Tropical geometry, p -adic integration, and uniformity. David - - PowerPoint PPT Presentation

Tropical geometry, p-adic integration, and uniformity.

David Zureick-Brown (Emory University) Joe Rabinoff (Georgia Tech) Eric Katz (Waterloo University)

Slides available at http://www.mathcs.emory.edu/~dzb/slides/

Special Session on Combinatorics and Algebraic Geometry Fall Western Sectional Meeting San Francisco State University Oct 26, 2014

Faltings’ theorem / Mordell’s conjecture

Theorem (Faltings, Vojta, Bombieri)

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) p-adic integration, and uniformity. Oct 26, 2014 2 / 25

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) p-adic integration, and uniformity. Oct 26, 2014 3 / 25

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).

Tools

p-adic integration and Riemann–Roch

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 4 / 25

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.

Tools

p-adic integration, Riemann–Roch, and Clifford’s theorem

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 5 / 25

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 (Still doesn’t prove uniformity)

#X sm(Fp) can contain an n-gon, for n arbitrarily large.

Tools

p-adic integration and arithmetic Riemann–Roch (K · Xp = 2g − 2)

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 6 / 25

Improved bad reduction bound

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.

Remark

Still doesn’t prove uniformity.

Tools

p-adic integration and Clifford’s theorem for graphs

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 7 / 25

Stoll’s hyperelliptic uniformity theorem

Theorem (Stoll)

Let X be a hyperelliptic curve of genus g and let r = rankZ JacX(Q). Suppose r < g − 2. Let X be a stable proper model of X. Then #X(Q) ≤ 8(r + 4)(g − 1) + max{1, 4r} · g

Tools

p-adic integration on annuli comparison of different analytic continuations of p-adic integration

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 8 / 25

Main Theorem (partial uniformity for non-hyperelliptic curves)

Theorem (Katz, Rabinoff, ZB)

Let X be any curve of genus g and let r = rankZ JacX(Q). Suppose r ≤ g − 2. Let d = 3(g+1)2 and let p ≥ 2g + d. Then #X(Q) ≤ 2gpd/2 + (2g − 2)(p2 + 2) + 2 · gg(6g − 6)(4g − 4).

Tools

p-adic integration on annuli comparison of different analytic continuations of p-adic integration Rabinoff’s bounds for Laurent series Tropical canonical bundle

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 9 / 25

Comments

Corollary ((Partially) effective Manin-Mumford)

There is an effective constant N(g) such that if g(X) = g, then # (X ∩ JacX,tors) (Q) ≤ N(g)

Corollary (In progress)

There is an effective constant N′(g) such that if g(X) = g > 3 and X has totally degenerate, trivalent reduction mod 2, then # (X ∩ JacX,tors) (C) ≤ N′(g)

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 10 / 25

Models – semistable example

y2 = (x(x − 1)(x − 2))3 − 5 = (x(x − 1)(x − 2))3 mod 5. Note: no point can reduce to (0, 0). Local equation looks like xy = 5

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 11 / 25

Models – semistable example (not regular)

y2 = (x(x − 1)(x − 2))3 − 54 = (x(x − 1)(x − 2))3 mod 5 Now: (0, 52) reduces to (0, 0). Local equation looks like xy = 54

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 12 / 25

Models – semistable example

y2 = (x(x − 1)(x − 2))3 − 54 = (x(x − 1)(x − 2))3 mod 5 Blow up. Local equation looks like xy = 53

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 13 / 25

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

y2 = (x(x − 1)(x − 2))3 − 54 = (x(x − 1)(x − 2))3 mod 5 Blow up. Local equation looks like xy = 5

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 14 / 25

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) p-adic integration, and uniformity. Oct 26, 2014 15 / 25

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) p-adic integration, and uniformity. Oct 26, 2014 16 / 25

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) p-adic integration, and uniformity. Oct 26, 2014 17 / 25

Analytic continuation of integrals

(Residue Discs.)

P ∈ X sm(FP), t : DP ∼ = pZp, ω|DP = f (t)dt

(Integrals on a disc.)

Q, R ∈ DP, R

Q

ω := t(R)

t(Q)

f (t)dt.

(Integrals between discs.)

Q ∈ DP1, R ∈ DP2, R

Q

ω := ?

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 18 / 25

Analytic continuation of integrals via Abelian varieties

(Integrals between discs.)

Q ∈ DP1, R ∈ DP2, R

Q

ω := ?

(Albanese map.)

ι: X ֒ → JacX, Q → [Q − ∞]

(Abelian integrals via functorality and additivity.)

R

Q

ι∗ω = ι(R)

ι(Q)

ω = [R−∞]

[Q−∞]

ω = [R−Q] ω = 1 n n[R−Q] ω

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 19 / 25

Analytic continuation of integrals via Frobenius

(Integrals between discs.)

Q ∈ DP1, R ∈ DP2, R

Q

ω := ?

(Abelian integrals via functorality and Frobenius.)

R

Q

ω = φ(Q)

Q

ω + φ(R)

φ(Q)

ω + R

φ(R)

ω

(Very clever trick (Coleman))

φ(R)

φ(Q)

ωi = R

Q

φ∗ω =

• j

R

Q

aijωj

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 20 / 25

Comparison of integrals

Facts

1 For X with good reduction, the Abelian and Coleman integrals

agree.

2 A mystery. The associated Berkovich curve is contractable. 3 For X with bad reduction they differ.

Theorem (Stoll)

There exist linear functions a(ω), c(ω) such that R

Q

ω − R

Q

ω = a(ω) (log(t(R)) − log(t(Q))) + c(ω) (t(Q) − t(R))

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 21 / 25

Why bother? Integration on Annuli (a trade off)

Assumption

Assume X/Zp is stable, but not regular. (Residue Discs.)

P ∈ X sm(Fp), t : DP ∼ = pZp, ω|DP = f (t)dt

(Residue Annuli.)

P ∈ X sing(Fp), t : DP ∼ = pZp − prZp, ω|DP = f (t, t−1)dt

(Integrals on an annulus are multivalued.)

R

Q

ω := t(R)

t(Q)

f (t, t−1)dt = · · · + a(ω) log t + · · ·

(Cover the annulus with discs)

Each analytic continuation implicitly chooses a branch of log.

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 22 / 25

Why bother? Integration on Annuli (a trade off)

(Abelian integrals.) Analytically continue via Albanese.

R

Q

ω := t(R)

t(Q)

f (t, t−1)dt = · · · + a(ω) logab t + · · ·

(Berkovich-Coleman integrals.) Analytically continue via Frobenius.

R

Q

ω := t(R)

t(Q)

f (t, t−1)dt = · · · + a(ω) logCol t + · · ·

(Stoll’s theorem.)

R

Q

ω − R

Q

ω = a(ω) (logab(r(R)) − logab(t(Q))) + c(ω) (t(Q) − t(R))

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 23 / 25

Stoll’s comparison theorem, tropical geometry edition

Theorem (Katz, Rabinoff, ZB)

The difference logCol − logab is the unique homomorphism that takes the value

• γ

ω

• n Trop(γ), where Trop: G(K) → T(K)/T(O).

T

• Λ

G

• JacX

B T = torus, Λ = discrete, and B = Abelian w/ good reduction.

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 24 / 25

Proof of uniformity

(Ignore the log and comparison terms (Stoll’s idea))

r < g − 2 and linear algebra allows one to find ω with no log term, and same abelian and Coleman integrals.

(Bounds on Annuli, via Rabinoff)

• ω is a Laurent series

# zeroes is bounded by the number of zeroes and poles of ω

(Global step)

• ω gives a section of the tropical canonical bundle on the dual graph.
• The order of the pole of ω at a node is the slope of the section of the

tropical canonical bundle on the corresponding edge of the dual graph.

David Zureick-Brown (Emory) p-adic integration, and uniformity. Oct 26, 2014 25 / 25