Motivic Periods, Coleman Functions, and the Unit Equation

Transcription

1 Motivic Periods, Coleman Functions, and the Unit Equation An Ongoing Project D. Corwin 1 I. Dan-Cohen 2 1 MIT 2 Ben Gurion University of the Negev April 10, 2018 Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

2 Table of Contents 1 Motivation: The Unit Equation 2 Motivic Periods 3 Polylogarithmic Cocycles and Integral Points 4 Recent and Current Computations Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

3 Table of Contents 1 Motivation: The Unit Equation 2 Motivic Periods 3 Polylogarithmic Cocycles and Integral Points 4 Recent and Current Computations Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

4 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 fundamental groups and p-adic analytic Coleman functions. Refined Problem (Chabauty-Kim Theory) Find p-adic analytic (Coleman) functions on X (Z p ) that vanish on X (Z). Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

5 Table of Contents 1 Motivation: The Unit Equation 2 Motivic Periods 3 Polylogarithmic Cocycles and Integral Points 4 Recent and Current Computations Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

6 Periods Let (Y, D, ω, γ) be a period datum, i.e., a smooth algebraic variety Y of dimension d over Q, a normal crossings divisor D in Y, an element ω Ω d (Y ), and an element γ H d (Y (C), D(C); Q). Definition A period is a complex number equal to an integral I (Y, D, ω, γ) := γ ω for some period datum (Y, D, ω, γ). Examples Algebraic numbers, 2πi, log(r), ζ(k), Li k (r), π, (for r Q >0 and k Z >0 ) Algebraic numbers: If α is a root of p(x) Q[x], let (Y, D, ω, γ) = (A 1, {p(z) = 0} {0}, dz, [0, α]). 2πi: (G m,, dz z, γ) with γ a counterclockwise loop around 0. log(r) for r Q >0 : (G m, {1, r}, dz z, [1, r]). Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

7 Polylogarithms as Periods As we will use them extensively, we now explicitly describe the period datum that gives rise to a polylogarithm. Let z P 1 \ {0, 1, }(Q) and γ : (0, 1] P 1 \ {0, 1, }(C) a path from 0 to z. For k 0, we set z i = γ(t i ) for i = 1,, k. One may write Li γ k (z) = dz 1 dz 2 dz k 1 z 1 z 2 z k 0 t 1 t 2 t k 1 One may obtain this integral from a period datum as follows: One first considers the variety Y = X k and the divisor D = {z 1 = 0} {z 1 = z 2 } {z 2 = z 3 } {z k 1 = z k } {z k = z}. To deal with the improperness of the integral, one must use a blowup procedure to resolve the singularity in the integrand. The latter follows from Deligne s theory of tangential basepoints. Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

8 Relations Between Periods One may deduce relations between periods using the following rules: Linearity: I (Y, D, ω, γ) is linear in ω and γ. Algebraic Change of Variables: If f : (Y 1, D 1 ) (Y 2, D 2 ) is a morphism of pairs over Q, f (ω 2 ) = ω 1, and f (γ 1 ) = γ 2, then I (Y 1, D 1, ω 1, γ 1 ) = I (Y 2, D 2, ω 2, γ 2 ). Stokes Theorem: Let (Y, D) be a pair as above, D the normalization of D, and D 1 the divisor with normal crossings in D coming from double points in D. If β Ω d 1 ( D) and γ H d (Y (C), D(C); Q), then I (Y, D, dβ, γ) = I ( D, D 1, β, δγ), where δ : H d (Y (C), D(C); Q) H d 1 ( D(C), D 1 (C); Q) is the boundary map. For example, one can deduce 6ζ(2) = π 2 in this way (see the Kontsevich-Zagier article mentioned in the last slide). Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

9 Motivic Periods Defintion The ring P m,+ of effective motivic periods is the formal Q-algebra generated by period data, modulo the relations in the previous slide. There is then a natural map I : P m,+ C given by integration. Conjecture (Kontsevich-Zagier) The map I : P m,+ C is injective. Examples We denote the corresponding motivic special values by (2πi) m, log m (r), ζ m (k), Li m k (r), However, we would like something more adapted to Coleman integration. Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

10 De Rham Periods Coleman integrals use de Rham cohomology (specifically, the Frobenius and Hodge filtration) but not Betti cohomology. We therefore need: Definition The ring P dr,+ of effective de Rham periods is a variant of P m,+ in which γ represents a (relative) de Rham homology class. Examples We similarly write log dr (r), ζ dr (k), Li dr k (r), where for γ we take the canonical de Rham path between any two points. Fact There is a subring P dr,+ MT (Z) Pdr,+ of effective mixed Tate de Rham periods over Z that contains all periods coming from unirational pairs (Y, D) with good reduction over Z. Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

11 Mixed Tate Periods From now on, we assume that Frac(Z) = Q. Furthermore, the Coleman version of ζ(2), written ζ p (2), is zero. Our Motivic Periods We will therefore work with O(π un 1 (Z)) := P dr,+ MT (Z)/ζdr (2) We let ζ u (r), log u (r), ζ u (k), Li u k(r), be the images in O(π1 un(z)) of the corresponding elements of P dr,+ MT (Z). Coleman integration gives a map per p : O(π1 un(z)) Q p for p Spec(Z). An inclusion Z Z induces an inclusion O(π1 un (Z)) O(πun 1 (Z )). In particular, O(π1 un (Q)) is the union of O(πun 1 ((Z[1/S])) over all S. Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

12 Why Motivic Periods The reason working with motivic periods rather than ordinary periods is useful is that they have a nice algebraic structure. Fact (Deligne, Goncharov, Voevodsky, Borel,...) O(π1 un (Z)) has the structure of a graded Hopf algebra, and as such is abstractly isomorphic to an explicit free shuffle algebra.assuming Frac(Z) = Q, it is the free shuffle algebra Q {{τ p } p S, {σ 2n+1 } n 1 }, where each τ p has degree 1, and σ 2n+1 has degree 2n + 1. As a graded vector space, it s the free non-commutative algebra in these generators. However, it s equipped with a commutative product denoted by X, which we describe more precisely in the next slide. Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

13 Free Shuffle Algebras Let I be an index set and d : I Z a function, and let Q {x i } i I be the graded vector space underlying the free non-commutative graded algebra over Q in the variables x i, where x i has degree d(i). We define a graded Hopf algebra structure on Q {x i } i I as follows: The group X(r, s) is the set of permutations σ of {1, 2,, r + s} such that σ(1) < σ(2) < < σ(r) and σ(r + 1) < σ(r + 2) < < σ(r + s). The product of two words w 1, w 2 in the x i s is given as follows: w 1 Xw 2 := σ X(l(w 1 ),l(w 2 )) σ(w 1 w 2 ), where l denotes length, and w 1 w 2 denotes concatenation. The coproduct is given by w := w 1 w 2 w 1 w 2 =w Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

14 Table of Contents 1 Motivation: The Unit Equation 2 Motivic Periods 3 Polylogarithmic Cocycles and Integral Points 4 Recent and Current Computations Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

15 Polylogarithms and Integral Points Definition Let π PL 1 (X ) := Spec(Q[logu, Li u 1, Li u 2, ]). Fact As before, let X = P 1 \ {0, 1, }. Let z X (Q). For each integer k, using the representation of the kth polylogarithm as an iterated integral, one can define a motivic period Li u k(z) O(π1 un(q)). It follows that each z X (Q) defines a homomorphism κ(z): O(π1 PL (X )) O(πun 1 (Q)) sending Liu k to Li u k(z). z X (Z) iff Image(κ(z)) O(π un 1 (Z)) Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

16 Hopf Algebra Structure There is furthermore a graded Hopf algebra structure on O(π1 PL (X )), in which Li u k has degree k, log u has degree 1, and the reduced coproduct is given by: k 1 Li u k = Li u k i (logu ) i. i! Fact i=1 For z X (Q), the homomorphism κ(z) is a homomorphism of graded Hopf algebras. In particular, k 1 Li u k(z) = Li u k i(z) (logu (z)) i. i! i=1 Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

17 Motivic Kim s Cutter For a prime p, this gives us a diagram: 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). Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

18 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 Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

19 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). Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

20 Table of Contents 1 Motivation: The Unit Equation 2 Motivic Periods 3 Polylogarithmic Cocycles and Integral Points 4 Recent and Current Computations Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

21 The Chabauty-Kim Ideal General goal: Compute elements of IPL Z and verify cases of Kim s conjecture, i.e., that they suffice to precisely determine the integral points. 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) 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. Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

22 Our Current Work Our current work revolves around improving the algorithm, extending to multiple polylogarithms, and verifying cases of Kim s conjecture. We need to use an explicit description of Z 1,Gm PL, which is known as the space of cocycles. We can compute the former in terms of any good abstract basis of (Z)), i.e., that expresses it as a free shuffle algebra. O(π un 1 We then need to apply per p, for which we must compute a good basis of O(π1 un (Z)) (up to a certain degree) as linear combinations of explicit polylogarithms of the form Li u k(z) for z X (Q) and k 0. More specifically, we are working on Z = Z[1/3] and Z = Z[1/6]. Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

23 Understanding Z 1,G m PL Fix an arbitrary cocycle c Z 1,Gm PL. For each nonnegative integer k and each word w in Σ := {{τ p } p S, {σ 2n+1 } n 1 } of length k (or 1 if k = 0), let φ w k (c) Q denote the associated matrix entry of c, so that in the notation above, we have c(li u k) = φ w k (c)w. w Then for 0 r k, τ 1,..., τ r Σ of degree 1, and σ Σ of degree k r, we have φ στ 1 τ r k (c) = φ τ 1 0 (c) φτr 0 (c)φσ k r (c), and all other coefficients vanish. Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

24 Computing the Image of the Bottom Arrow in Kim s Cutter We write π1 un (Z) := Spec(O(πun 1 (Z))). We note that a cocycle is just a homomorphism of group schemes π1 un (Z) πpl 1 (X ). It is conjectured that per p is injective.this means we can focus instead on computing the image of Z 1,Gm PL π un 1 (Z) π PL 1 (X ) π un 1 (Z), where the map is cocycle evaluation times the identity. For any n, one may replace π1 PL (X ) by its finite-dimensional graded quotient π1 PL(X ) n := Spec(Q[log u, Li u 1,, Li u n]). We may similarly write Z 1,Gm PL,n (R) = Hom GrHopf(O(π1 PL(X ) n), O(π1 un (Z))) R). We are then reduced to computing the image of a map between finite-dimensional varieties over the function field of π1 un (Z), meaning we can algorithmically compute its image. Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

25 Basis Computations for A(Z[1/6]) To simplify notation, we let A = A(Z[1/6]) denote O(π un 1 (Z[1/6])). We let A n denote the nth graded piece. The abstract description as a free shuffle algebra shows that dim(a 0 ) = 1, dim(a 1 ) = 2, dim(a 2 ) = 4, dim(a 3 ) = 9, and dim(a 4 ) = 20. In fact, A is a free polynomial algebra on infinitely many generators, so we only need to find such generators. There are two in degree 1, one in degree 2, three in degree 3, and five in degree 4. Basic tool: use the reduced coproduct. It s injective in degrees 2 and 4 and has a kernel of dimension one in degree 3, generated by ζ u (3). Procedure: Inductively on n, write down motivic periods of the form Li u n(z) for z X (Z), apply, check dependence in lower degree. The non-injectivity of for n = 3 requires use of p-adic approximation to determine rational multiples of ζ u (3). Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

26 Basis Computations for A(Z[1/3]) As X (Z[1/3]) =, we cannot use periods of the form Li u k(z) for z X (Z[1/3]). Instead, we must find elements of A(Z[1/3]) as special elements of A(Z[1/6]). Using the abstract description of A(Z[1/6]) as Q {{τ p } p 6, {σ 2n+1 } n 1 }, the subring A(Z[1/3]) corresponds to the vector space generated by those words without any τ 2 s. More specifically, we have found that: σ 3 τ 3 = Liu 3(3) 3 52 Liu 3(9) This has allowed us to find a new element of IPL Z for Z = Z[1/3]. We have used it to verify Kim s conjecture in this case for p = 5, 7. Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

27 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 Our definition of motivic periods comes from Periods, Kontsevich and Zagier ( Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28

28 Thank You! Corwin, Dan-Cohen Motivic Periods, Coleman Functions, and the Unit Equation April 10, / 28