Diophantine and p -adic geometry David Zureick-Brown joint with - - PowerPoint PPT Presentation

Diophantine and p-adic geometry

David Zureick-Brown joint with Eric Katz (Waterloo) and Joe Rabinoff (Georgia Tech)

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

Spring Lecture Series, Fayetteville, AR April 6, 2018

Mordell Conjecture

Example

−y2 = (x2 − 1)(x2 − 2)(x2 − 3) This is a cross section of a two holed torus. The genus is the number of holes.

Conjecture (Mordell); Theorem (Faltings, Bombieri, Vojta)

A curve of genus g ≥ 2 has only finitely many rational solutions.

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 2 / 30

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 University) Diophantine and p-adic geometry April 6, 2018 3 / 30

Uniformity numerics

g 2 3 4 5 10 45 g Bg(Q) 642 112 126 132 192 781 16(g + 1)

Remark

Elkies studied K3 surfaces of the form y2 = S(t, u, v) with lots of rational lines, such that S restricted to such a line is a perfect square.

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 4 / 30

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

This can be used to provably compute X(Q).

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 5 / 30

Example (Gordon, Grant)

y 2 = x(x − 1)(x − 2)(x − 5)(x − 6)

Analysis

1

rankZ JacX(Q) = 1, g = 2

2

X(Q) contains {∞, (0, 0), (1, 0), (2, 0), (5, 0), (6, 0), (3, ±6), (10, ±120).}

3

#X(F7) = 8

4

10 ≤ #X(Q) ≤ #X(F7) + 2g − 2 = 10 This determines X(Q).

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 6 / 30

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 University) Diophantine and p-adic geometry April 6, 2018 7 / 30

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 (p-adic Rolle’s (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 University) Diophantine and p-adic geometry April 6, 2018 8 / 30

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 University) Diophantine and p-adic geometry April 6, 2018 9 / 30

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 (p-adic Rolle’s (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 University) Diophantine and p-adic geometry April 6, 2018 10 / 30

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 University) Diophantine and p-adic geometry April 6, 2018 11 / 30

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 University) Diophantine and p-adic geometry April 6, 2018 12 / 30

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 University) Diophantine and p-adic geometry April 6, 2018 13 / 30

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 University) Diophantine and p-adic geometry April 6, 2018 14 / 30

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 University) Diophantine and p-adic geometry April 6, 2018 15 / 30

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 University) Diophantine and p-adic geometry April 6, 2018 16 / 30

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. 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 p-adic Rolle’s on hyperelliptic annuli

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 17 / 30

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 University) Diophantine and p-adic geometry April 6, 2018 18 / 30

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 University) Diophantine and p-adic geometry April 6, 2018 19 / 30

Analytic continuation of integrals via Frobenius

(Integrals between discs.)

Q ∈ DP1, R ∈ DP2, R

Q

ω := ?

(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∗ωi = dfi +

• j

R

Q

aijωj

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 20 / 30

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; Katz–Rabinoff–Zureick-Brown)

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(R) − t(Q)]

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 21 / 30

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(R) + · · ·

(Cover the annulus with discs)

Each analytic continuation implicitly chooses a branch of log.

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 22 / 30

Why bother? Integration on Annuli (a trade off)

(Abelian integrals.) Analytically continue via Albanese.

R

Q

ω = 0 if R, Q ∈ X(Q), ω ∈ V

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

R

Q

ω := t(R)

t(Q)

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

(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 University) Diophantine and p-adic geometry April 6, 2018 23 / 30

Berkovich picture

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 24 / 30

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)an

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

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 25 / 30

Main Theorem (partial uniformity for curves)

Theorem (Katz, Rabinoff, ZB)

Let X be any curve of genus g and let r = rankZ JacX(Q). Suppose r < g − 2. Then #X(Q) ≤ 84g2 − 98g + 28

Tools

p-adic integration on annuli comparison of different analytic continuations of p-adic integration Non-Archimedean (Berkovich) structure of a curve [BPR] Combinatorial restraints coming from the Tropical canonical bundle p-adic Rolle’s on annuli for arbitrary curves

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 26 / 30

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

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

The second corollary is a big improvement

1 It requires working over a non-discretely valued field. 2 The bound only depends on the reduction type. 3 Integration over wide opens (c.f. Coleman) instead of discs and annuli. David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 27 / 30

Baker-Payne-Rabinoff and the slope formula

(Dual graph Γ of XFp) (Contraction Theorem) τ : X an → Γ. (Combinatorial harmonic analysis/potential theory) f a meromorphic function on X an F := (− log |f |)

• Γ

associated tropical, piecewise linear function div F combinatorial record of the slopes of F (Slope formula) τ∗ div f = div F

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 28 / 30

Berkovich picture

David Zureick-Brown (Emory University) Diophantine and p-adic geometry April 6, 2018 29 / 30