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, y 2 = x 2 g +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 over 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 = 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). 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 = 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 . 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 = 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 (Still doesn’t prove uniformity) # X sm ( F p ) can contain an n -gon, for n arbitrarily large. Tools p -adic integration and arithmetic Riemann–Roch ( K · X p = 2 g − 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 = 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 . 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 = rank Z Jac X ( Q ) . Suppose r < g − 2 . Let X be a stable proper model of X. Then # X ( Q ) ≤ 8( r + 4)( g − 1) + max { 1 , 4 r } · 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 = rank Z Jac X ( Q ) . Suppose r ≤ g − 2 . Let d = 3 ( g +1) 2 and let p ≥ 2 g + d. Then # X ( Q ) ≤ 2 gp d / 2 + (2 g − 2)( p 2 + 2) + 2 · g g (6 g − 6)(4 g − 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 ∩ Jac X , 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 ∩ Jac X , tors ) ( C ) ≤ N ′ ( g ) David Zureick-Brown (Emory) p -adic integration, and uniformity. Oct 26, 2014 10 / 25
Models – semistable example = ( x ( x − 1)( x − 2)) 3 − 5 y 2 = ( 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) = ( x ( x − 1)( x − 2)) 3 − 5 4 y 2 = ( x ( x − 1)( x − 2)) 3 mod 5 Now: (0 , 5 2 ) reduces to (0 , 0). Local equation looks like xy = 5 4 David Zureick-Brown (Emory) p -adic integration, and uniformity. Oct 26, 2014 12 / 25
Models – semistable example = ( x ( x − 1)( x − 2)) 3 − 5 4 y 2 = ( x ( x − 1)( x − 2)) 3 mod 5 Blow up. Local equation looks like xy = 5 3 David Zureick-Brown (Emory) p -adic integration, and uniformity. Oct 26, 2014 13 / 25
Models – semistable example (regular at (0,0)) = ( x ( x − 1)( x − 2)) 3 − 5 4 y 2 = ( 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 ⊂ 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) p -adic integration, and uniformity. Oct 26, 2014 15 / 25
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) p -adic integration, and uniformity. Oct 26, 2014 16 / 25
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) p -adic integration, and uniformity. Oct 26, 2014 17 / 25
Analytic continuation of integrals ( Residue Discs. ) P ∈ X sm ( F P ) , t : D P ∼ = p Z p , ω | D P = f ( t ) dt ( Integrals on a disc. ) � R � t ( R ) Q , R ∈ D P , ω := f ( t ) dt . Q t ( Q ) ( Integrals between discs. ) � R Q ∈ D P 1 , R ∈ D P 2 , ω := ? Q David Zureick-Brown (Emory) p -adic integration, and uniformity. Oct 26, 2014 18 / 25
Analytic continuation of integrals via Abelian varieties ( Integrals between discs. ) � R Q ∈ D P 1 , R ∈ D P 2 , ω := ? Q ( Albanese map. ) ι : X ֒ → Jac X , Q �→ [ Q − ∞ ] ( Abelian integrals via functorality and additivity. ) � R � ι ( R ) � [ R −∞ ] � [ R − Q ] � n [ R − Q ] ω = 1 ι ∗ ω = ω = ω = ω n Q ι ( Q ) [ Q −∞ ] 0 0 David Zureick-Brown (Emory) p -adic integration, and uniformity. Oct 26, 2014 19 / 25
Analytic continuation of integrals via Frobenius ( Integrals between discs. ) � R Q ∈ D P 1 , R ∈ D P 2 , ω := ? Q ( Abelian integrals via functorality and Frobenius. ) � R � φ ( Q ) � φ ( R ) � R ω = ω + ω + ω φ ( Q ) φ ( R ) Q Q ( Very clever trick (Coleman) ) � φ ( R ) � R � R � ω i = φ ∗ ω = a ij ω j φ ( Q ) Q Q j 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 � R ω − ω = a ( ω ) (log( t ( R )) − log( t ( Q ))) + c ( ω ) ( t ( Q ) − t ( R )) Q Q 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 / Z p is stable , but not regular. ( Residue Discs. ) P ∈ X sm ( F p ) , t : D P ∼ = p Z p , ω | D P = f ( t ) dt ( Residue Annuli. ) P ∈ X sing ( F p ) , t : D P ∼ = p Z p − p r Z p , ω | D P = f ( t , t − 1 ) dt ( Integrals on an annulus are multivalued. ) � R � t ( R ) f ( t , t − 1 ) dt = · · · + a ( ω ) log t + · · · ω := Q t ( Q ) ( 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
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