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.


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).

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