BUNTES Wrap Up of Non-Archimedean Local Heights (Sachi)

Section 6.9 Wrap Up of Non-Archimedean Local Heights (Sachi)

This will be a reminder / recap / overview of where we are at.

Subsection 6.9.1 Recap of Initial Motivation

Big motivation, finding infinite order points on elliptic curves, leads us to Gross-Zagier.

If \(J\) is the Jacobian of \(X_0(N)\text{,}\) \(\Delta \lt 0\) a fundamental discriminant of an imaginary quadratic field \(K\text{.}\)

\begin{equation*} s \colon \Cl_K \xrightarrow\sim \Gal HK\text{.} \end{equation*}

For any \(\mathcal A \in \Cl_K\text{,}\) we define the partial theta series

\begin{equation*} \theta _{\mathcal A} (z) = \frac{1}{2u} + \sum _{a \subseteq \ints K,a\in \mathcal A} q^{\norm(a)} = \frac{1}{2u} + \sum_{n \ge 1} r_{\mathcal A}(n)q^n\text{.} \end{equation*}

\begin{equation*} r_{\mathcal A}(n) = \#\text{integral ideals in }\mathcal A\text{ of norm }n\text{.} \end{equation*}

This series defines a modular form of weight 1 and level \(\Gamma _1(\Delta )\) with character

\begin{equation*} \epsilon (n) = \legendre{\Delta }{n} \colon \ZZ \to \{\pm 1\}\text{.} \end{equation*}

For any \(f \in \sum a_n q^n \in S_2(\Gamma _0(N))^{\text{new}}\) we define

\begin{equation*} L_{\mathcal A} (f,s) = \sum_{n \ge 1, (n, \Delta N_f) = 1} \legendre\Delta n^{1-2s} \sum_{n\ge 1}a_n r_{\mathcal A}(n)n^{-s}\text{.} \end{equation*}

Theorem 6.9.1. Gross-Zagier.

The series

\begin{equation*} g_{\mathcal A} (z) = \sum_{m \ge 0} \pair c{T_m c^{s(A)}} q^m \end{equation*}

is a modular form of weight 2 and level \(\Gamma _0(N)\) and

\begin{equation*} (f,g_{\mathcal A }) = \frac{u^2 \sqrt\Delta }{8\pi ^2} L'_{\mathcal A}(f,1) \end{equation*}

where \(\pair \cdot \cdot \) is the Néron-Tate height pairing on

\begin{equation*} J(H)\times J(H) \to \RR\text{.} \end{equation*}

\begin{equation*} c = (x) - (\infty ) \end{equation*}

\(x\) a Heegner point over \(H\text{.}\)

Recall?: The Shimura correspondence

Theorem 6.9.2. Kohnen-Shimura.

Let \(\epsilon \in \{\pm 1\}\) then

\begin{equation*} \dim S_{k+\frac{1}{2}}^\epsilon (\Gamma_0 (4N)) = \dim S_{2k}^\epsilon (\Gamma _0(N)) \end{equation*}

and for each Hecke eigenform

\begin{equation*} f = \sum_{n\ge 1} a_n q^n\in S^\epsilon _{2k} (\Gamma _0(N)) \end{equation*}

there is a 1-dimensional space of forms \(g \in S_{k+1/2}^\epsilon (\Gamma _0(4N))\) whose fourier coefficients \(c_m\) are related by

\begin{equation*} a_n c_m= \sum _{d|n} \legendre{-m}d d^{k-1}c_{mn^2/d^2}\text{.} \end{equation*}

Remark 6.9.3.

If \(f \) is a modular form attached to \(E\) an elliptic curve then \(g\) is weight \(3/2\text{.}\)

Recall: To compute \(\pair ab\) compute as a sum of local height pairings. Néron-Tate local height for \(v\) a place of \(H\) has properties

bi-additive, symmetric, continuous
\begin{equation*} a=\sum_P m_P P,\,b = \divisor f \end{equation*}
with disjoint support then
\begin{equation*} \pair ab_v = \sum_P m_P |\log |f(P)||_v\text{.} \end{equation*}

Subsection 6.9.2 Heights

Let \(v\) be a non-archimidean place, assume \(m\) is prime to \(N\text{.}\) If \(v|p\) a place of \(H\) then \(H_v\) the completion \(\Lambda _v\) ring of integers and \(\pi \) uniformizer, \(\Lambda _v/\pi \) residue field of cardinality \(q\text{.}\) \(W\) the completion of the maximal unramified extension of \(\Lambda _v\text{.}\)

\begin{equation*} \pair ab_v = -(A.B) _v \log q \end{equation*}

where \(A,B\) are divisors on some regular model of \(X\) over a DVR (like \(\Lambda _v\)) and \(A\) is fibral.

Working with \(c = (x)-(\infty )\) \(d = (x)-(0)\text{.}\)

\begin{equation*} \pair c{T_m d^\sigma }_v = (\underline x. T_m \underline x^\sigma )\log q\text{.} \end{equation*}

So we need to compute a regular model for \(X_0(N)/\ZZ\text{.}\) We need to identify components of \(T_m \underline x^\sigma \text{.}\) Need to compute RHS explicitly to show

\begin{equation*} = \frac 12 \sum_{n\ge 1} \#\Hom _{W/\pi ^n} (\underline x ,\underline x^\sigma )_{\deg m}\text{.} \end{equation*}

Subsection 6.9.3 Brief sketch of regular model

Recall pts on \(X_0(N)\) correspond to cyclic isogenies

\begin{equation*} \psi \colon E \to E' \end{equation*}

of degree \(N\text{.}\) The Heegner points have \(\End(E) = \End(E') = \ints\) an order in \(K\text{.}\) Similarly consider generalized elliptic curves and cyclic isogenies of degree \(N\text{.}\)

These components are isomorphic to \(X_0(M) \otimes \ZZ/p\text{.}\) They intersect at supersingular points \(E\xrightarrow \phi E'\) where both are supersingular. We have a good understanding of where the cusps are.

Subsection 6.9.4 Homomorphsims

\(S\) complete local ring, \(k\) algebraically closed field

\begin{equation*} \underline x = (\phi \colon E \to E') \end{equation*}

\begin{equation*} \underline y = (\psi \colon F \to F') \end{equation*}

points on \(X_0(N)(S)\) then homomorphisms \(\underline x \to \underline y\) are \(f\colon E\to F, f'\colon E'\to F'\) such that \(f' \phi = f \psi\text{.}\) The set of such has a group structure inherited from \(F,F'\text{.}\) This is a right module under \(\End_S(\underline x)\) by composition.

\begin{equation*} \End_S(\underline x) = \ZZ, \text{ order in im quad }, \text{ order in indef. quat. alg.} \end{equation*}

\begin{equation*} \deg(f,f') = \deg f = \deg f'\text{.} \end{equation*}

To show above

\begin{equation*} (c.T_m d^\sigma ) = (\underline x. T_m \underline x^\sigma ) - (\underline x . T_m 0) - (\infty .T_m \underline x^\sigma ) + (\infty .T_m 0) \end{equation*}

3 terms on right are 0.

Main difficulty. \(m\) prime to \(N\) and \(r_{\mathcal A} (m) = 0\text{.}\)

\begin{equation*} (\underline x. T_m \underline x ^\sigma ) = \frac 12 \sum_{n\ge 1} \# \Hom _{W/\pi ^n}(\underline x ^\sigma , \underline x)_{\deg m} \end{equation*}

Remark 6.9.4.

This is a finite sum as \(x, T_mx^\sigma \) are relatively prime divisors there are no degree \(m\) isogenies from \(x^\sigma \) to \(x\text{.}\) For large \(n\) therefore \(\Hom_{W/\pi ^n,\deg m} = \emptyset\text{.}\)

we denote by \(h_n\) this RHS summand.

Proof.

When \(p\)-splits Deuring lifting gives

\begin{equation*} \Hom_W (\underline x^\sigma ,\underline x) = \Hom_{W/\pi ^n}(\underline x^\sigma , \underline x) \end{equation*}

for all \(n\text{.}\) As \(r_{\mathcal A}(m)= 0\) we have no elements of degree \(m\text{.}\)

If \(p\) is non-split

\begin{equation*} \End_W(\underline x) = \ints \end{equation*}

an order in a quaternion algebra.

\begin{equation*} h_n(\underline x^\sigma , \underline x) _{\deg m} = \sum_{\underline y\in T_m \underline x} h_n(\underline y, \underline x)_{\deg 1} \end{equation*}

Moral, can compute the fourier coefficients of \(g_{\mathcal A}\text{.}\)