey paper)
Example
X : y 2 = x6 + 8x5 + 22x4 + 22x3 + 5x2 + 6x + 1
1 Points reducing to Q̃ = (0, 1) are given by
x = p · t, where t ∈ Zp
y =√
x6 + 8x5 + 22x4 + 22x3 + 5x2 + 6x + 1 = 1 + x2 + · · ·
2
∫ Pt(0,1)
xdx
y=
∫ t0
(x − x3 + · · · )dx
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 19 / 38
Stoll’s idea: use multiple ω
(Coleman, via Newton Polygons) Number of zeroes of∫ω in a
residue class DP is ≤ 1 + nP , where nP = # (divω ∩ DP)
Let ñP = minω∈V # (divω ∩ DP)(2 examples) r ≤ g − 2, ω1, ω2 ∈ V
(Stoll’s bound)∑
ñP ≤ 2r . (Recall dimQp V ≥ g − r)
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 20 / 38
Stoll’s bound – proof (D =∑
ñPP)
(Wanted)
dim H0(XFp ,K − D) ≥ g − r ⇒ deg D ≤ 2r
(Clifford)
H0(XFp ,K − D ′) 6= 0 ⇒ dim H0(XFp ,D ′) ≤1
2deg D ′ + 1
(D′ = K−D)
dim H0(XFp ,K − D) ≤1
2deg(K − D) + 1
(Assumption)g − r ≤ dim H0(XFp ,K − D)
(Recall dimQp V ≥ g − r)
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 21 / 38
Complications when XFp is singular
1 ω ∈ H0(X ,Ω) may vanish along components of XFp ;2 i.e. H0(XFp ,K − D) 6= 0 6⇒ D is special;3 rank(K − D) 6= dim H0(XFp ,K − D)− 1
Summary
The relationship between dim H0(XFp ,K − D) and deg D is lesstransparent and does not follow from geometric techniques.
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 22 / 38
Rank of a divisor
Definition (Rank of a divisor is)
1 r(D) = −1 if |D| is empty.2 r(D) ≥ 0 if |D| is nonempty3 r(D) ≥ k if |D − E | is nonempty for any effective E with deg E = k .
Remark
1 If X is smooth, then r(D) = dim H0(X ,D)− 1.2 If X is has multiple components, then r(D) 6= dim H0(X ,D)− 1.
Remark
Ingredients of Stoll’s proof only use formal properties of r(D).
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 23 / 38
Formal ingredients of Stoll’s proof
Need:
(Clifford) r(K − D) ≤ 12 deg(K − D)
(Large rank) r(K − D) ≥ g − r − 1
(Recall, V ⊂ H0(XQp ,Ω1X ), dimQp V ≥ g − r)
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 24 / 38
Semistable case
Idea: any section s ∈ H0(X ,D) can be scaled to not vanish on acomponent (but may now have zeroes or poles at other components.)
Divisors on graphs:
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 25 / 38
Semistable case
Idea: any section s ∈ H0(X ,D) can be scaled to not vanish on acomponent (but may now have zeroes or poles at other components.)
Divisors on graphs:
-2 1 -2 0
1
1
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 26 / 38
Semistable case
Idea: any section s ∈ H0(X ,D) can be scaled to not vanish on acomponent (but may now have zeroes or poles at other components.)
Divisors on graphs:
-2 1 -2 0
1
1
-2 1 0 0
0
0
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 27 / 38
Divisors on graphs
Definition (Rank of a divisor is)
1 r(D) = −1 if |D| is empty.2 r(D) ≥ 0 if |D| is nonempty3 r(D) ≥ k if |D − E | is nonempty for any effective E with deg E = k .
1 3
-1
0
1 1
0
1
Remark
r(D) ≥ 0
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 28 / 38
Divisors on graphs
Definition (Rank of a divisor is)
1 r(D) = −1 if |D| is empty.2 r(D) ≥ 0 if |D| is nonempty3 r(D) ≥ k if |D − E | is nonempty for any effective E with deg E = k .
1 3
-2
0
1 1
-1
1
Remark
r(D) ≥ 1
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 29 / 38
Semistable case – line bundles
Let X be a curve over Zp with semistable special fiber XFp =⋃
Xi .
Definition (Divisor associated to a line bundle)
Given L ∈ Pic X , define a divisor on Γ by∑v∈V (Γ)
(degLXi )vXi .
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 30 / 38
Semistable case – line bundles
Let X be a curve over Zp with semistable special fiber XFp =⋃
Xi .
Definition (Divisor associated to a line bundle)
Given L ∈ Pic X , define a divisor on Γ by∑v∈V (Γ)
(degLXi )vXi .
Example: L = ωX , XFp totally degenerate (g(Xi ) = 0)
0 0 0
1
1
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 31 / 38
Semistable case – line bundles
Let X be a curve over Zp with semistable special fiber XFp =⋃
Xi .
Definition (Divisor associated to a line bundle)
Given L ∈ Pic X , define a divisor on Γ by∑v∈V (Γ)
(degLXi )vXi .
Example: L = O(H) (H a “horizontal” divisor on X )
-2 1 -2 0
1
1
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 32 / 38
Semistable case – line bundles
Let X be a curve over Zp with semistable special fiber XFp =⋃
Xi .
Definition (Divisor associated to a line bundle)
Given L ∈ Pic X , define a divisor on Γ by∑v∈V (Γ)
(degLXi )vXi .
Example: L = O(Xi ),
Xi
0 -2 0
1
1
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 33 / 38
Divisors on graphs
Definition
For D ∈ Div Γ, rnum(D) ≥ k if |D − E | is non-empty for every effective Eof degree k .
Theorem (Baker, Norine)
Riemann-Roch for rnum.
Clifford’s theorem for rnum.
Specialization: rnum(D) ≥ r(D).Formal corollary: X (Q) ≤ #X sm(Fp) + 2r (for X totally degenerate).
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 34 / 38
Semistable case – main points
Xi
0 -2 0
1
1
Remark (Main points)
1 Chip firing is the same as twising by O(Xi ).2 If ∃s ∈ H0(X ,L) and div s =
∑Hi +
∑niXi , then
L ⊗O(−n1X1)⊗ · · · ⊗ O(−nkXk)
specializes to an effective divisor on Γ.
3 The firing sequence (n1, . . . , nn) wins the chip firing game.
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 35 / 38
Semistable but not totally degenerate – abelian rank
Problems when g(Γ) < g(X ). (E.g. rank can increase after reduction.)
Definition (Abelian rank rab)
Let L ∈ X have specialization D ∈ Div Γ. Then rab(L) ≥ k if1 |D − E | is nonempty for any effective E with deg E = k , and2 for every LE specializing to E , there exists some (n1, . . . , nk) such
thatL′ := L ⊗ L−1E ⊗O(n1X1)⊗ · · · ⊗ O(nkXk)
has effective specialization and such that H0(Xi ,L′Xi ) 6= 0 for everycomponent Xi .
-2 1 0 0
0
0
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 36 / 38
Main Theorem – abelian rank
Theorem (Katz-ZB)
Clifford’s theorem: rab(K − D) ≤ 12 deg(K − D)Specialization: rab(K − D) ≥ g − r .Formal corollary: X (Q) ≤ #X sm(Fp) + 2r (for semistable curves.)
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 37 / 38
Final remarks
Remark
Also prove: semistable case ⇒ general case.
Remark (Néron models)
1 Suppose L ∈ PicX and deg(L|Xp
)= 0.
2 rnum(L) = 0 if and only if L|Xp ∈ Pic0Xp .3 rab(L) = 0 if and only if the image of L|Xp in Pic0X̃p is the identity.
Remark (Toric rank)
1 Can also define rtor – additionally require that sections agree at nodes
2 rtor incorporates the toric part of Néron model
David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 38 / 38