Lecture 1: Modular Class Groups

Lecture 2: Number Rings

A famous result of Leonhard Euler says that his so-called “convenient numbers” $N$ have the property that a positive integer $n$ has a unique representation of the form $n=x^2+Ny^2$ with $\gcd(x^2,Ny^2)=1$ if and only if $n$ is a prime, a prime power, twice one of these, or a power of 2. The set of known 65 convenient numbers is ${ 1,2,3,4,5,6,7,8,9,10,12,13,15,\dots,1848 }$, and it is conjectured that these are all of them. So, when we look at this set, we see that 11 is the first “inconvenient” number, and therefore we consider the natural question which positive integers have a representation of the form $n=x^2+11 y^2$ with $\gcd(x,11y)=1$.

Our approach is split into two parts. First we introduce the modular class group $G_{11}$ of level 11 and give a detailed description of its structure. We show that there are four conjugacy classes of elliptic elements of order 2, we provide concrete matrices representing these elliptic elements, and we give an explicit representation of $G_{11}$ using them. Then we conjugate the first of these matrices, namely $t_1=\binom{0, 1}{-1, 0}$, by the elements of $G_{11}$ and get matrices whose top right entry is of the form $x^2+11 y^2$. Conversely, we construct elliptic elements $A_n(\ell)$ of order 2 in $G_{11}$ which are conjugate to one of the generators. Then the matrices conjugate to $t_1$ are the ones we are interested in, and we find a set of candidate numbers $C$ such that $C=S_1 \cup S_2$, where $S_1$ is the set we want to characterise. Thus the task is reduced to distinguishing between $S_1$ and $S_2$.

This problem is addressed in the second part of the talk using number rings in $K=\mathbb{Q}(\sqrt{-11})$. The ring of integers of this number field is $\mathcal{O}_K=\mathbb{Z}[(-11+\sqrt{-11})/2]$, and the more natural ring $\mathbb{Z}[\sqrt{-11}]$ is its order of conductor 2. By realizing the elements of $S_1$ and $S_2$ as norms of elements in $\mathcal{O}_K$, we get some of their basic properties. The main theorem provides a precise description of the primitive representations $n=x^2+11 y^2$ into four classes, where cubic numbers and prime numbers are two classes which admit separate, detailed descriptions. For the prime numbers in $S_1$, we need to use some consequences of ring class field theory for $\mathbb{Z}[\sqrt{-11}]$, but all other results are largely self-contained.