Rational points on curves and tropical geometry. David Zureick-Brown - - PowerPoint PPT Presentation

▶
rational points on curves and tropical geometry
SMART_READER_LITE
LIVE PREVIEW

Rational points on curves and tropical geometry. David Zureick-Brown - - PowerPoint PPT Presentation

Dec 13, 2023 223 likes •616 views

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 Curves

slide-1
SLIDE 1

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 Curves BIRS April 3, 2014

slide-2
SLIDE 2

Faltings’ theorem

Theorem (Faltings)

Let X be a smooth curve over Q with genus at least 2. Then X(Q) is finite.

Example

For g ≥ 2, y2 = 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

slide-3
SLIDE 3

Uniformity

Problem

1 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 g

  • ver 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

slide-4
SLIDE 4

Coleman’s bound

Theorem (Coleman)

Let X be a curve of genus g and let r = rankZ JacX(Q). Suppose p > 2g is a prime of good reduction. Suppose r < g. Then #X(Q) ≤ #X(Fp) + 2g − 2.

Remark

1 A modified statement holds for p ≤ 2g or for K = 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

slide-5
SLIDE 5

Stoll’s bound

Theorem (Stoll)

Let X be a curve of genus g and let r = rankZ JacX(Q). Suppose p > 2g is 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

slide-6
SLIDE 6

Bad reduction bound

Theorem (Lorenzini-Tucker, McCallum-Poonen)

Let X be a curve of genus g and let r = rankZ JacX(Q). Suppose p > 2g is 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 and X is hyperelliptic.

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 6 / 38

slide-7
SLIDE 7

Main Theorem

Theorem (Katz-ZB)

Let X be a curve of genus g and let r = rankZ JacX(Q). Suppose p > 2g is 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

slide-8
SLIDE 8

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 sm

5

(F5) = 5

3

7 ≤ #X(Q) ≤ #X sm

5

(F5) + 2 · 1 = 7 This determines X(Q).

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 8 / 38

slide-9
SLIDE 9

Non-example

y2 = 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 sm

5

(F5) + 2 · 1 = 20

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 9 / 38

slide-10
SLIDE 10

Models (X /Zp)

y2 = 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

slide-11
SLIDE 11

Models – not regular

y2 = 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

slide-12
SLIDE 12

Models – not regular (blow up)

y2 = x6 + 52 = x6 mod 5 Blow up.

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 12 / 38

slide-13
SLIDE 13

Models – semistable example

y2 = (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

slide-14
SLIDE 14

Models – semistable example (not regular)

y2 = (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

slide-15
SLIDE 15

Models – semistable example

y2 = (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

slide-16
SLIDE 16

Models – semistable example (regular at (0,0))

y2 = (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

slide-17
SLIDE 17

Main Theorem

Theorem (Katz-ZB)

Let X be a curve of genus g and let r = rankZ JacX(Q). Suppose p > 2g is 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

slide-18
SLIDE 18

Chabauty’s method

(p-adic integration) There exists V ⊂ H0(XQp, Ω1

X) with

dimQp V ≥ g − r such that, Q

P

ω = 0 ∀P, Q ∈ X(Q), ω ∈ V (Coleman, via Newton Polygons) Number of zeroes in a residue disc DP is ≤ 1 + nP, where nP = # (div ω ∩ 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

slide-19
SLIDE 19

Example (from McCallum-Poonen’s survey paper)

Example

X : y2 = x6 + 8x5 + 22x4 + 22x3 + 5x2 + 6x + 1

1 Points reducing to

Q = (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 = t (x − x3 + · · · )dx

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 19 / 38

slide-20
SLIDE 20

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 = # (div ω ∩ DP) Let nP = minω∈V # (div ω ∩ DP) (2 examples) r ≤ g − 2, ω1, ω2 ∈ V (Stoll’s bound) nP ≤ 2r.

(Recall dimQp V ≥ g − r) David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 20 / 38

slide-21
SLIDE 21

Stoll’s bound – proof (D = nPP)

(Wanted) dim H0(XFp, K − D) ≥ g − r ⇒ deg D ≤ 2r (Clifford) H0(XFp, K − D′) = 0 ⇒ dim H0(XFp, D′) ≤ 1 2 deg D′ + 1 (D′ = K − D) dim H0(XFp, K − D) ≤ 1 2 deg(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

slide-22
SLIDE 22

Complications when XFp is singular

1 ω ∈ H0(X, Ω) may vanish along components of XFp; 2 i.e. H0(XFp, K − D) = 0 ⇒ D is special; 3 rank(K − D) = dim H0(XFp, K − D) − 1

Summary

The relationship between dim H0(XFp, K − D) and deg D is less transparent and does not follow from geometric techniques.

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 22 / 38

slide-23
SLIDE 23

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 nonempty 3 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) = 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

slide-24
SLIDE 24

Formal ingredients of Stoll’s proof

Need:

(Clifford) r(K − D) ≤ 1

2 deg(K − D)

(Large rank) r(K − D) ≥ g − r − 1 (Recall, V ⊂ H0(XQp, Ω1

X), dimQp V ≥ g − r)

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 24 / 38

slide-25
SLIDE 25

Semistable case

Idea: any section s ∈ H0(X, D) can be scaled to not vanish on a component (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

slide-26
SLIDE 26

Semistable case

Idea: any section s ∈ H0(X, D) can be scaled to not vanish on a component (but may now have zeroes or poles at other components.) Divisors on graphs:

  • 2

1

  • 2

1 1

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 26 / 38

slide-27
SLIDE 27

Semistable case

Idea: any section s ∈ H0(X, D) can be scaled to not vanish on a component (but may now have zeroes or poles at other components.) Divisors on graphs:

  • 2

1

  • 2

1 1

  • 2

1

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 27 / 38

slide-28
SLIDE 28

Divisors on graphs

Definition (Rank of a divisor is)

1 r(D) = −1 if |D| is empty. 2 r(D) ≥ 0 if |D| is nonempty 3 r(D) ≥ k if |D − E| is nonempty for any effective E with deg E = k.

1 3

  • 1

1 1 1

Remark

r(D) ≥ 0

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 28 / 38

slide-29
SLIDE 29

Divisors on graphs

Definition (Rank of a divisor is)

1 r(D) = −1 if |D| is empty. 2 r(D) ≥ 0 if |D| is nonempty 3 r(D) ≥ k if |D − E| is nonempty for any effective E with deg E = k.

1 3

  • 2

1 1

  • 1

1

Remark

r(D) ≥ 1

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 29 / 38

slide-30
SLIDE 30

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)

(deg LXi)vXi.

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 30 / 38

slide-31
SLIDE 31

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)

(deg LXi)vXi. Example: L = ωX , XFp totally degenerate (g(Xi) = 0)

1 1

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 31 / 38

slide-32
SLIDE 32

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)

(deg LXi)vXi. Example: L = O(H) (H a “horizontal” divisor on X )

  • 2

1

  • 2

1 1

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 32 / 38

slide-33
SLIDE 33

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)

(deg LXi)vXi. Example: L = O(Xi),

Xi

  • 2

1 1

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 33 / 38

slide-34
SLIDE 34

Divisors on graphs

Definition

For D ∈ Div Γ, rnum(D) ≥ k if |D − E| is non-empty for every effective E

  • f 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

slide-35
SLIDE 35

Semistable case – main points

Xi

  • 2

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

slide-36
SLIDE 36

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 if

1 |D − E| is nonempty for any effective E with deg E = k, and 2 for every LE specializing to E, there exists some (n1, . . . , nk) such

that L′ := L ⊗ L−1

E ⊗ O(n1X1) ⊗ · · · ⊗ O(nkXk)

has effective specialization and such that H0(Xi, L′

Xi) = 0 for every

component Xi.

  • 2

1

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 36 / 38

slide-37
SLIDE 37

Main Theorem – abelian rank

Theorem (Katz-ZB)

Clifford’s theorem: rab(K − D) ≤ 1

2 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

slide-38
SLIDE 38

Final remarks

Remark

Also prove: semistable case ⇒ general case.

Remark (N´ eron models)

1 Suppose L ∈ PicX and deg

  • L|Xp
  • = 0.

2 rnum(L) = 0 if and only if L|Xp ∈ Pic0

Xp.

3 rab(L) = 0 if and only if the image of L|Xp in Pic0

  • Xp 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 N´

eron model

David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 38 / 38