HARBATER–MUMFORD COMPONENTS AND TOWERS OF MODULI SPACES

Abstract

A method of choice for realizing finite groups as regular Galois groups over $\mathbb{Q}(T)$ is to find $\mathbb{Q}$-rational points on Hurwitz moduli spaces of covers. In another direction, the use of the so-called patching techniques has led to the realization of all finite groups over $\mathbb{Q}_p(T)$. Our main result shows that, under some conditions, these $p$-adic realizations lie on some special irreducible components of Hurwitz spaces (the so-called Harbater–Mumford components), thus connecting the two main branches of the area. As an application, we construct, for every projective system $(G_n)_{n\geq0}$ of finite groups, a tower of corresponding Hurwitz spaces $(\mathcal{H}_{G_n})_{n\geq0}$, geometrically irreducible and defined over some cyclotomic extension of $\mathbb{Q}$, which admits projective systems of $\mathbb{Q}_p^{\mathrm{ur}}$-rational points for all primes $p$ not dividing the orders $|G_n|$ ($n\geq0$).