BUNTES Weil pairings (Maria)

Section 1.10 Weil pairings (Maria)

Subsection 1.10.1 Weil pairings on elliptic curves

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*}

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*}

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*}

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*}

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*}

Subsection 1.10.2 Weil pairings on abelian varieties

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

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*}

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*}

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*}

Proposition 1.10.1.

The Weil \(e_m\)-pairing has the following properties

\(e_m\) is bilinear
\begin{equation*} e_m(a_1+a_2,a') = e_m(a_1,a')e_m(a_2, a') \end{equation*}

\begin{equation*} e_m(a,a'_1+a'_2) = e_m(a,a'_1)e_m(a,a'_2) \end{equation*}
\(e_m\) is non-degenerate: if \(e_m(a,a') = 1 \forall a\in A\lb m\rb\) then \(a' = 0\) (and likewise for the reverse).
\(e_m\) is Galois-invariant... but we assume \(\overline k =k\) so we ignore this.
\(e_m\) is compatible
\begin{equation*} e_{mm'} (a,a')^{m'} = e_m(m'a, m'a') \ \forall a \in A[mm'], a'\in A^\vee [mm'] \end{equation*}
\((mm',\characteristic k) = 1\)

Corollary 1.10.2.

There exists a bilinear non-degenerate (Galois invariant) pairing

\begin{equation*} e_l = e \colon T_lA \times T_lA^\vee \to T_l\mu \end{equation*}

\begin{equation*} ((a_n), (a'_n)) \mapsto (e_{l^n}(a_,a'_n)) \end{equation*}

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*}

Notation.

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

Proposition 1.10.3.

For a homomorphism \(\alpha \colon A \to B\)

\begin{equation*} e(a,\alpha^\vee(b)) = e(\alpha (a), b) \forall a \in T_lA,\,b\in T_l B \end{equation*}
\begin{equation*} e^{\alpha^\vee \lambda \alpha}(a,a') = e ^\lambda (\alpha(a), \alpha(a')) \end{equation*}
for \(a,a' \in T_l(A)\text{,}\) \(\lambda \in \Hom(B,B^\vee)\text{.}\)
\begin{equation*} e^{\alpha^* \sheaf L} (a,a') = e^{\sheaf L}(\alpha (a),\alpha(a')) \end{equation*}
\(a,a'\in T_lA\) \(\sheaf L\in \Pic(B)\text{.}\)
\begin{equation*} \Pic A \to \Hom( \bigwedge^2 T_lA , T_l\mu) \end{equation*}

\begin{equation*} \sheaf L \mapsto e^{\sheaf L} \end{equation*}
is a homomorphism (in particular \(e^{\sheaf L}\) is skew-symmetric).

Proof.

\(a = (a_n) \in T_lA\) \(b\in (b_n) \in T_l B^\vee\) 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
\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*}
\begin{equation*} \lambda_{\alpha^* \sheaf L} = \alpha^\vee \lambda_{\sheaf L} \alpha \end{equation*}
Follows from \(\lambda_{\sheaf L\otimes \sheaf L'} = \lambda_{\sheaf L} + \lambda_{\sheaf L'}\text{.}\)

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 \(e^\lambda (a,a') = \zeta^{-E(a,a')}\)

Subsection 1.10.3 Results about polarizations

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

Theorem 1.10.5. 13.4.

Let \(\alpha\colon A\to B\) be an isogeny of degree prime to \(\characteristic k\) and \(\lambda \in \NS(A)\) then \(\lambda = \alpha^* \lambda '\) for \(\lambda ' \in \NS(B) \iff \forall l |\deg(\alpha)\) \(l\) prime there exists a skew-symmetric form \(f\colon T_lB\times T_lB \to T_l\mu\) s.t. \(e^\lambda(a,a') = f(\alpha(a), \alpha(a'))\) for all \(a,a' \in T_l(A)\text{.}\)

Proof.

Milne 1986 16.4

Corollary 1.10.6. 13.5.

\(l \ne \characteristic (k)\) \(\lambda \in \NS(A) \) is divisible by \(l^n \iff e^\lambda\) is divisible by \(l^n\) in \(\Hom (\bigwedge^2 T_lA, T_l \mu)\text{.}\)

Proof.

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

Lemma 1.10.7. 13.7.

Let \(\sheaf P\) be the Poincaré sheaf on \(A\times A^\vee\) then

\begin{equation*} e^{\sheaf P} ((a,b),(a',b')) = \frac{e(a,b')}{e(a',b)} \end{equation*}

for all \(a,a' \in T_l A\text{,}\) \(b,b' \in T_l A^\vee\text{.}\)

Proof.

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*}

Proposition 1.10.8. 13.6.

Assume \(\characteristic k \ne l,2\) then a homomorphism \(\lambda\colon A\to A^\vee\) is \(\lambda = \lambda_{\sheaf L}\) for some \(\sheaf L \in \Pic A\) iff \(e^\lambda\) is skew-symmetric.

Proof.

Case.

Clear.

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 \(\lambda = \lambda_{\sheaf L'}\text{.}\)

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.\(mb = a'\text{.}\)

Check: this is well defined.

Note 1.10.10.

\(e^\lambda\) is skew-symmetric.

Proposition 1.10.11. 13.8.

\(\alpha\colon A \to B\) is an isogeny of degree prime to \(p\text{,}\) \(\lambda\colon A\to A^\vee\) polarization then \(\lambda = \alpha^* \lambda', \,\lambda' \colon B\to B^\vee\) polarization iff

\begin{equation*} \ker(\alpha) \subset \ker \lambda \end{equation*}

\begin{equation*} e^{\lambda} \text{ is trivial on } \ker (\alpha)\times \ker(\alpha) \end{equation*}

Note 1.10.12.

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

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

Corollary 1.10.13. 13.10.

\(A\) an abelian variety, \(\lambda \colon A \to A^\vee\) is a polarization with \((\deg (\lambda), p) = 1\) then \(A\) is isogenous to a principally polarized abelian variety.

Proof.

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.

Corollary 1.10.14. 13.11.

Let \(\lambda\) be a polarization of \(A\) s.t. \(\ker (\lambda) \subseteq A\lb m \rb\) for some \((m,p)=1\text{.}\) If \(\exists \alpha \colon A \to A\) s.t. \(\alpha(\ker (\lambda)) \subseteq \ker(\lambda)\) and \(\alpha^\vee \lambda \alpha = - \lambda\) on \(A\lb m^2\rb\) then \(A\times A^\vee\) is principally polarized.

Theorem 1.10.15. 13.12 (Zarhin's trick).

For any abelian variety \(A\) \((A\times A^\vee)^4\) is principally polarized.

Proof.

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.

Corollary 1.10.16. 13.13.

Let \(k\) be a finite field, then for each \(g \in \ZZ\) there exist only finitely many isomorphism classes of abelian varieties of dimension \(g\) over \(k\text{.}\)

Proof.

\(A/k\) 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).