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Section 6.6 Deuring's theory of lifts (Angus)

Notations: \(X = X_0(N)/\QQ\text{,}\) \(x = (E \to E')\) a heegner point of discriminant \(D\text{,}\) with CM by \(\ints_K\text{.}\) \(H\) the Hilbert class field of \(K\text{,}\) \(v\) a place of \(p\text{.}\)

\begin{equation*} c = (x) - (\infty ),\, d = (x) - (0) \end{equation*}
\begin{equation*} \operatorname{Art}_K \colon \Cl_K \to \Gal H K \end{equation*}
\begin{equation*} \mathscr A \to \sigma \end{equation*}
\begin{equation*} r_{\mathscr A}(m) = \# \text{integral ideals of }\ints_K\text{ in the class of }\mathscr A \text{ of norm }m \end{equation*}
\begin{equation*} \Lambda _v = \text{ring of integers in the completion }H_v \end{equation*}
\begin{equation*} W = (\Lambda _v^{nr})^\wedge \end{equation*}
\begin{equation*} \pi _v, k_v, q_v \end{equation*}
\begin{equation*} \mathcal X/\Lambda _v \end{equation*}

a model of \(X\) over \(\Lambda _v\text{.}\)

Meta-Goal

Understand

\begin{equation*} g_{\mathscr A}(z) = \sum_{m=1}^\infty \pair c {T_m d^\sigma } e^{2\pi i m z} \end{equation*}

strategy is to decompose

\begin{equation*} \pair a b = \sum_v \pair ab_v \end{equation*}

so far, seen the archimidean \(v\) case. Today, nonarchimidean.

Following GZ and Michigan seminar.

Subsection 6.6.1 Generalities on nonarchimidean local heights

Remark 6.6.2.

\(\mathcal X\) is an arithmetic surface.

Today we will begin the proof of this proposition in the case that \(p\) is split in \(K\text{.}\) In this case LHS and RHS are both 0.

Subsection 6.6.2 Deuring's theory of lifts

Let \(E/F\) be an elliptic curve over a number field with CM by \(K\text{,}\) (\(\End (E) = \ints_K\)). We'll begin by studying the reductions \(\overline E \pmod p\text{.}\)

Definition 6.6.5.

Let \(\overline E /\FF_q\) be an elliptic curve. We say \(\overline E\) is ordinary if \(\overline E\lb p\rb (\overline F_q) = \ZZ/p\text{,}\) supersingular if this group is 0.

Remark 6.6.7.

One criterion for \(\phi \colon C_1 \to C_2\) to be separable is that

\begin{equation*} \phi ^* \colon \Omega _{C_2} \to \Omega_{C_1} \end{equation*}

is nonzero.

\(p\ints K = \ideal p \ideal p'\text{,}\) sat \(\wp / \ideal p\text{.}\) Let \(m\) be the order of \(\ideal p\) in \(\Cl_K\text{,}\) so that

\begin{equation*} \ideal p ^m = (\mu ) , (\ideal p')^m = (\mu ') \end{equation*}

change by units if necessary so that

\begin{equation*} \mu \mu ' = p^m \end{equation*}

then

\begin{equation*} [\mu '] \in \End(E) \end{equation*}

Let \(\omega \in \Omega _E\) and note

\begin{equation*} [\mu ']^* \omega = \mu '\omega \end{equation*}

since \(\mu ' \not \in \ideal p\text{,}\) \(\lb \mu '\rb ^*\omega \not \equiv 0 \pmod {\ideal p}\text{.}\) So \(\lb \mu '\rb \in \End(\overline E)\) is separable and of \(p\)-power degree. This implies

\begin{equation*} \lb p\rb \end{equation*}

is not purely inseparable so \(\overline E \) is ordinary.

Consider \(\overline E\) ordinary,

\begin{equation*} \ints_K \otimes \ZZ_p \simeq \End(\overline E) \otimes \ZZ_p \to \End_{\ZZ_p}(T_p(\overline E)) \simeq \ZZ_p \end{equation*}

tensoring with \(\QQ\) gives,

\begin{equation*} K\otimes \QQ_p \to \QQ_p \end{equation*}

the LHS is 2-dimensional over \(\QQ_p\text{,}\) so this map cannot be an injection. So \(K\otimes \QQ_p\) cannot be a field so \(p\) splits in \(K\text{.}\)

Definition 6.6.9.

Let \(E/K\) be an ordinary elliptic curve over a perfect field of characteristic \(p\text{.}\) A canonical lift is an elliptic curve

\begin{equation*} \mathscr E/W(K) \end{equation*}

s.t. the connected-etale sequence

\begin{equation*} 0\to\mathscr E [p^\infty ]^0 \to \mathscr E [p^\infty ] \to \mathscr E[p^\infty ]^{et} \to 0 \end{equation*}

splits.

First note that if we have a lift then we can trivially lift \(\alpha _0 = \lb n\rb \text{.}\) So we can reduce to the case

  1. \(\ker (\alpha _0)\) is cyclic.
  2. \(p\nmid \deg(\alpha _0)\text{.}\)

now let \(n = \deg(\alpha _0)\text{.}\) Let \(j\) be transcendental over \(\QQ\) and

\begin{equation*} E(j)/\QQ(j) \end{equation*}

and elliptic curve with that \(j\)-invariant. Let \(C_1, \ldots, C_{\psi (n)}\) be the cyclic order \(n\) subgroups of \(E(j)\) and consider the isogenies

\begin{equation*} E(j) \to E(j)/C_i = E(j_i) \end{equation*}

(\(\psi\) is the Dedekind \(\psi\) function \(\psi(n) = n\prod_{p|n}(1+1/p)\)). Fact: Each \(j_i\) is integral over \(\ZZ\lb j\rb \)/ Consider \(\ZZ\lb j,j_1,\ldots, j_n\rb \) and its integral closure \(R\) in \(\QQ(j,j_1, \ldots, j_n)\text{.}\) We have a map

\begin{equation*} \ZZ[j] \to \overline \FF_p \end{equation*}
\begin{equation*} j \mapsto j(E_0) \end{equation*}

which can be extended to

\begin{equation*} R \xrightarrow\phi \overline \FF_p \end{equation*}

and let

\begin{equation*} \ideal m = \ker (\phi ) \end{equation*}

we have \(\overline {E(j)} \cong E_0 \pmod{\ideal m}\text{.}\) Consider the reductions

\begin{equation*} \overline C_i,\overline{E(j_i)}\text{.} \end{equation*}

Since \(p\nmid n\) the reduction is injective on \(n\)-torsion. So \(\overline C_i\) cover all the cyclic order \(n\) subgroups of \(E_0\text{.}\) This for some \(i\) we have \(\ker(\alpha _0) = \overline C_i\text{,}\) so

\begin{equation*} E(j) \to E(j_i) \end{equation*}

reduces to \(\alpha _0\text{.}\) Note:

\begin{equation*} \overline{E(j)} \cong \overline{E(j_i)} \implies (p, j- j_i) \subseteq \ideal m\text{.} \end{equation*}

Pick a minimal prime over \((j - j_i)\) in \(R\) and let \(\ideal q\) be an extension to \(\overline R\) (the integral closure of \(R\) in \(\overline{\QQ(j)}\text{.}\)) Note \(\ideal q \cap \ZZ = 0\) else \(\ideal q | q\) and thus be height \(\ge 2\text{.}\) So \(\overline E/ \ideal q\) is an integral extension of \(\ZZ\) and

\begin{equation*} E(j)_{\ideal q} \simeq E(j_i)_{\ideal q} \end{equation*}

Let \(F = \Frac (\overline R/\ideal q)\text{,}\) \(E = E(j)_{\ideal q}\text{,}\) \(\ideal p = \ideal m /\ideal q\text{,}\) let \(\alpha \) be the composition

\begin{equation*} \alpha \colon E(j)_{\ideal q} \to E(j_i)_{\ideal q} \xrightarrow\sim E(j)_{\ideal q}\text{.} \end{equation*}

So \(\alpha \equiv \alpha _0 \circ \sigma\) for \(\sigma \in \Aut(E_0)\text{.}\) We can lift automorphisms because \(\pm1 \) lift trivially and the only other possibilities are \(j(E_0) = 0,1728\) and these lift as \(E\colon y^2 = x^3 - 1,E\colon y^2 = x^3-x\) respectively.

If \(E_0\) is ordinary \(\End(E_0) = \ints_K = \ZZ+ \tau _0\ZZ\) then applying Deuring lifting to \((E_0, \tau _0)\) gives \((E, \tau )\) i.e.

\begin{equation*} \End(E) = \ZZ+ \tau \ZZ \simeq \ints_K\text{.} \end{equation*}

Subsection 6.6.3 Beginning of the proof of the Prop

Deuring lifting implies that \(\End_W(\underline x) \simeq \End_{W/\pi _v}(\underline x)\text{.}\) Serre-Tate gives that Deuring lifting implies that \(\Hom_W(\underline x^\sigma , \underline x) \simeq \Hom_{W/\pi _v}(\underline x^\sigma , \underline x)\text{.}\) The LHS is then zero via computing the intersection.