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Section 6.7 Serre-Tate theory (Alex)

Subsection 6.7.1 Intro/background

We will work in generality following Katz's Serre-Tate Local Moduli [64]. Note that Hida [61] also has a modernized exposition of the same. The original “source” material is Woods hole notes, sketchy at best.

Recall that one thing we will try and do is to prove the following formula

\begin{equation*} (\underline x. T_m \underline x^\sigma ) = \frac12 \sum_{n=1}^\infty \#\Hom_{W/\pi _v^n}(\underline x^\sigma , \underline x)_{\deg m} \end{equation*}

To do this we want to understand more about the special nature of Heegner points, representing pairs of CM elliptic curves. Angus told us last time about how to lift curves together with an endomorphism from \(\FF_q\) to a number field. The set of all lifts to positive characteristic of a given curve over a finite field, can be thought of as deformations of the given curve. The aim is to describe these deformations in terms of a simpler object.

We will work in generality, because it isn't really any harder, and makes it a bit clearer in some cases. That is we will work with abelian schemes which are higher dimensional generalizations of elliptic curves (e.g. products of elliptic curves, weil restrictions, or jacobians of higher genus curves). You can replace abelian scheme with elliptic curve if you like and restrict to dimension \(g =1\text{.}\) . We let \(R\) be a ring, and define the category

\begin{equation*} \mathrm{AbSch}(R) = \{\text{abelian schemes over} R\}\text{.} \end{equation*}

We will fix \(W\) a complete DVR with residue field \(\overline \FF_p\) (i.e. \(W\) could be the Witt vectors of \(\overline \FF_p\)). Complete means that

\begin{equation*} W = W_\infty =\varprojlim_{m} \underbrace{W/p^m}_{=W_m} \end{equation*}

The for \(m =1,2,\ldots, \infty \) we let \(R\) be a base ring Given \(R\) ring we can reduce to the residue field, but this map is many to one, what is the smallest amount of data needed to recover \(A/R\) an abelian scheme from \(A_0/R_0\text{?}\)

Our rings today will probably all be complete local \(R\)-algebras.

Given any abelian scheme \(A/R\) over any ring we can form its \(p\)-divisible group, also known as a Barsotti-Tate group

\begin{equation*} A[p^\infty ] \end{equation*}

this is “p-divisible” as given any \(p\)-power torsion point on \(A\) its division by \(p\) is also \(p\)-power torsion. Formally the definition is

Definition 6.7.1.

A \(p\)-divisible group \(G\) over \(R\) of height \(h\) is an inductive system

\begin{equation*} G = (G_v, i_v),\,v \ge 0 \end{equation*}

where each \(G\) is a finite group scheme over \(R\) of order \(p^{vh}\) and for each \(v\ge0\)

\begin{equation*} 0 \to G_v \xrightarrow{i_v} G_{v+1} \xrightarrow{p^v} G_{v+1} \end{equation*}

is exact, so \(i_v\) is the kernel of \(p^v\text{.}\)

Example 6.7.2.

For normal abelian groups (i.e. constant group schemes) we must have

\begin{equation*} G_v = (\ZZ/p^v)^h \end{equation*}

with

\begin{equation*} \lim G_v = (\QQ_p/\ZZ_p)^h\text{.} \end{equation*}
Example 6.7.3.

For abelian varieties \(A\) of dimension \(d\) we have

\begin{equation*} (A[p^v], i_v\colon A[p^v] \hookrightarrow A[p^{v+1}]) \end{equation*}

of height \(h =2d\text{.}\) Note this is true even in the supersingular case!

Given a map of rings \(R \to R_0\) let the category of deformations be

\begin{equation*} \mathrm{Def}(R,R_0) =\{(A_0, G, \epsilon ) : A_0/R_0\text{ abelian scheme},\,G/R\text{ a }p\text{-divisible group},\,\epsilon \colon G_0 \to A_0 [p^\infty ]\} \end{equation*}

\(\epsilon \) an isom of \(p\)-divisible groups ove \(R_0\text{.}\) So these are abelian schemes over the “small” ring and a choice of compatible \(p\)-divisible group over the big ring.

With this setting we have a nice map as follows If \(R\) is a ring with \(p\) nilpotent, \(I \subseteq R\) a nilpotent ideal and \(R_0 = R/I\) then

\begin{equation*} \mathrm{AbSch}(R) \to \mathrm{Def}(R,R_0) \end{equation*}
\begin{equation*} A \mapsto (A_0, A[p^\infty ],A[p^\infty ]\otimes R_0 \simeq A_0 [p^\infty ])\text{.} \end{equation*}

Thus the set of deformations of a fixed \(A_0\) corresponds to deformations of \(A_0\lb p^\infty \rb \text{.}\)

This is a kinda ridiculous theorem, it tells us that all the information in an abelian variety over \(R\) is contained in the reduction to \(R_0\) except the \(p^\infty\) torsion and the information of how this fits together.

Hence to study the abelian varieties over \(R\) reducing to a given \(A_{R_0}/R_0\) we can just study the \(p\)-divisible groups over \(R\) with an isomorphism to \(A_{R_0}\lb p^\infty\rb\text{.}\)

Subsection 6.7.2 Drinfeld's proof of Serre-Tate

Drinfeld's proof cleverly extracts the content common to both things we are lifting, the abelian scheme and the \(p\)-divisible group.

Let \(R\) be a local \(W_m\)-alg. \(I \subseteq R\) a nilpotent ideal with nilpotency index \(\nu + 1\text{,}\) let \(R_0 = R/I\text{.}\) \(N = p^t\) an integer s.t. \(N I = 0\text{.}\) Given an \(R\)-algebra \(A\) we might consider

\begin{equation*} A/IA = A \otimes R_0 \end{equation*}

and also

\begin{equation*} A/\ideal m_A\text{.} \end{equation*}

So given a functor from \(R\)-algebras to an arbitrary abelian category

\begin{equation*} G \colon R\text{-alg}\to C \end{equation*}

we have two natural subfunctors

\begin{equation*} G_I \colon A \mapsto \ker( G(A) \to G(A\otimes R_0)) \end{equation*}
\begin{equation*} \widehat G(A) \colon A \mapsto \ker(G(A) \to G(A/\ideal m_A))\text{,} \end{equation*}

note that

\begin{equation*} G_I \subseteq \widehat G\text{.} \end{equation*}

What are formal groups?

Definition 6.7.5. Formal groups.

A \(n\)-dimensional formal group over a ring \(R\) is a power series

\begin{equation*} F(x,y) = (x_1, \ldots, x_n) + (y_1,\ldots,y_n) + O(\text{degree 2 terms}) \in (R[[x_1,\ldots, x_n,y_1,\ldots, y_n]])^n \end{equation*}

that is associative in the sense that

\begin{equation*} F(F(x,y),z) = F(x,F(y,z))\text{.} \end{equation*}

The formal group is commutative if \(F(x,y) = F(y,x)\text{.}\)

Given an abelian variety we can get a 1-dimensional formal group by completing at the origin. E.g. for an elliptic curve

\begin{equation*} y^2 = x^3 + ax +b \end{equation*}

we can express \(x = x(t) =t^{-2} + \cdots\text{,}\) \(y = -t^{-3} + \cdots\)

\begin{equation*} \frac{1}{t^{2}} - at^{2} - bt^{4} - a^{2}t^{6} - 3 a bt^{8} + \left(-2 a^{3} - 2 b^{2}\right)t^{10} - 10 a^{2} bt^{12} + \left(-5 a^{4} - 15 a b^{2}\right)t^{14} + \left(-35 a^{3} b - 7 b^{3}\right)t^{16} + \left(-14 a^{5} - 84 a^{2} b^{2}\right)t^{18} + O(t^{20}) \end{equation*}
\begin{equation*} \frac{-1}{t^{3}} + at + bt^{3} + a^{2}t^{5} + 3 a bt^{7} + \left(2 a^{3} + 2 b^{2}\right)t^{9} + 10 a^{2} bt^{11} + \left(5 a^{4} + 15 a b^{2}\right)t^{13} + \left(35 a^{3} b + 7 b^{3}\right)t^{15} + \left(14 a^{5} + 84 a^{2} b^{2}\right)t^{17} + \left(126 a^{4} b + 84 a b^{3}\right)t^{19} + O(t^{20}) \end{equation*}

then the group law in terms of \(t\) is

\begin{equation*} t_{1} + t_{2} + \left(-2 a\right) t_{1}^{4} t_{2} + \left(-4 a\right) t_{1}^{3} t_{2}^{2} + \left(-4 a\right) t_{1}^{2} t_{2}^{3} + \left(-2 a\right) t_{1} t_{2}^{4} + \left(-3 b\right) t_{1}^{6} t_{2} + \left(-9 b\right) t_{1}^{5} t_{2}^{2} + \left(-15 b\right) t_{1}^{4} t_{2}^{3} + \left(-15 b\right) t_{1}^{3} t_{2}^{4} + \left(-9 b\right) t_{1}^{2} t_{2}^{5} + \left(-3 b\right) t_{1} t_{2}^{6} + \left(-2 a^{2}\right) t_{1}^{8} t_{2} + \left(8 a^{2}\right) t_{1}^{6} t_{2}^{3} + \left(16 a^{2}\right) t_{1}^{5} t_{2}^{4} + \left(16 a^{2}\right) t_{1}^{4} t_{2}^{5} + \left(8 a^{2}\right) t_{1}^{3} t_{2}^{6} + \left(-2 a^{2}\right) t_{1} t_{2}^{8} + O(t_{1}, t_{2})^{10}\text{.} \end{equation*}

In general an \(n\)-dimensional abelian variety gives an \(n\)-dimensional formal group.

Given a complete local ring \(R\) we can evaluate by substituting \(t\) for anything in the maximal ideal. So a formal group \(G\) defines a functor

\begin{equation*} G\colon \text{complete local }R\text{-algebras} \to \text{AbGrp} \end{equation*}
\begin{equation*} G(A) = (\ideal m_A)^n\text{with multiplication by }G\text{.} \end{equation*}

We need to show that \(\lb N\rb a = 0\) for any \(a \in G_I(A) \subseteq G(A)\) for which \(a_i \in I\) for all \(i\text{.}\) An element of \(G_I(A)\) has coordinates in \(IA\) and \(NR = 0\) so we have

\begin{equation*} (\lb N\rb a)_i = N a_i + \text{h.o.t.} \in N(IA) + (IA)^2 = (IA)^2 \end{equation*}

as \(R\) and hence \(A\) is \(N\) torsion, this gives inductively that

\begin{equation*} ([N^\nu] a)_i \in (IA)^{2\nu} =0 \end{equation*}

as \(I^{\nu + 1} =0 \text{.}\)

Definition 6.7.7.

Given a covariant functor

\begin{equation*} G \colon \text{complete local} R-alg \to \text{AbGrp} \end{equation*}

which for any faithfully flat finite type \(A \hookrightarrow C\) we have

\begin{equation*} G(A)\hookrightarrow G(C) \end{equation*}

and “the sheaf condition” w.r.t \(A\hookrightarrow C\text{.}\) Is called an fppf abelian sheaf.

Example 6.7.8.
\begin{equation*} G(A) = E(A) \end{equation*}

for \(E\) an abelian variety.

  1. \(p\)-divisibility implies that if \(pf = 0\) so \(pf(x)= 0\) for all \(x\) then if \(py = x\) we have
    \begin{equation*} f(x) = pf(y) =0 \end{equation*}
    so \(f = 0\text{.}\)
  2. We can write
    \begin{equation*} 0 \to H_I \to H \to H_0 \to 0 \end{equation*}
    so that by left exactness of hom
    \begin{equation*} 0 \to \Hom(G, H_I) \to \Hom(G, H) \to \Hom(G, H_0) = \Hom(G_0, H_0) \end{equation*}
    so the second map is what we want so we want
    \begin{equation*} \im(\Hom(G, H_I) \to \Hom(G,H)) = 0 \end{equation*}
    rhs p-tors free, and \(H_I\) is killed by \(N^\nu \) (using formality of \(\hat H \) here and another lemma I didn't really state).
  3. Uniqueness follows from 2. so we just lift
    \begin{equation*} f_0 \in \Hom(G_0, H_0) \end{equation*}
    to \(y \in H(A)\text{.}\)

Proof of serre tate:

As above \(N\) is a p-power killing \(I\text{,}\) \(\nu \) an integer such that \(I^{\nu + 1} = 0\text{.}\) We can apply Drinfeld to each of \(A, A', A\lb p^\infty \rb, A'\lb p^\infty \rb , A_0\lb p^\infty \rb , A'_0 \lb p^\infty \rb \text{.}\)

We show our functor is fully faithful ie.

\begin{equation*} \operatorname{Hom}_{\mathcal{A}}\left(A, A^{\prime}\right) \rightarrow \operatorname{Hom}_{D E F}\left(\left(A_{0}, A\left[p^{\infty}\right], \text { id }_{A_{0}}\right),\left(A_{0}^{\prime}, A^{\prime}\left[p^{\infty}\right], \text { id }_{A_{0}^{\prime}}\right)\right) \end{equation*}

part 2. with \(G =A\text{,}\) \(H = A'\) gives inj as an abvar is a p-div abelian fppf sheaf.

To show surjectivity apply part 3. of drinfeld with \(G = A\text{,}\) \(H = A'\) to get a lift from each \(f_0\in \Hom(A_0, A_0')\) of \(N^\nu f_0\) to

\begin{equation*} g = `` N^\nu f'' \in \Hom(A, A') \end{equation*}

to satisfy part 4 we need that \(g\) kills \(A\lb N^\nu \rb \text{.}\) We have \(N^\nu f = g\) on \(A\lb p^\infty \rb \) and as \(N\) is a \(p\)-power in fact \(A \lb N^\nu \rb \subseteq A\lb p^\infty \rb \) is killed by \(N^\nu f\text{.}\)

To prove essential surjectivity onto \((A_0, D, \phi )\text{,}\) we lift \(A_0\) to \(X\) arbitrarily, and must match up the \(p\)-divisible group and iso. We have an isom \(\alpha _0 \lb p^\infty \rb \to A_0 \lb p^\infty \rb \text{.}\) And so a lift

\begin{equation*} g\colon X[p^\infty ] \to D \end{equation*}

\(N^\nu \alpha _0\) applying the lemma to \(G = X_0\lb p^\infty \rb \text{,}\) \(H = D\text{.}\) So we get an isogeny \(g\) and we quotient by the kernel.