Hurwitz spaces and Inverse Galois Theory
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917399472635904 |
|---|---|
| author | Dèbes, Pierre |
| author_facet | Dèbes, Pierre |
| contents | Hurwitz spaces which parametrize branched covers of the line play a prominent role in inverse Galois theory. This paper surveys fifty years of works in this direction with emphasis on recent advances. Based on the Riemann-Hurwitz theory of covers, the geometric and arithmetic setup is first reviewed, followed by the semi-modern developments of the 1990--2010 period: large fields, compactification, descent theory, modular towers. The second half of the paper highlights more recent achievements that have reshaped the arithmetic of Hurwitz spaces, notably via the systematic study of the ring of components. These include the construction of components defined over ${\mathbb Q}$, and the Ellenberg-Venkatesh-Westerland approach to rational points over finite fields, applied to the Cohen-Lenstra heuristics and the Malle conjecture over function fields ${\mathbb F}_q(T)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06532 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hurwitz spaces and Inverse Galois Theory Dèbes, Pierre Number Theory Algebraic Geometry Primary 12F12, 14H10, Secondary 14H30, 12E30, 11Gxx Hurwitz spaces which parametrize branched covers of the line play a prominent role in inverse Galois theory. This paper surveys fifty years of works in this direction with emphasis on recent advances. Based on the Riemann-Hurwitz theory of covers, the geometric and arithmetic setup is first reviewed, followed by the semi-modern developments of the 1990--2010 period: large fields, compactification, descent theory, modular towers. The second half of the paper highlights more recent achievements that have reshaped the arithmetic of Hurwitz spaces, notably via the systematic study of the ring of components. These include the construction of components defined over ${\mathbb Q}$, and the Ellenberg-Venkatesh-Westerland approach to rational points over finite fields, applied to the Cohen-Lenstra heuristics and the Malle conjecture over function fields ${\mathbb F}_q(T)$. |
| title | Hurwitz spaces and Inverse Galois Theory |
| topic | Number Theory Algebraic Geometry Primary 12F12, 14H10, Secondary 14H30, 12E30, 11Gxx |
| url | https://arxiv.org/abs/2601.06532 |