Motivic Periods, Coleman Functions, and Diophantine Equations

Transcription

1 Motivic Periods, Coleman Functions, and Diophantine Equations An Ongoing Project D. 1 1 MIT April 10, 2018 Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

2 Motivation: The Unit Equation Let Z be an integer ring with a finite set of primes inverted (= O k [1/S]) and X = P 1 \ {0, 1, }. Theorem There are finitely many x, y Z such that x + y = 1 Equivalently, X (Z) <. Originally proven by Siegel using Diophantine approximation around Problem Find X (Z) for various Z, or even find an algorithm. In 2004, Minhyong Kim gave a proof in the case k = Q using a non-abelian version of Chabauty s method. Refined Problem (Chabauty-Kim Theory) Find p-adic analytic (Coleman) functions on X (Z p ) that vanish on X (Z). Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

3 The Chabauty-Kim Ideal The method produces a set I Z PL of Q p-valued functions on X (Z p ) that vanish on X (Z) (for any p Spec(Z)). This is an ideal in the ring of Coleman functions, a certain set of p-adic analytic functions on X (Z p ) with especially nice properties. In particular, any such function has finitely many zeroes (which proves finiteness of X (Z)). A conjecture of Kim and others states that the set of all common zeroes of elements of IPL Z is X (Z). Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

4 Recent Results Theorem (Dan-Cohen, Wewers, 2013) For Z = Z[1/2] and p 2, the following Coleman function is in I Z PL : 24 log p (2)ζ p (3)Li p 4 (z) + 8 ) (log p (2) Li p4 7 (12 ) log p (z)li p 3 (z) ( logp (2) ) 7 Lip 4 (1 2 ) + logp (2)ζ p (3) log p (z) 3 log p (1 z) Dan-Cohen and Wewers furthermore verified cases Kim s conjecture using the above function and various p. In 2015, Dan-Cohen posted a preprint showing that this could be made into an algorithm, whose halting is conditional on certain well-known conjectures. Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

5 Recent Results, cont. Theorem (C, Dan-Cohen, 2017) For Z = Z[1/3] and p 2, 3, the following Coleman function is in I Z PL : ( ζ p (3) log p (3)Li p 18 4 (z) 13 Lip 4 (3) 3 ( ζ p (3) log p (3) 4 (logp (z)) 3 Li p 1 (z) 24 ) 52 Lip 4 (9) log p (z)li p 3 (z) )), ( Lip 4 (3) 3 52 Lip 4 (9) Our current goal is to verify Kim s conjecture in this case. In other words, as X (Z[1/3]) =, we must show that this function has no zeroes, at least for various p. Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

6 Classical Chabauty s Method The classical Chabauty-Coleman method allows one to find p-adic analytic functions on a hyperbolic curve X that vanish on its Z p points. One embeds X into a semi-abelian variety J and considers the diagram: X (Z) X (Z p ) J(Z) J(Z p ) The method works when rank Z (J(Z)) < dim(j), in which case the bottom horizontal arrow has non-dense image. In particular, functions on J(Z p ) that vanish on J(Z) pullback to functions on X (Z p ) that vanish on X (Z). The theory of Coleman shows that such functions can be expressed explicitly in terms of p-adic integrals. Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

7 Towards non-abelian Chabauty The non-abelian version of Chabauty s method is designed to apply when rank Z (J(Z)) < dim(j) fails. One may in fact replace J(Z) by a corresponding Selmer group, which is a subset of H 1 (Gal(Q/Q), H 1 (J Q, Q p )). Noting that X and J have the same H 1, this can be written purely in terms of H 1 (X Q, Q p ) as an abstract Galois representation. The non-abelian version replaces H 1 (X Q, Q p ) by a quotient of π 1 (X Q, Q p ). In fact, we use a motivic version of the above, which simplifies the situation. Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

8 The Motivic Galois Group One works with two important motivic objects. The first is the motivic Galois group π1 un (Z), a pro-algebraic group over Q whose ring of functions A(Z) := O(π1 un (Z)) is known as the ring of mixed Tate motivic periods over Z. A(Z) is naturally a graded Hopf algebra, and deep theorems in arithmetic geometry give the abstract structure of A(Z). A typical element of A(Z) is given by a formal special value of a polylogarithm, i.e., an element of the form Li u k(z) for z X (Z) and k Z 1. There is a map per p : A(Z) Q p sending a formal special value to its corresponding p-adic value. Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

9 The Motivic Fundamental Group Fact The other object is the (polylogarithmic) motivic fundamental group π1 PL (X ). As a scheme, we have π1 PL(X ) := Spec(Q[logu, Li u 1, Li u 2, ]), but its coordinate ring also has the structure of a graded Hopf algebra. The coproduct is given by the formula k 1 Li u k = Li u k i (logu ) i. i! i=1 For each z X (Z), we have a ring homomorphism κ(z): O(π PL 1 (X )) A(Z) sending Liu k to Li u k(z). For z X (Z), the homomorphism κ(z) is a homomorphism of graded Hopf algebras. Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

10 Motivic Kim s Cutter For a prime p, this gives us a diagram, known as Kim s Cutter: X (Z) X (Z p ) κ Hom GrHopf (O(π1 PL (X )), O(πun 1 (Z))) per p Hom Alg (O(π1 PL(X )), Q p) We recall the integration map per p : O(π un 1 (Z)) Q p for p Spec(Z). This induces Hom GrHopf (O(π1 PL (X )), O(πun 1 (Z))) per p Hom Alg (O(π1 PL(X )), Q p). In addition, an arbitrary z X (Z p ) induces a homomorphism O(π1 PL(X )) Q p sending Li u k to Li p k (z). Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

11 Motivic Kim s Cutter, cont. X (Z) X (Z p ) κ Hom GrHopf (O(π1 PL (X )), O(πun 1 (Z))) per p Hom Alg (O(π1 PL(X )), Q p) The above diagram is known as Kim s Cutter. We may upgrade the bottom horizontal morphism to a map of schemes, as follows: We define a scheme Z 1,Gm PL over Q by Z 1,Gm PL (R) = Hom GrHopf (O(π1 PL (X )), O(πun 1 (Z))) R) for a Q-algebra R. The bottom arrow may then be viewed as a map of Q p -schemes Z 1,Gm PL Q p π PL 1 (X ) Q p Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

12 Motivic Kim s Cutter, cont. X (Z) X (Z p ) κ Z 1,Gm per p PL Q p π1 PL(X ) Q p Dimension counts show that the bottom horizontal arrow is non-dominant, which is what proves Siegel s theorem. Therefore, there is a nonzero ideal I Z PL O(πPL 1 (X )) Q p vanishing on the image of the bottom arrow, known as the (polylogarithmic) Chabauty-Kim ideal. The right-hand vertical map is Coleman analytic, so elements of I Z PL pull back to Coleman functions on X (Z p ) that vanish on X (Z). Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

13 Our Current Work One key step in computing functions in IPL Z is computing basis for A(Z) in low degrees in terms of explicit special values of polylogarithms. For example, our result with Dan-Cohen involved showing that a certain special element of A(Z[1/3]) could be expressed as Liu 3(3) 3 52 Liu 3(9) The above computation, combined with a better method for calculating Kim s cutter in terms of A(Z), is part of a forthcoming paper. In future work with Owen Patashnick and others, we hope to extend these methods to hyperbolic curves other than X = P 1 \ {0, 1, }, with the goal of finding an effective Mordell. The corresponding motivic theory is significantly more complicated and largely conjectural, yet probably computable. Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

14 Useful References The following are on arxiv: Mixed Tate Motives and the Unit Equation, Ishai Dan-Cohen and Stefan Wewers Mixed Tate Motives and the Unit Equation II, Ishai Dan-Cohen Single-Valued Motivic Periods, Francis Brown Motivic Periods and P 1 \ {0, 1, }, Francis Brown Notes on Motivic Periods, Francis Brown Integral Points on Curves and Motivic Periods, Francis Brown Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15

15 Thank You! Motivic Periods, Coleman Functions, and Diophantine Equations April 10, / 15