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Section 6.4 Archimedean Local Heights I (Aash)

Breuil-Conrad-Diamond-Taylor proved modularity of elliptic curves over \(\QQ\text{.}\) Gross-Zagier assume this so we can now state results unconditionally.

\begin{equation*} Cl_K \leftrightarrow \Gal HK \end{equation*}
\begin{equation*} A \leftrightarrow \sigma \end{equation*}

the artin map.

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

\(x\) a Heegner point and \(\pair \cdot\cdot\) is the global height pairing on \(J(H) \otimes \CC\text{.}\)

\(J\) is the Jacobian of \(X_0(N)\text{,}\) \(K = \QQ(\sqrt{D})\text{,}\) class number \(h\text{.}\)

\(H/K\) is the hilbert class field and \(2u\) is the number of roots of unity in \(K\text{.}\)

Where \(L_A\) is a twisted \(L\)-function related to a component theta function, i.e.

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

Also

Subsection 6.4.1 Height Pairings

\begin{equation*} | \cdot |_v \colon H_v^\times \to \RR_+^\times \end{equation*}
\begin{equation*} |\alpha |_v = \alpha \bar \alpha \end{equation*}

if \(H_v \cong \CC\) or \(q_v^{-v(\alpha )}\) if \(v\) is non-archimidean.

Neron's theory gives us a unique symbol on relatively primes divisors (divisors whose supports are disjoint). This pairing when defined splits up as

\begin{equation*} \pair ab = \sum_v \pair ab_v \end{equation*}
\begin{equation*} g_A(z) = \sum_{m\ge1} \pair c{ T_m c^\sigma } e^{2\pi i m z} \end{equation*}
\begin{equation*} c=(x) - (\infty ) \end{equation*}
\begin{equation*} d = (x) - (0) \end{equation*}

\((0)-(\infty )\) is of finite order in \(J(\QQ)\text{.}\)

\begin{equation*} \pair c{T_m c^\sigma } = \pair c {T_m d^\sigma } \end{equation*}
Remark 6.4.3.
\begin{equation*} r_A(m) = 0,\, N \gt1 \end{equation*}

implies \(c,T_m d^\sigma\) are relatively prime.

If \(S\) is a compact Riemann surface then there exists a partially defined

\begin{equation*} \pair \cdot \cdot \colon \Div^0(S) \times \Div^0(S) \to \RR \end{equation*}

which satisfies

  1. \(\pair ab\) is defined when \(a,b\) have disjoint support.
  2. \(\pair \cdot\cdot\) is bi-additive and symmetric whenever it is defined.
  3. If \(f\) is meromorphic on \(S\) and
    \begin{equation*} a= \sum_i n_i x_i \end{equation*}
    \begin{equation*} \pair{\divisor (f)} a = \sum n_i \log|f(x_i)|^2 \end{equation*}
  4. \begin{equation*} \pair a {\sum_j m_j (y_j)} \end{equation*}
    is continuous on \(S\smallsetminus |a|\) w.r.t each \(y_j\text{.}\) Where
    \begin{equation*} |a| \end{equation*}
    is the support of \(a\text{.}\)
Uniqueness.

Considering the difference of two symbols satisfying this then then it descends to the Jacobian as the values on \(\divisor(f)\) cancel.

Therefore

\begin{equation*} J\to \RR \end{equation*}
\begin{equation*} b\mapsto \pair b a \end{equation*}

is a continuous homorphism. Therefore the image is 0 (as 0 is the only compact function).

Existence.

Fix \(x_0, y_0 \in S\)

\begin{equation*} G(x,y) = \pair{(x) - (x_0)}{ (y) - (y_0)} \end{equation*}

where \(x\ne y,y\ne x_0,x\ne y_0\text{,}\) \(G\) is a Green's function

Biadditivity

\begin{equation*} \implies \pair ab = \sum_{i,j} n_i m_j G(x_i, y_j) \end{equation*}

\(a = \sum n_i(x_i)\text{,}\) \(b = \sum m_j(y_j)\text{,}\) \(y_0 \not \in |a|,x_0 \not \in |b|\text{.}\)

Conversely given \(G(x,y)\) we can define a symbol \(\pair \cdot \cdot\) if for fixed \(x \ne y_0\) the function

\begin{equation*} y \mapsto G(x,y) \end{equation*}

on \(S\smallsetminus\{x,x_0\}\) is:

  1. continuous
  2. harmonic, i.e.
    \begin{equation*} \Delta _y^2 G(x,y) = 0\text{.} \end{equation*}
  3. has logarithmic singularities of residue \(+1,-1\) at \(y=x,y=x_0, x=y_0\text{.}\)
Remark 6.4.4.

\(f\) has logarithmic singularities at \(z_0\) if

\begin{equation*} f(z) - \alpha \log|\rho (z)|^2 \end{equation*}

is continuous near \(z_0\text{,}\) \(\rho \) is holomorphic near \(z_0\) and vanishing to order \(1\) at \(z_0\text{.}\)

\(\alpha \) is called the residue of this singularity. \(\rho \) is the uniformizing parameter near \(z_0\text{.}\) Same symmetric condition on \(x\text{.}\)

So this is well defined, continuous and bi-additive if

\begin{equation*} (|a| \cup\{x_0\}) \cap (|b| \cup \{y+0\} ) = \emptyset \end{equation*}

we want to extend to \(|a|\cap |b| = \emptyset\text{.}\)

Sufficient to show

\begin{equation*} G(x_1,y) -G(x_2,y) \end{equation*}

makes sense as \(y\to y_0\text{,}\) \(x_1, x_2 \not \subset |b| \cup\{y_0\}\text{.}\)

\begin{equation*} G(x_i, y) = - \log|\rho |^2 + c_i + O(\rho (y)) \end{equation*}

therefore

\begin{equation*} G(x_1,y) - G(x_2,y) \to c_1 - c_2 \end{equation*}

as \(y \to x_0\text{.}\) Therefore this is well defined and continuous by hypothesis 3. on \(G(x,y)\text{.}\)

\(\pair\cdot\cdot\) is defined and continuous and bi-additive now, consider

\begin{equation*} (f) = \sum_{j=1}^ k m_j (y_j) \end{equation*}

a principal divisor, \(x_0 \not \in |(f)|\)

\begin{equation*} \delta \colon x\mapsto \pair {(x) - (x_0) }{f} - \left(\log|f(x)|^2 - \log|f(x_0)|^2\right) \end{equation*}
\begin{equation*} = \sum m_j G(x,y_j) - \left( \log|f(x)| ^2 - \log|f(x_0)|^2\right) \end{equation*}

is harmonic for \(x \in S - \{u_0, y_k\}\) and continuous everywhere so the difference is constant.

\begin{equation*} \pair {\sum n_i(x_i)}{(f)} - \sum n_i \log|f(x_i)|^2 \end{equation*}
\begin{equation*} =\sum n_i \delta (x_i) = \sum n_i C = 0\text{.} \end{equation*}

If we take \(G\) with the given hypothesis as \(\pair\cdot\cdot\text{.}\) \(S = X_0(N)(\CC)\text{,}\) \(x_0 = \infty \text{,}\) \(y_0 = 0\text{.}\) Conditions on \(G\) needed:

  • G1 , \(G\) is a real valued continuous harmonic function on
    \begin{equation*} E = \{ (z,z') \in \HH^2 : z\not \in \Gamma _0(N) z'\} \end{equation*}
    such that \(G(\gamma z, \gamma 'z') = G(z,z')\) for all
    \begin{equation*} (z,z')\in E, \gamma ,\gamma '\in \Gamma _0(N)\text{.} \end{equation*}
  • G2 , Fix \(z\in \HH\)
    \begin{equation*} G(z,z') = e_z\log|z- z'|^2 + O(1) \end{equation*}
    as \(z'\to z\text{,}\) where \(e_z\) is the order of the stabilizer in \(\Gamma _0(N)\text{.}\)
  • G3 , For \(z\in \HH\) fixed
    \begin{equation*} G(z,z') = 4\pi y' + O(1) \end{equation*}
    as \(z' = x' + iy' \to \infty \) and \(G(z,z') = O(1)\) at other cusps.
  • G4 , For \(z'\in \HH\) fixed
    \begin{equation*} G(z,z') = 4\pi y/N|z|^2 + O(1) \end{equation*}
    as \(z = x + iy \to 0 \) and \(G(z,z') = O(1)\) at other cusps.

G2,G3,G4 come from uniformizing parameters , at \(\infty \) \(e^{2\pi i z} \leftrightarrow \rho \text{,}\) non-cusp : \(|z' -z|^{e_z} \leftrightarrow \rho \) , at 0 \(e^{-2\pi i z}/|z|^2\text{.}\) applies the logarithmic singularity hypothesis on \(G\text{.}\)

\begin{equation*} G(z,z') = \lim_{s\to 1} \left( G_{N,s} (z,z') + 4 \pi E_N(w_N z, s) + 4 \pi E_N(z', s) + \frac{K_N}{s-1}\right) + c \end{equation*}