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
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, y 2 = x 2 g +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 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 over 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 = rank Z Jac X ( Q ) . Suppose p > 2 g is a prime of good reduction. Suppose r < g. Then # X ( Q ) ≤ # X ( F p ) + 2 g − 2 . Remark 1 A modified statement holds for p ≤ 2 g 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
Stoll’s bound Theorem (Stoll) Let X be a curve of genus g and let r = rank Z Jac X ( Q ) . Suppose p > 2 g is a prime of good reduction. Suppose r < g. Then # X ( Q ) ≤ # X ( F p ) + 2 r . 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 = rank Z Jac X ( Q ) . Suppose p > 2 g is a prime. Suppose r < g. Let X be a regular proper model of X. Then # X ( Q ) ≤ # X sm ( F p ) + 2 g − 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
Main Theorem Theorem (Katz-ZB) Let X be a curve of genus g and let r = rank Z Jac X ( Q ) . Suppose p > 2 g is a prime. Let X be a regular proper model of X. Suppose r < g. Then # X ( Q ) ≤ # X sm ( F p ) + 2 r . David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 7 / 38
Example (hyperelliptic curve with cuspidal reduction) = ( x − 50)( x − 9)( x − 3)( x + 13)( x 3 + 2 x 2 + 3 x + 4) − 2 · 11 · 19 · 173 · y 2 = x ( x + 1)( x + 2)( x + 3)( x + 4) 3 mod 5 . Analysis X ( Q ) contains 1 {∞ , (50 , 0) , (9 , 0) , (3 , 0) , ( − 13 , 0) , (25 , 20247920) , (25 , − 20247920) } # X sm ( F 5 ) = 5 2 5 7 ≤ # X ( Q ) ≤ # X sm ( F 5 ) + 2 · 1 = 7 3 5 This determines X ( Q ). David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 8 / 38
Non-example = x 6 + 5 y 2 = x 6 mod 5 . Analysis 1 X ( Q ) ⊃ {∞ + , ∞ − } 2 X sm ( F 5 ) = {∞ + , ∞ − , ± (1 , ± 1) , ± (2 , ± 2 3 ) , ± (3 , ± 3 3 ) , ± (4 , ± 4 3 ) } 3 2 ≤ # X ( Q ) ≤ # X sm ( F 5 ) + 2 · 1 = 20 5 David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 9 / 38
Models ( X / Z p ) = x 6 + 5 y 2 = x 6 mod 5 . Note: no Z p -point can reduce to (0 , 0). David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 10 / 38
Models – not regular = x 6 + 5 2 y 2 = x 6 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) = x 6 + 5 2 y 2 = x 6 mod 5 Blow up. David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 12 / 38
Models – semistable example = ( x ( x − 1)( x − 2)) 3 + 5 y 2 = x 6 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) = ( x ( x − 1)( x − 2)) 3 + 5 4 y 2 = x 6 mod 5 Now: (0 , 5 2 ) reduces to (0 , 0). Local equation looks like xy = 5 4 David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 14 / 38
Models – semistable example = ( x ( x − 1)( x − 2)) 3 + 5 4 y 2 = x 6 mod 5 Blow up. Local equation looks like xy = 5 3 David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 15 / 38
Models – semistable example (regular at (0,0)) = ( x ( x − 1)( x − 2)) 3 + 5 4 y 2 = x 6 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 = rank Z Jac X ( Q ) . Suppose p > 2 g is a prime. Let X be a regular proper model of X. Suppose r < g. Then # X ( Q ) ≤ # X sm ( F p ) + 2 r . David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 17 / 38
Chabauty’s method ( p -adic integration ) There exists V ⊂ H 0 ( X Q p , Ω 1 X ) with dim Q p V ≥ g − r such that, � Q ω = 0 ∀ P , Q ∈ X ( Q ) , ω ∈ V P ( Coleman, via Newton Polygons ) Number of zeroes in a residue disc D P is ≤ 1 + n P , where n P = # (div ω ∩ D P ) ( Riemann-Roch ) � n P = 2 g − 2. ( Coleman’s bound ) � P ∈ X ( F p ) (1 + n P ) = # X ( F p ) + 2 g − 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 = x 6 + 8 x 5 + 22 x 4 + 22 x 3 + 5 x 2 + 6 x + 1 1 Points reducing to � Q = (0 , 1) are given by x = p · t , where t ∈ Z p √ x 6 + 8 x 5 + 22 x 4 + 22 x 3 + 5 x 2 + 6 x + 1 = 1 + x 2 + · · · y = � P t � t xdx ( x − x 3 + · · · ) dx = 2 y (0 , 1) 0 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 D P is ≤ 1 + n P , where n P = # (div ω ∩ D P ) Let � n P = min ω ∈ V # (div ω ∩ D P ) ( 2 examples ) r ≤ g − 2, ω 1 , ω 2 ∈ V ( Stoll’s bound ) � � n P ≤ 2 r . (Recall dim Q p V ≥ g − r ) David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 20 / 38
Stoll’s bound – proof ( D = � � n P P ) (Wanted) dim H 0 ( X F p , K − D ) ≥ g − r ⇒ deg D ≤ 2 r (Clifford) H 0 ( X F p , K − D ′ ) � = 0 ⇒ dim H 0 ( X F p , D ′ ) ≤ 1 2 deg D ′ + 1 ( D ′ = K − D ) dim H 0 ( X F p , K − D ) ≤ 1 2 deg( K − D ) + 1 ( Assumption ) g − r ≤ dim H 0 ( X F p , K − D ) (Recall dim Q p V ≥ g − r ) David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 21 / 38
Complications when X F p is singular 1 ω ∈ H 0 ( X , Ω) may vanish along components of X F p ; 2 i.e. H 0 ( X F p , K − D ) � = 0 �⇒ D is special; 3 rank( K − D ) � = dim H 0 ( X F p , K − D ) − 1 Summary The relationship between dim H 0 ( X F p , 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
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 H 0 ( X , D ) − 1. 2 If X is has multiple components, then r ( D ) � = dim H 0 ( 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: r ( K − D ) ≤ 1 (Clifford) 2 deg( K − D ) (Large rank) r ( K − D ) ≥ g − r − 1 (Recall, V ⊂ H 0 ( X Q p , Ω 1 X ) , dim Q p V ≥ g − r ) David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 24 / 38
Semistable case Idea : any section s ∈ H 0 ( 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
Semistable case Idea : any section s ∈ H 0 ( X , D ) can be scaled to not vanish on a component (but may now have zeroes or poles at other components.) Divisors on graphs : 1 1 -2 0 -2 1 David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 26 / 38
Semistable case Idea : any section s ∈ H 0 ( X , D ) can be scaled to not vanish on a component (but may now have zeroes or poles at other components.) Divisors on graphs : 1 1 -2 0 -2 1 0 1 0 0 -2 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 nonempty 3 r ( D ) ≥ k if | D − E | is nonempty for any effective E with deg E = k . 0 1 1 3 1 1 -1 0 Remark r ( D ) ≥ 0 David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, 2014 28 / 38
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