Rational points on curves and tropical geometry.

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1 Rational points on curves and tropical geometry. David Zureick-Brown (Emory University) Eric Katz (Waterloo University) Slides available at Specialization of Linear Series for Algebraic and Tropical Curves BIRS April 3, 2014

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, y 2 = x 2g has only finitely many solutions with x, y Q. David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

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 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, / 38

4 Coleman s bound Theorem (Coleman) Let X be a curve of genus g and let r = rank Z Jac X (Q). Suppose p > 2g is a prime of good reduction. Suppose r < g. Then Remark #X (Q) #X (F p ) + 2g 2. 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, / 38

5 Stoll s bound Theorem (Stoll) Let X be a curve of genus g and let r = rank Z Jac X (Q). Suppose p > 2g is a prime of good reduction. Suppose r < g. Then #X (Q) #X (F p ) + 2r. David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

6 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 > 2g is a prime. Suppose r < g. Let X be a regular proper model of X. Then #X (Q) #X sm (F p ) + 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, / 38

7 Main Theorem Theorem (Katz-ZB) Let X be a curve of genus g and let r = rank Z Jac X (Q). Suppose p > 2g is a prime. Let X be a regular proper model of X. Suppose r < g. Then #X (Q) #X sm (F p ) + 2r. David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

8 Example (hyperelliptic curve with cuspidal reduction) y 2 = (x 50)(x 9)(x 3)(x + 13)(x 3 + 2x 2 + 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, ), (25, )} 2 #X sm 5 (F 5 ) = #X (Q) #X sm 5 (F 5 ) = 7 This determines X (Q). David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

9 Non-example y 2 = x = 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 5 (F 5 ) = 20 David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

10 Models (X /Z p ) y 2 = x = 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, / 38

11 Models not regular y 2 = x = x 6 mod 5 Now: (0, 5) reduces to (0, 0). David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

12 Models not regular (blow up) y 2 = x = x 6 mod 5 Blow up. David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

13 Models semistable example y 2 = (x(x 1)(x 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, / 38

14 Models semistable example (not regular) y 2 = (x(x 1)(x 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, / 38

15 Models semistable example y 2 = (x(x 1)(x 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, / 38

16 Models semistable example (regular at (0,0)) y 2 = (x(x 1)(x 2)) = x 6 mod 5 Blow up. Local equation looks like xy = 5 David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

17 Main Theorem Theorem (Katz-ZB) Let X be a curve of genus g and let r = rank Z Jac X (Q). Suppose p > 2g is a prime. Let X be a regular proper model of X. Suppose r < g. Then #X (Q) #X sm (F p ) + 2r. David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

18 Chabauty s method (p-adic integration) There exists V H 0 (X Qp, Ω 1 X ) with dim Qp 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 D P is 1 + n P, where n P = # (div ω D P ) (Riemann-Roch) n P = 2g 2. (Coleman s bound) P X (F p) (1 + n P) = #X (F p ) + 2g 2. David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

19 Example (from McCallum-Poonen s survey paper) Example X : y 2 = x 6 + 8x x x 3 + 5x 2 + 6x Points reducing to Q = (0, 1) are given by x = p t, where t Z p y = x 6 + 8x x x 3 + 5x 2 + 6x + 1 = 1 + x Pt (0,1) xdx y = t 0 (x x 3 + )dx David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

20 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 ñ P = min ω V # (div ω D P ) (2 examples) r g 2, ω 1, ω 2 V (Stoll s bound) ñ P 2r. (Recall dim Qp V g r) David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

21 Stoll s bound proof (D = ñ P P) (Wanted) dim H 0 (X Fp, K D) g r deg D 2r (Clifford) H 0 (X Fp, K D ) 0 dim H 0 (X Fp, D ) 1 2 deg D + 1 (D = K D) dim H 0 (X Fp, K D) 1 deg(k D) (Assumption) g r dim H 0 (X Fp, K D) (Recall dim Qp V g r) David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

22 Complications when X Fp is singular 1 ω H 0 (X, Ω) may vanish along components of X Fp ; 2 i.e. H 0 (X Fp, K D) 0 D is special; 3 rank(k D) dim H 0 (X Fp, K D) 1 Summary The relationship between dim H 0 (X Fp, 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, / 38

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 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, / 38

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 H 0 (X Qp, Ω 1 X ), dim Q p V g r) David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

25 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, / 38

26 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, / 38

27 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, / 38

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 Remark r(d) 0 David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

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 Remark r(d) 1 David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

30 Semistable case line bundles Let X be a curve over Z p with semistable special fiber X Fp = X i. Definition (Divisor associated to a line bundle) Given L Pic X, define a divisor on Γ by (deg L Xi )v Xi. v V (Γ) David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

31 Semistable case line bundles Let X be a curve over Z p with semistable special fiber X Fp = X i. Definition (Divisor associated to a line bundle) Given L Pic X, define a divisor on Γ by (deg L Xi )v Xi. v V (Γ) Example: L = ω X, X Fp totally degenerate (g(x i ) = 0) David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

32 Semistable case line bundles Let X be a curve over Z p with semistable special fiber X Fp = X i. Definition (Divisor associated to a line bundle) Given L Pic X, define a divisor on Γ by (deg L Xi )v Xi. v V (Γ) Example: L = O(H) (H a horizontal divisor on X ) David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

33 Semistable case line bundles Let X be a curve over Z p with semistable special fiber X Fp = X i. Definition (Divisor associated to a line bundle) Given L Pic X, define a divisor on Γ by (deg L Xi )v Xi. Example: L = O(X i ), v V (Γ) X i 1 David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

34 Divisors on graphs Definition For D Div Γ, r num (D) k if D E is non-empty for every effective E of degree k. Theorem (Baker, Norine) Riemann-Roch for r num. Clifford s theorem for r num. Specialization: r num (D) r(d). Formal corollary: X (Q) #X sm (F p ) + 2r (for X totally degenerate). David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

35 Semistable case main points X i 1 Remark (Main points) 1 Chip firing is the same as twising by O(X i ). 2 If s H 0 (X, L) and div s = H i + n i X i, then L O( n 1 X 1 ) O( n k X k ) specializes to an effective divisor on Γ. 3 The firing sequence (n 1,..., n n ) wins the chip firing game. David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

36 Semistable but not totally degenerate abelian rank Problems when g(γ) < g(x ). (E.g. rank can increase after reduction.) Definition (Abelian rank r ab ) Let L X have specialization D Div Γ. Then r ab (L) k if 1 D E is nonempty for any effective E with deg E = k, and 2 for every L E specializing to E, there exists some (n 1,..., n k ) such that L := L L 1 E O(n 1X 1 ) O(n k X k ) has effective specialization and such that H 0 (X i, L X i ) 0 for every component X i David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

37 Main Theorem abelian rank Theorem (Katz-ZB) Clifford s theorem: r ab (K D) 1 2 deg(k D) Specialization: r ab (K D) g r. Formal corollary: X (Q) #X sm (F p ) + 2r (for semistable curves.) David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

38 Final remarks Remark Also prove: semistable case general case. Remark (Néron models) 1 Suppose L Pic X and deg ( L Xp ) = 0. 2 r num (L) = 0 if and only if L Xp Pic 0 X p. 3 r ab (L) = 0 if and only if the image of L Xp in Pic 0 Xp is the identity. Remark (Toric rank) 1 Can also define r tor additionally require that sections agree at nodes 2 r tor incorporates the toric part of Néron model David Zureick-Brown (Emory) Rational points and tropical geometry. April 3, / 38

Diophantine and tropical geometry

Diophantine and tropical geometry David Zureick-Brown joint with Eric Katz (Waterloo) and Joe Rabinoff (Georgia Tech) Slides available at http://www.mathcs.emory.edu/~dzb/slides/ University of Colorado-Boulder