Like families of stars in a constellation not visible to the naked eye, families of arithmetic objects parameterized by integral or rational points on varieties can be counted in special circumstances.
If $K$ is a field extension of finite degree over the rationals $Q$, its absolute discriminant is a measure of the size of the lattice of algebraic integers $O_K$ in $K$. For $n\in Z_{\geq 1}$ and $X>0$, let $N_n(X)$ be the family of such (number field) extensions of fixed degree $n$ and absolute discriminant bounded by $X$. $N_n(X)$ is finite and for any fixed value of $n$, the size of $N_n(X)$ is expected to grow linearly in $X$. This is easy to prove for $n=2$, and was shown by Davenport and Heilbronn for $n=3$ and by Bhargava for $n=4,5$. In this talk we take a close look at the method of Ellenberg and Venkatesh which yields an upper bound $\mid N(n,X)\mid << (C_n X)^{exp(c\sqrt{\log n})}$ for any value of $n$ (here, $C_n$ is a constant which depends on $n$ alone and $c$ is an absolute constant).

(clarification: The theme has no intended implications to astronomy or star-gazing.)