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Section: Research Program

Arithmetic Geometry: Curves and their Jacobians

Theme: 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 $𝒳$ over a field $𝐊$ is defined by an equation

$𝒳:{F}_{𝒳}\left(x,y\right)=0\phantom{\rule{1.em}{0ex}}\text{where}\phantom{\rule{4.pt}{0ex}}{F}_{𝒳}\in 𝐊\left[x,y\right].$

(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 ${g}_{𝒳}$ of $𝒳$ is a non-negative integer classifying the essential geometric complexity of $𝒳$; it depends on the degree of ${F}_{𝒳}$ and on the number of singularities of $𝒳$. The curve $𝒳$ is associated in a functorial way with an algebraic group ${J}_{𝒳}$, called the Jacobian of $𝒳$. The group ${J}_{𝒳}$ has a geometric structure: its elements correspond to points on a ${g}_{𝒳}$-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 $𝒳$.

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.