## Section: Research Program

### Arithmetic Geometry: Curves and their Jacobians

*Arithmetic Geometry* is the meeting point of algebraic geometry and
number theory: that is, the study of geometric objects defined over
arithmetic number systems (such as the integers and finite fields).
The fundamental objects for our applications
in both coding theory and cryptology
are curves and their Jacobians over finite fields.

An algebraic *plane curve* $\mathcal{X}$ over a field
$\mathbf{K}$ is defined by an equation

(Not every curve is planar—we may have more variables, and more
defining equations—but from an algorithmic point of view,
we can always reduce to the plane setting.)
The *genus* ${g}_{\mathcal{X}}$ of $\mathcal{X}$
is a non-negative integer classifying the essential geometric complexity
of $\mathcal{X}$;
it depends on the degree of ${F}_{\mathcal{X}}$
and on the number of singularities of $\mathcal{X}$.
The simplest curves with nontrivial Jacobians are
curves of genus 1,
known as *elliptic curves*;
they are typically defined by equations of the form
${y}^{2}={x}^{3}+Ax+B$.
Elliptic curves are particularly important given their central
role in public-key cryptography over the past two decades.
Curves of higher genus are important in both cryptography and coding theory.

The curve $\mathcal{X}$ is associated in a functorial way
with an algebraic group ${J}_{\mathcal{X}}$,
called the *Jacobian* of $\mathcal{X}$.
The group ${J}_{\mathcal{X}}$ has a geometric structure:
its elements correspond to points on a ${g}_{\mathcal{X}}$-dimensional
projective algebraic group variety. Typically,
we do not compute with the equations defining this projective variety:
there are too many of them, in too many variables, for this to be
convenient. Instead, we use fast algorithms based on the
representation in terms of classes of formal sums of points on
$\mathcal{X}$.