# Elliptic curve model of the finite field extension

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).