From cc9bb99ac57aaa8d87e2a349d43e4b82f3762068 Mon Sep 17 00:00:00 2001 From: arnaucube Date: Wed, 21 Jun 2023 08:57:41 +0200 Subject: [PATCH] Add weil reciprocity example, weil pairing properties, small typo in hypernova notes --- notes_hypernova.pdf | Bin 359311 -> 359174 bytes notes_hypernova.tex | 10 ++++------ weil-pairing.pdf | Bin 222050 -> 225307 bytes weil-pairing.tex | 43 ++++++++++++++++++++++++++++++++++++++----- 4 files changed, 42 insertions(+), 11 deletions(-) diff --git a/notes_hypernova.pdf b/notes_hypernova.pdf index d52b0c31b826655ead5678fec59573e8c94c4a86..f24a0f639670d09faf2b2f8770470ed634612262 100644 GIT binary patch delta 5483 zcmV-x6_o0a_Z5cr6|h-Q0X381SSd+cS#KLR5PsLM*uKeV#FE21MvA6D8}tI2#3)+C zesFEsQS8`_?IKNoeTRE#xti5-t@V*Y2oT#6cgVRue#0H|>@?%qiwQ4(lgv;ae`hA8 zoSx1v=2wfO@Q0cn&-b}d7mJ(45+9!|OZ{IMac6 zuGZH)Khk-JUVjYy<7ay_poK~ZkbD?|Z` zEerBUjPM00>doVQOi)Z_3b4u)c;YT86L&X3I0_wZLx>La)JsCQV*BEEVE?*~N5e_n z5mcb8I9;{kA1(?>R>qnsU5mQDng4}ArQK8!vqCUtJ@yH)e{jRNs~OU-lpzuBR6XJg 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ord_P(r) \cdot P = 0$$ +$$\sum_{P \in E(\Bbbk)} ord_P(r) \cdot P = 0$$ -\begin{definition}{Support} - $$\sum_P n_P[P], ~\forall P \in E(\Bbbk) \mid n_P \neq 0$$ +\begin{definition}{Support of a divisor} + $$\sum_P n_P[P], ~\forall P \in E(\Bbbk) ~\text{s.t.}~ n_P \neq 0$$ \end{definition} \begin{definition}{Principal divisor} - iff $deg(D)=0$ and $sum(D)=0$ + iff + $$deg(D)=0$$ + $$sum(D)=0$$ \end{definition} $D \sim D'$ iff $D - D'$ is principal. @@ -169,11 +171,24 @@ $D \sim D'$ iff $D - D'$ is principal. \section{Weil reciprocity} \begin{theorem}{(Weil reciprocity)} - Let $E/ \Bbbk$ be an e.c. over an alg. closed field. If $r,~s \in \Bbbk\setminus \{0\}$ are rational functions whose divisors have disjoint support, then + Let $E/ \Bbbk$ be an e.c. over an algebraically closed field. If $r,~s \in \Bbbk\setminus \{0\}$ are rational functions whose divisors have disjoint support, then $$r(div(s)) = s(div(r))$$ \end{theorem} Proof. (todo) +\paragraph{Example} +\begin{align*} + p(x)=x^2 - 1,&~ q(x)=\frac{x}{x-2}\\ + div(p)&= 1 \cdot [1] + 1 \cdot [-1] - 2 \cdot [\infty]\\ + div(q)&= 1 \cdot [0] - 1 \cdot [2]\\ + &\text{(they have disjoint support)}\\ + p(div(q)) &= p(0)^1 \cdot p(2)^{-1}= (0^2 - 1)^1 \cdot (2^2 - 1)^{-1} = \frac{-1}{3}\\ + q(div(p)) &= q(1)^1 \cdot q(-1)^1 - q(\infty)^2\\ + &= (\frac{1}{1-2})^1 \cdot (\frac{-1}{-1-2})^1 \cdot (\frac{\infty}{\infty - 2})^2 = \frac{-1}{3} +\end{align*} + +so, $p(div(q))=q(div(p))$. + \section{Generic Weil Pairing} Let $E(\Bbbk)$, with $\Bbbk$ of char $p$, $n$ s.t. $p \nmid n$. @@ -236,6 +251,24 @@ with $S \neq \{O, P, -Q, P-Q \}$. \section{Properties} +\begin{enumerate}[i.] + \item $e_n(P, Q)^n = 1 ~\forall P,Q \in E[n]$\\ + ($\Rightarrow~ e_n(P,Q)$ is a $n^{th}$ root of unity) + \item Bilinearity + $$e_n(P_1+P_2, Q) = e_n(P_1, Q) \cdot e_n(P_2, Q)$$ + $$e_n(P, Q_1+Q_2) = e_n(P, Q_1) \cdot e_n(P, Q_2)$$ + \emph{proof:} + recall that $e_n(P,Q)=\frac{g(S+P)}{g(S)}$, then, + \begin{align*} + e_n(P_1, Q) &\cdot e_n(P_2, Q) = \frac{g(P_1 + S)}{g(S)} \cdot \frac{g(P_2 + P_1 + S)}{g(P_1 + S)}\\ + &\text{(replace $S$ by $S+P_1$)}\\ + &= \frac{g(P_2 + P_1 + S)}{g(S)} = e_n(P_1+P_2, Q) + \end{align*} + \item Alternating + $$e_n(P, P)=1 ~\forall P\in E[n]$$ + \item Nondegenerate + $$\text{if}~ e_n(P,Q)=1 ~\forall Q\in E[n],~ \text{then}~ P=0$$ +\end{enumerate} \section{Exercises}