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Section 1.10 Weil pairings (Maria)

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Subsection 1.10.1 Weil pairings on elliptic curves

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Start with elliptic curves, later repeat for abelian varieties. E/k an elliptic curve, m\ge 2\text{,} if \characteristic(k) = p \gt 0 (m,p) = 1\text{.} The Weil e_m-pairing e_m \colon E\lb m\rb \times E\lb m \rb \to \mu_m is defined as follows: Fix T\in E\lb m \rb then f\in \overline k (E) s.t. \divisor(f) = m(T) - m(0)\text{.} Fix T' \in E with mT' = T and g\in \overline k(E) s.t. \divisor(g) = \lb m \rb^*(T) = \lb m \rb^*(0)= \sum_{R\in E\lb m \rb} (T+R) - (R)\text{.} Check \divisor (f\circ \lb m \rb) = \divisor(g^m)\text{,} hence

\begin{equation*} f\circ [m] = c g^m \end{equation*}

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so can assume f\circ \lb m \rb = g^m\text{.} For s \in E\lb m \rb\text{,} x\in E\text{:}

\begin{equation*} g(x + s) = f([m]x + [m]s) = f([m]x) = g(x)^m \end{equation*}
\begin{equation*} \frac{g(\cdot + s)^m}{g(\cdot)} \colon E \to \PP^1 \end{equation*}

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is then a constant function, since not surjective. So we define

\begin{align*} e_m\colon E[m]\times E[m] \amp\to \mu_m\\ (s,t)\amp \mapsto \frac{g_t(x+s)}{g_t(x)} \end{align*}

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will state many properties later, but for now. e_m is compatible:

\begin{equation*} e_{mm'} (a,a')^{m'} = e_m(m'a, m'a') \ \forall a,a' \in E[mm'] \end{equation*}

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so for any l\ne \characteristic(k) prime we can combine e_{l^n}-pairings into an l-adic Weil pairing on T_l E

\begin{equation*} e \colon T_l E\times T_lE \to T_l \mu = \ZZ_l(1) \end{equation*}
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Subsection 1.10.2 Weil pairings on abelian varieties

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Story will be broadly similar to before but we must use the dual, which doesn't appear in the presentation for elliptic curves.

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Let A/k be an abelian variety k = \overline k\text{.} We construct a Weil e_m-pairing

\begin{align*} e_m \colon A[m]\times A^\vee [m] \amp\to \mu_m\\ (a,a') \amp\mapsto \frac{g\circ t_a(x)}{g(x)} = \frac{g(x+a)}{g(x)} \end{align*}

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Fix a\in A\lb m\rb,\,a'\in A^\vee\lb m\rb say a' corresponds to \sheaf L and a divisor D then \sheaf L^m and m_A^* \sheaf L are trivial so \exists f,g \in k(A) s.t.

\begin{equation*} \divisor (f) = m D \end{equation*}
\begin{equation*} \divisor (g) = m_A^* D \end{equation*}

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again we have

\begin{equation*} \divisor( f\circ m_A) = \divisor (g^m) \end{equation*}
\begin{equation*} g(x+a)^m = g(x)^m \end{equation*}

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For a homomorphism \lambda \colon A \to A^\vee we define

\begin{equation*} e_m^\lambda \colon A[m]\times A [m] \to \mu_m \end{equation*}
\begin{equation*} (a,a') \mapsto e_m(a, \lambda(a')) \end{equation*}
\begin{equation*} e_m \colon T_lA\times T_l A \to T_l \mu \end{equation*}
\begin{equation*} (a,a') \mapsto e_m(a, \lambda(a'))\text{.} \end{equation*}
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Notation.

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If \lambda = \lambda_{\sheaf L} e^{\sheaf L} = e^{\lambda_{\sheaf L}}\text{.}

  1. \(a = (a_n) \in T_lA\) \(b\in (b_n) \in T_l B^\vee\) fix a divisor \(D \) on \(B\) representing \(b_n\) and \(g\in k(B)\) s.t. \(\divisor (h) = (l^n_B)^* D\text{.}\) Then \(\alpha^* D\) represents \(\alpha^\vee(b_n)\) so:
    \begin{equation*} \divisor(g\circ \alpha) = \alpha^*\divisor (g) = \alpha^*(l^n_B)^* D = (l_A^n)^* \alpha^* D\text{.} \end{equation*}
    So
  2. \begin{equation*} e^{\alpha^{\vee}\lambda \alpha} (a,a') = e(a, \alpha^\vee\lambda \alpha(a')) = e(\alpha(a),\lambda (\alpha(a'))) = e^\lambda (\alpha(a), \alpha(a'))\text{.} \end{equation*}
  3. \begin{equation*} \lambda_{\alpha^* \sheaf L} = \alpha^\vee \lambda_{\sheaf L} \alpha \end{equation*}
  4. Follows from \(\lambda_{\sheaf L\otimes \sheaf L'} = \lambda_{\sheaf L} + \lambda_{\sheaf L'}\text{.}\)
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Example 1.10.4. Computation over \CC.

A/\CC be an abelian variety

\begin{equation*} 0\to \ZZ \to \sheaf O_A \xrightarrow{e^{2\pi i (\cdot)}} \sheaf O^\times \to 0 \end{equation*}

induces

\begin{equation*} H^1(A(\CC), \ZZ) \to H^1(A(\CC), \sheaf O) \to H^1(A(\CC), \sheaf O^\times) \simeq \Pic A \to H^2(A(\CC), \ZZ) \end{equation*}

and

\begin{equation*} H^1(A(\CC), \sheaf O)/ H^1(A(\CC), \ZZ) \simeq A^\vee(\CC) = \Pic^0(A) \end{equation*}

so we get an exact sequence

\begin{equation*} 0 \to \NS(A) \to H^2 (A(\CC),\ZZ) \to H^2(A(\CC),\sheaf O_A) \end{equation*}
\begin{equation*} \lambda \mapsto E_\lambda \end{equation*}

then we can regard E_\lambda as a skew-symmetric 2-form on H_1(A(\CC), \ZZ)\text{.} Mumford pg. 237 proves

\begin{equation*} \xymatrix{ H_1(A(\CC), \ZZ) \times H_1(A(\CC), \ZZ) \ar[r] \ar[d] & \ZZ\ni m \ar[d] \\ T_l \times T_l \ar[r] & T_l \mu \ni \zeta^m } \end{equation*}

commutes with - sign so e^\lambda (a,a') = \zeta^{-E(a,a')}

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Subsection 1.10.3 Results about polarizations

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k = \overline k p = \characteristic (k) \ge 0\text{.}

Milne 1986 16.4

Apply theorem 13.4 with \(\alpha = l^n\text{.}\)

Milne 1986 16.7. Use:

\begin{equation*} (1+ \lambda_{\sheaf L})^* \sheaf P \cong m^* \sheaf L \otimes p^* \sheaf L^{-1} \otimes q^* \sheaf L^{-1} \end{equation*}
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Case.

\(e^\lambda\) is skew-symmetric, define \(\sheaf L = (1 \times \lambda)^* \sheaf P\) then \(\forall a,a' \in T_l A\)

\begin{equation*} e(a,\lambda_{\sheaf L} (a') ) = e^{\sheaf L}(a,a') = e^{(1\times \lambda)^* \sheaf P} (a,a') = e^{\sheaf P}((a,\lambda (a)), (a',\lambda(a'))) = \frac{e(a,\lambda (a'))}{ e(a',\lambda(a))} \end{equation*}
\begin{equation*} = \frac{e^\lambda(a,a')}{ e^\lambda(a',a)} = (e^\lambda(a,a'))^2 = e(a,2\lambda (a')) \end{equation*}

so \(2\lambda = \lambda_{\sheaf L}\text{.}\) So by corollary 13.5 \(\lambda_{\sheaf L} = 2\lambda_{\sheaf L'}\) for some \(\sheaf L' \in \Pic A\) so \(\lambda = \lambda_{\sheaf L'}\text{.}\)

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Definition 1.10.9.

For a polarization \lambda \colon A\to A^\vee define

\begin{equation*} e^{\lambda} \colon \ker(\lambda)\times \ker(\lambda) \to \mu_m \end{equation*}
\begin{equation*} (a,a')\mapsto e_m(a,\lambda(b)) \end{equation*}

where m kills \ker(\lambda) and b \in A s.t.mb = a'\text{.}

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Check: this is well defined.

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Note 1.10.10.

e^\lambda is skew-symmetric.

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Note 1.10.12.

If \lambda = \alpha^* \lambda' then

\begin{equation*} \deg(\lambda) = \deg (\lambda') \deg(\alpha)^2\text{.} \end{equation*}

Fix \(l |\deg(\lambda)\) prime. Choose a subgroup \(N\subseteq \ker \lambda\) of order \(l\) let \(\alpha \colon A\to A/N = B\) \(N\) is cyclic and \(e^\lambda\) is skew-symmetric so \(e^{\lambda}\) is trivial on \(N\times N\) so \(B\) has a polarization of degree \(\deg(\lambda) / l^2\) by 13.8.

Fix \(\lambda \colon A\to A^\vee\) polarization, assume \(\ker (\lambda) \subseteq A\lb m \rb\) \((m, p) = 1\) there exists \(a,b,c,d \in \ZZ\) s.t. \(a^2 + b^2 + c^2 + d^2 = m^2 - 1 = -1 \pmod {m^2}\) then

\begin{equation*} \begin{pmatrix} a\amp -b \amp-c\amp -d \\ b\amp a \amp d \amp -c \\ c\amp -d \amp a \amp b \\ d \amp c \amp -b \amp a\end{pmatrix} \end{equation*}

works.

\(A/k\) an abelian variety of dimension \(g\text{,}\) so \((A\times A^\vee)^4\) is an abelian variety of dimension \(8g\) with a principal polarization so using theorem 11.2 there are finitely many (up to \(\simeq\)) of those. Also \((A\times A^\vee)^4\) has finitely many direct factors (theorem 15.3).