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 − y 2 = ( x 2 − 1)( x 2 − 2)( x 2 − 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 over 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 B g ( Q ) 642 112 126 132 192 781 16( g + 1) Remark Elkies studied K3 surfaces of the form y 2 = 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 = 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 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 rank Z Jac X ( Q ) = 1 , g = 2 1 X ( Q ) contains 2 {∞ , (0 , 0) , (1 , 0) , (2 , 0) , (5 , 0) , (6 , 0) , (3 , ± 6) , (10 , ± 120) . } # X ( F 7 ) = 8 3 10 ≤ # X ( Q ) ≤ # X ( F 7 ) + 2 g − 2 = 10 4 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 = 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 University) Diophantine and p -adic geometry April 6, 2018 7 / 30
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 ( p -adic Rolle’s (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 University) Diophantine and p -adic geometry April 6, 2018 8 / 30
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 University) Diophantine and p -adic geometry April 6, 2018 9 / 30
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 ( p -adic Rolle’s (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 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 = 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 University) Diophantine and p -adic geometry April 6, 2018 11 / 30
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 University) Diophantine and p -adic geometry April 6, 2018 12 / 30
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 University) Diophantine and p -adic geometry April 6, 2018 13 / 30
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 University) Diophantine and p -adic geometry April 6, 2018 14 / 30
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 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 = 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 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 = rank Z Jac X ( Q ) . Suppose r < g − 2 . 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 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 ( 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 University) Diophantine and p -adic geometry April 6, 2018 18 / 30
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 University) Diophantine and p -adic geometry April 6, 2018 19 / 30
Analytic continuation of integrals via Frobenius ( Integrals between discs. ) � R Q ∈ D P 1 , R ∈ D P 2 , ω := ? Q ( 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 i = df i + a ij ω j φ ( Q ) Q Q j 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 � R ω − ω = a ( ω ) [log( t ( R )) − log( t ( Q ))] + c ( ω ) [ t ( R ) − t ( Q )] Q 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 / 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 ( R ) + · · · ω := Q t ( Q ) ( 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
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