4
$\begingroup$

Recently Thorsten Kleinjung and Benjamin Wesolowski published https://arxiv.org/abs/1906.10668 There is the definition of an elliptic curve model of a finite field extension:

Definition 2.1 (Elliptic curve model). Consider a prime power $q$ and an integer $n > 1$. Suppose there is an ordinary elliptic curve $E$ defined over $\mathbb F_q$, a rational point $Q\in E(\mathbb F_q)$ and an irreducible divisor $I$ of degree $n$ over $\mathbb F_q$ such that for any $f\in \mathbb F_q(E)$, one has $f\circ \phi_q\equiv f\circ \tau_Q \mod I$, where $\phi_q$ is the q-Frobenius and $\tau_Q$ is the translation by $Q$. Then, $\mathbb F_q[I]\cong \mathbb F_{q^n}$, and we call $(E,Q,I)$ a $(q,n)$-elliptic curve model of the field $\mathbb F_{q^n}$.

Tell me what is the meaning of maps equivalence modulo divisor, please. And in what book can I find the detailed presentation of the topic. Thanks.

$\endgroup$
1
$\begingroup$

It works essentially like arithmetic modulo a polynomial, except $D$ represents the roots of the polynomial instead of the polynomial itself. Let $P,f,g \in \mathbf F_q[x]$. Then $f \equiv g \mod P$ means that $P$ divides $f-g$, which can also be interpreted as the roots of $P$ are also roots of $f-g$, with multiplicities. So the congruence only depends on the roots of $P$ and their multiplicities, i.e., the divisor of $P$.

A general way to interpret congruence modulo a divisor is as follows: if $D$ is a positive divisor, and $f$ and $g$ are functions with no pole at $D$, then $f \equiv g \mod D$ means that $\mathrm{div}^+(f-g) \geq D$ (where $\mathrm{div}^+(h)$ is the positive part of $\mathrm{div}(h)$).

This can be seen as a ring quotient: let $\mathcal O_D \subset \mathbf F_q(E)$ be the subring of functions with no pole at $D$, and $\mathfrak m_D \subset \mathcal O_D$ the ideal of functions $f$ such that $\mathrm{div}^+(f) \geq D$ (i.e., $f$ is a zero at each point of $D$, with appropriate multiplicity). The quotient $\mathcal O_D / \mathfrak m_D$ is the ring of functions 'modulo $D$', and when $D$ is an irreducible divisor over $\mathbf F_q$, it is a field (the residue field, that is the $\mathbf F_q[I] \cong \mathbf F_{q^n}$ in the elliptic curve model).

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.