Notes on the Gauss-Manin Connection

Glenn Stevens

§0. The Gauss-Manin Connection.

Let p > 2 be a prime and N ≥ 4. The p-adic modular curve X_{1}(Np) will be denoted X. We let X = W_{0} ∪ W_{∞} denote the usual decomposition of X as the union of two wide open sets. Hence W_{i} (i = 0 or ∞) is a wide open neighborhood of the ordinary component Z_{i} containing the cusp i (= 0 or ∞) and the intersection W = W_{∞} ∩ W_{0} is the union the supersingular annuli. We have Z_{∞} := W_{∞} \ W and Z_{0} := W_{0} \ W. Let X(v) be the largest connected wide open neighborhood of Z_{∞} on which the canonical subgroup is defined. Let π : E ⟶ X denote the universal generalized elliptic curve over X and let H := H_{DR}^{1}(E/X) denote the De Rham cohomology sheaf on X. Let ∇ : H ⟶ H denote the Gauss-Manin connection on H. One knows from Katz that H has a natural decomposition as

H = \underline{ω}^{−1} ⊕ \underline{ω}

where \underline{ω} := π_{∗}Ω_{E/X}^{1}(log(cusps)). Now choose two sections X, Y of H(X(v)) as follows. Choose X ∈ \underline{ω}(X(v)) ⊆ H(X(v)) corresponding to the weight one Eisenstein series E_{1}, and let Y be the generator of \underline{ω}^{−1}(X(v)) for which the cup product of X and Y is 1. It follows then that the q-expansions of X and Y at the ∞ cusp are

\begin{aligned}X & = E_{1}(q)ω_{can}, \\ Y & = E_{1}(q)^{−1}η_{can}. \end{aligned}

There is a matrix ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) of regular 1-forms on X(v) such that

∇ ( \begin{matrix}X \\ Y \end{matrix} ) = ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ( \begin{matrix}X \\ Y \end{matrix} )

Since ⟨X, Y ⟩ = 1 we have ⟨∇X, Y ⟩ + ⟨X, ∇Y ⟩ = 0. This is equivalent to the assertion α + δ = 0. Hence the above matrix has trace zero. In particular, we may write ∇ in terms of the Lie algebra sl(2) as

∇ = (αh + βx + γy)

where h := ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), x = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ), and y := ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ).

§1. Representations of sl(2) and the Gauss-Manin connection.

Our basic observation is that the above description of ∇ gives us a natural way of making vector bundles with connection on X out of representations of sl(2), as follows.

Consider a category R of representations of sl(2). I won’t give a precise definition of what this category ought ot be right now, but I’ll need to give one at some point, since I want to allow representations to be infinite dimensional, and to have coefficients which are functions on the space X defined below. The simplest thing to do would be to require the objects to be Banach modules, but I think this may be too restrictive.

Also consider a category C of quasi-coherent O-modules with connection on Z_{∞}^{†}. Again, I should be more precise, but I’m not ready to commit to a particular definition of what an object in this category should be. For now, let me just say that I want to include O^{†}-modules of infinite rank. For example, we could fix a radius v of overconvergence and then consider orthonormalizable O(X(v))-Banach modules. This is a good model to start with, but again, I think it is too restrictive.

In the last section we saw how the choice of a basis for H over Z_{∞}^{†} gives rise to an element

∇ ∈ sl(2) ⊗ O^{†}

Now we make the simple observation that ∇ gives rise to a functor from R to C in the obvious way. Namely, if V is an object of R then

\begin{matrix}∇ : V ˆ⊗O^{†} & ⟶ V ˆ⊗Ω^{†} \\ v ⊗ 1 & ⟼ h(v) ⊗ α + x(v) ⊗ β + y(v) ⊗ γ \end{matrix}

defines an object of C.

I have to add another word of caution. In practice, I will often want to take for V the dual of a Banach space. In that case, I will want the above ˆ⊗ product to be a completed tensor with respect to the weak topology on V, not the strong (Banach space) topology. The moral of the story is that it will be pretty clear what we want to do in examples, but I’m having some trouble saying it at the right level of generality.

§2. Verma modules.

Let X := Hom(Z_{p}^{×}, C_{p}^{×}). We view Z as embedded in X by n ↦ (t ↦ t^{n}). The building blocks of our Verma modules will be the formal symbols

X^{[κ_{1}]}Y^{κ_{2}}, (κ_{1}, κ_{2}) ∈ X^{2}.

The notation is intended to suggest that we are taking something like the κ_{1}-divided power of X and the usual κ_{2}-power of Y. More precisely, if k_{1} (resp. k_{2}) is the weight of κ_{1} (resp. κ_{2}), then we want X^{[κ_{1}]} (resp. Y^{κ_{2}}) to behave like the k_{1}-divided power of X (resp. the k_{2}-power of Y) with respect to differential operators. Here k_{1}, k_{2} are the weights of κ_{1}, κ_{2} respectively. I prefer to keep the κ_{i}’s in the notation because it will be more natural later on when we exponentiate to obtain an action of the group

G_{n} := \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL(2, Z_{p}) | \begin{matrix}c ≡ 0 mod p \\ b ≡ 0 mod p^{n} \end{matrix} \}

for some sufficiently large n.

For κ ∈ X define V_{κ}, V_{κ}^{−}, and D_{κ} by

V_{κ} := { ∑_{n=0}^{∞} λ_{n}X^{[κ−n]}Y^{n} | λ_{n} ∈ C_{p}, lim_{n→∞} |n!λ_{n}| = 0} ,

V_{κ}^{−} := { ∑_{n=1}^{∞} λ_{n}X^{[κ+n]}Y^{[−n]} | λ_{n} ∈ C_{p}, |(n − 1)!λ_{n}| = O(1)} ,

D_{κ} := { ∑_{n=0}^{∞} λ_{n}X^{n}Y^{κ−n} | λ_{n} ∈ C_{p}, |n!λ_{n}| = O(1)} ,

where we let sl(2) act by

\begin{aligned}x & := Y \frac{∂}{∂X} \\ y & := X \frac{∂}{∂Y} \\ h & := X \frac{∂}{∂X} − Y \frac{∂}{∂Y} . \end{aligned}

More generally, if S ⊆ X is a rigid subspace of X we may define V_{S}, V_{S}^{−}, and D_{S} similarly, by allowing the coefficients λ_{n} to be rigid functions on S satisfying the stated conditions pointwise on S. (Here again, I’m not sure what the “right” condition should be).

If κ ∈ X is an arithmetic character of weight k ≥ 0 then we may also define

V_{κ} := C_{p}[X, Y]_{k},

the finite dimensional space of homogeneous polynomials of degree k, with the action of sl(2) induced by the above formulas for x, y, h.

Proposition. If κ ∈ X is an arithmetic character of weight k ≥ 0, then there is a natural exact sequence

0 ⟶ V_{κ} ⟶ D_{κ} ⟶ V_{κ}^{−} ⟶ 0,

where the map V_{κ} ⟶ D_{κ} is defined in the obvious way, and the map D_{κ} ⟶ V_{κ}^{−} is defined termwise by

X^{n}Y^{κ−n} ⟼ \{ \begin{matrix}\frac{(−1)^{n−k−1}n!}{(n−k−1)!} · X^{[κ+n−k]}Y^{[k−n]} & if n > k; \\ 0 & otherwise. \end{matrix}

Proof. The proof is a straightforward calculation (assuming I’ve stated this correctly).

Remark. It is important to notice that the map D_{κ} ⟶ V_{κ}^{−} is defined only at arithmetic points κ and is not interpolizable on any affinoid subdomain S ⊆ X by a map D_{S} ⟶ V_{S}^{−}.

§3. Quasi-coherent O^{†}-modules and the Gauss-Manin connection.

Now we apply the functor of §1 to the sl(2)-modules described in §2 to obtain O^{†}- modules V_{κ}, V_{κ}^{−}, D_{κ} on Z_{∞}^{†} corresponding to the sl(2)-modules V_{κ}, V_{κ}^{−}, D_{κ} respectively. Here again, I haven’t written down a precise definition of these things. Presumeably, κ should encode both the weight and the nebentype. To keep things simple for now I will take κ in the identity component of X and close enough to the trivial character so that the nebentype is trivial. Eventually, we will have to change point of view and replace X^{κ} (which corresponds to the family E_{1}^{κ}, whatever that means) by the family E_{κ}, κ ∈ X.

If κ is arithmetic, we also let H_{κ} be the coherent O^{†}-module associated to V_{κ}. This is the sheaf you have been considering in your papers.

We are interested in the long exact cohomology sequence associated to the short exact sequence in the proposition of the last section.

Theorem. Let κ be an arithmetic character in X and let M_{κ}^{†} denote the space of overconvergent modular forms of weight κ. Then we have natural isomorphisms

\begin{aligned}ν : M_{−κ}^{†} & ⟶^{∼} H^{0}(Z_{∞}^{†}, D_{κ}) \\ μ : M_{2+κ}^{†} & ⟶^{∼} H^{0}(Z_{∞}^{†}, V_{κ}^{−}) \end{aligned}

Moreover, the cohomology sequence attached to the short exact sequence of the last proposition corresponds to an exact sequence

0 ⟶ H^{0}(Z_{∞}^{†}, H_{κ}) ⟶ M_{−κ}^{†} ⟶^{θ^{k+1}} M_{2+κ}^{†} ⟶^{ω} H^{1}(Z_{∞}^{†}, H_{κ}) ⟶ ⋯

where θ is the usual θ-operator and ω is the composition of Kodaira-Spencer and the standard map (H_{κ} ⊗ Ω^{1})(Z_{∞}^{†}) ⟶ H^{1}(Z_{∞}^{†}, H_{κ})

Proof Sketch: The existence of the isomorphism ν is a consequence of transversality of ∇, exactly as in your papers. (The module V_{κ} is filtered (not as sl(2)-module, just as O-module) and ∇ defines a linear isoomorphism from F il^{r}/F il^{r+1} to F il^{r−1}/F il^{r} for every r > −k.) A similar argument proves the existence of μ. The exact sequence is an easy consequence of the definitions of ν and μ.

§4. Frobenius Structure.

Let φ be the canonical lifting of Frobenius to H and let F : φ^{∗}H ⟶ H be the corresponding horizontal isomorphism. Then (H, ∇, φ, F) is an overconvergent F -crystal on Z_{∞}. Then one can also define a Frobenius structure on (D_{κ}, ∇) for any κ (not necessarily arithmetic. We just need to define a horizontal isomorphism

F : φ^{∗}D_{κ} ⟶ D_{κ}.

For this, we let e be the overconvergent function defined by

e(q) = E_{1}(q^{p})/E_{1}(q).

Then we may define

F (X^{n}Y^{κ−n}) = p^{n}e^{2n−κ}X^{n}Y^{κ−n}.

Moreover, if S is a rigid subspace of X then this frobenius is analytic in κ ∈ S. Thus we also obtain a Frobenius on the family D_{S} of Verma modules:

F : φ^{∗}D_{S} ⟶ D_{S}.

If κ is arithmetic, we can also define a Frobenius structure on V_{κ} and on V_{κ}^{−}. However, it should be emphasized that it is not possible to define a Frobenius structure on V_{κ} nor on V_{κ}^{−} unless k ∈ Z. In particular we do not obtain a family of Frobenii on the V_{κ}. However, there does exist a “Verschiebung”-structure on these families.

Proposition. For any arithmetic point κ ∈ X the exact sequence of the last proposition commutes with Frobenius.