Crystalline Cohomology and Frobenius

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1 Crystalline Cohomology and Frobenius Drew Moore References: Berthelot s Notes on Crystalline Cohomology, discussions with Matt Motivation Let X 0 be a proper, smooth variety over F p. Grothendieck s etale cohomology furnishes us a way to speak of the cohomology of X 0 suitably with: coefficients in a field OR l-torsion coefficients, l p For example, if there exists a proper smooth variety X/Q with good reduction = X 0 at p, then H i Betti(X(C), Z l ) = H i ét(x 0/Fp, Z l ) as Z l -modules for l p. But what about p-torsion? Answer: H i Betti(X(C), Z p ) = H i crys(x 0 /Z p ) But crystalline cohomology gives even more. It doesn t just fill in the gap left at p by etale cohomology. My main goal for this talk is to give an example of crystalline cohomology furnishing a Frobenius operator on the solutions to differential equations. (Frobenius acting on differential equations is an idea that predates crystalline cohomology.) 2 Betti Cohomology with C-coefficients The construction of etale cohomology, though esoteric at first glance, is very much motivated by Betti cohomology and transcendental techniques.

2 Similarly, Grothendieck used transcendental ideas to motivate the search for another cohomology theory: one can say that the de Rham theorem motivates crystalline cohomology. So let s return to Betti cohomology for X(C). What is the Betti cohomology of the complex manifold X(C)? Let LS denote the category of local systems on X(C). It is the category of locally constant sheaves of complex vector spaces. Main example: C, the constant sheaf C(U) = C π 0(U). It is locally constant because it is constant! Global sections Γ: LS C-v.sp. is a left exact functor. Betti cohomology is its right derived functor. That is, H i Betti(X(C), C) := R i Γ(C) It is true that HBetti i is computable via cochains - you use the presheaf/sheaf of cochains. 3 The de Rham Theorem and Riemann-Hilbert Correspondence The de Rham theorem can be explained homologically as well. The key input is the Poincare lemma: 0 C O X d Ω X Ω 2 X... ( ) is an exact sequence of sheaves. This allows one to compute cohomology of C in terms of differential forms. The exactness of ( ) should really be though of as a quasi-isomorphism of C with the de Rham complex (C) (0 O X d Ω X Ω 2 X... ) The left side is in the transcendental world of local systems. The right side is in the algebraic world of vector bundles (locally free O X -modules). The complex on the RHS can be completely reconstructed from O X d Ω X using the Leibniz rule. So we can think of C and {O X sense. d Ω X } corresponding, in some 2

3 This is the trivial example of the Riemann-Hilbert correspondence. It gives an equivalence of categories between LS, and V IC - the category of vector bundles with integrable connection. If E is a vector bundle, a connection gives a way to differentiate sections e E with respect to a vector field ξ of X. Namely, it is a C-linear map satisfying a Liebniz rule. multiplication by ξ. : E E Ω X The section ξ (e) E is given by composing with inner Saying a connection is integrable is equivalent to saying that a certain sequence analogous to ( ) is exact. The Riemann Hilbert correspondence states there is an equivalence: LS X V IC X where (E, ) V IC is taken to the local system E = E =0. It was important that we assumed X was proper and smooth for the Riemann-Hilbert correspondence. A version exists for X only smooth - you restrict to the full subcategory of V IC given by those regular at. Example. The kernel of O X d Ω is the local system of locally constant functions C. Example. Let X = G m = Spec C[x, x ]. Set E = O X. Fix some c C \ Z. Define the integrable connection : O X Ω (f) = df c f dx x So we are looking to solutions to the equation x f (x) = c f(x) for c a constant. We conclude: E = E =0 is the local system of branches of kx c, for k C any constant. Since we chose c / Z, E is not the trivial local system. In the case that c = /2, the monodromy around 0 is multiplication by - that is, if I analytically continue z one full rotation around 0 C, i get to z. In general (c anything), the monodromy around 0 is multiplication by e 2πic. 3

4 Example. Let X = P \ {0,, } = Spec C[λ, λ, λ ]. Let E = O2 X, thought of as column vectors. We define a connection : OX 2 OX 2 OX Ω X = (Ω X ) 2 by For example, = d + 2(λ ) ( ) /λ dλ ( ) ( ) /2(λ ) = 0 /2λ(λ ) Wtf is this? Well if E λ is the Legendre family y 2 = x(x )(x λ) over X, then the de Rham cohomology of E λ is spanned by dx/y and x dx/y. The above bundle is OX 2 = dx O X y O X x dx y. Specifically, for fixed λ 0, the functions H (E λ0, Z) γ λ0 H (E λ0, Z) γ λ0 γ λ0 γ λ0 dx y x dx y express these meromorphic -forms as cohomology classes (well defined because dx/y is holomorphic, and xdx/y has residue zero at its pole). If we differentiate them as λ varies, we get the above equations. is the Gauss-Manin connection for the Legendre family. That is, I have ( ) d dx = dλ γ λ y dx 2(λ ) γ λ y x dx 2λ(λ ) γ λ y The previous example illustrates the following important fact: Suppose f : X S is smooth and proper. Then the local system on S of periods (or the Betti cohomology local system) of f correspond (under RH) to the relative de Rham cohomology of f with its Gauss Manin connection. 4 Idea of Crystalline Cohomology in Char. 0 Let X [] be the first order fuzz around X = (X) X X. The functions on X [] are generated by those of the form f(x) and f(x + dx) = f + df where 4

5 f is a function on X x = (x,..., x n ) are coordinates on X dx = (dx,..., dx n ) are infinitesimals - i.e. dx 2 i = 0 Via the two projections the functions O X [] X [] X X X on X [] form a sheaf of O X -algebras in 2 ways: one via multiplication by f(x). The left O X -alg structure. (Just saying that I idenitfy functions on X with functions on X [] via the first projection.) one via multiplication by f(x + dx). The right O X -alg structure. (The second projection.) Since calculus is about the difference between f(x) and f(x + dx), in order to do calculus on a vector bundle E, want to compare O X [] OX E and E OX O X [] In particular, we have the following: Proposition. A connection is equivalent to an isomorphism of vector bundles over X [] : ɛ: O X [] E E O X [] which is the identity when we restrict to X X [] (i.e. modulo the infinitesimals). Given ɛ, we have for e a section of E, (e) = ɛ( e) e This (barely) hints at the following important idea: vector bundles descent data with integrable for infinitesimal connection thickenings 5 Crystalline Summary Setup: S 0 is the base scheme in characteristic p. E.g., S 0 = Spec F p or S 0 = Spec F p 5

6 S is a PD-thickening of S 0 in the p-direction. E.g., S = Spec Z/p n Z or S = Spec W n (F p ). (One can also think S = Spec Z p though this isn t exactly correct.) X 0 S 0 is proper and smooth. We are interested in X 0 s crystalline cohomology. (E.g., X 0 is a characteristic p elliptic curve.) In this setup, one can define the crystalline site of X 0 /S. The open sets of X/S are U P D Ũ X 0 S 0 S infinitesimal neighborhoods of X 0 over S. Here, U Ũ is a PD-thickening. (Think U U [].) The covers of U Ũ are collections (U i Ũi) such that the U i cover U. Since we have a site, can define sheaves, and so we can define cohomology. A brief summary: Most important sheaf: The sheaf of functions O X/S (U Ũ) = O Ũ Most important computational tool: Comparison with de Rham cohomology. If X/S is smooth and X 0 = X S S 0, then H i crys(x 0 /S) := RΓ i crys(o X0 /S) = H i DR(X/S) This crystalline cohomology is an O S -module with integrable connection. (In the case that S is Spec Z p, the integrable connection is not really important.) Most important additional structure that Crystalline cohomology has that de Rham cohomology doesn t: Functoriality of the crystalline topos. If I have a picture (call 6

7 it f ) f 0 Y 0 X 0 T 0 T S 0 S f Then I can pull back sheaves on Y 0 /T to X 0 /S via f and I can push forward sheaves on X 0 /S to Y 0 /T. The usual example: let ϕ 0 : S 0 S 0 be Frobenius, and let ϕ: S S be any lift of Frobenius. Then: ϕ 0 X 0 X 0 S 0 S Get a map S 0 S F ϕ : ϕ H i (X 0 /S) H i (X 0 /S) which (by functoriality) commutes with the integrable connection. As a consequence: Frobenius acts on the solutions to differential equations. Note: de Rham cohomology also has functoriality - but only for things defined in the characteristic 0. Another important computational tool: Fibers of the crystal H i (X 0 /S) can be computed via Teichmuller lifts. ϕ 6 The Legendre Example Consider: S = Spf Z p [λ, λ, λ ] 7

8 E λ the (total space of the) Legendre family y 2 = x(x )(x λ) over S. S 0 and E λ,0 their special fibers. E the elliptic curve y 2 = x 3 x with CM by Z[i]. Its special fiber E,0. We study the relative crystalline cohomology H i crys(e λ,0 /S). It is an F -crystal on S. Since E λ is a smooth lifting of E λ,0, the crystal part of F -crystal has already been seen: it s just the de Rham cohomology with the Gauss Manin connection. What about the Frobenius? We will first understand what happens at λ =, which is easier to understand because E has CM by Z[i]. If p = αα splits in Z[i], then E has good ordinary reduction at p. The Frobenius action on the crystalline cohomology H (E /Z p ) has eigenvalues with p-adic valuations 0 and. If p is inert in Z[i], then E has good supersingular reduction at p. The Frobenius action has eigenvalues both with p-adic valuations /2. So now we go back to the differential equation given by the Gauss Manin equation, and solve in a neighborhood of λ =. We are looking for functions f(λ), g(λ) such that ( f(λ) dx ) dx + g(λ)x = 0 y y Equivalently, { 2λ(λ )f (λ) + λf(λ) g(λ) = 0 2(λ )g (λ) + f(λ) g(λ) = 0 Since we want to expand this at λ =, we find { 2(λ + ) 2 f (λ) 6(λ + )f (λ) + 4f (λ) + (λ + )f(λ) f(λ) g(λ) = 0 Abstractly set 2(λ + )g (λ) 4g (λ) + f(λ) g(λ) = 0 f(λ) = a n (λ + ) n g(λ) = b n (λ + ) n The n-th derivative at λ = of the first equation is given by For the second equation, we get 2(n )a n 6na n + 4(n + )a n+ + a n a n b n = 0 2nb n 4(n + )b n+ + a n b n = 0 8

9 Then, we can solve recursively for a n and b n. We get: { a n = 4n (2(n 2)a n 2 6(n )a n + a n 2 a n b n ) b n = 4n (2(n )b n + a n b n ) The surprising result: a0 = b0 = 0 aa = {-2:0, :0, 0:a0} bb = {-2:0, :0, 0:b0} def a(n): if not n in aa: aa[n] = -(2 * (n-2) * a(n-2) - 6 * (n-) * a(n-) + a(n-2) - a(n-) - b(n-))/(4 * n) return aa[n] def b(n): if not n in bb: bb[n] = (2 * (n-) * b(n-) + a(n-) - b(n-))/(4 * n) return bb[n] def denom_a(n): return a(n).denominator().factor() denom_a(000) ^2494 * 3 * 47 * 7 * 83 * 07 * 39 * 63 * 9 * 99 * 223 * 227 * 239 * 307 * 3 * 33 * 49 * 43 * 439 * 443 * 463 * 467 * 479 * 487 * 49 * 499 * 683 * 69 * 79 * 727 * 739 * 743 * 75 * 787 * 8 * 823 * 827 * 839 * 859 * 863 * 883 * 887 * 907 * 9 * 99 * 947 * 967 * 97 * 983 * 99 Figure : Sage code computing the factorization of the denominator of a

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