Ober­sem­in­ar "Num­ber The­ory and Arith­met­ic­al Stat­ist­ics": Dr. Be­ranger Seguin (Pader­born), Cov­ers and ri­gid­ity in in­verse Galois the­ory

Titel: Covers and rigidity in inverse Galois theory

Abstract: Celebrated bridges between analytic geometry and algebraic geometry lead to an equivalence of categories between finite
extensions of ℂ(T) and finite ramified covers of the Riemann sphere (i.e., the complex projective line). These covers are
well-understood, and this correspondence directly implies a positive answer to the inverse Galois problem over ℂ(T), as
well as a classification of the corresponding extensions.

The regular inverse Galois problem is the question of whether every finite group can be realized as the Galois group of a
finite extension of ℚ(T) which is regular (i.e., with no non-rational elements algebraic over ℚ). This question can be
reframed in terms of covers of the line: can one find geometrically connected Galois generically étale covers of the projective
line over ℚ which have a given automorphism group? Equivalently: among all connected Galois ramified covers of the complex
projective line with a specific Galois group, are there ones whose field of definition is ℚ?

Few methods exist to study this descent question. However, if G is a group with trivial center, one such method is the
so-called "rigidity criterion": one can deduce that a Galois connected cover of the projective line with Galois group G is
defined over ℚ from the fact that it is uniquely determined by a few elementary geometric invariants. This criterion can,
in turn, be checked group-theoretically using character theory. This method, and variants thereof, are used in the proof that
25 of the 26 sporadic finite simple groups are Galois groups over ℚ(T) (and, incidentally, over ℚ).

In this talk, we will explain the geometric ideas leading to the rigidity criterion and give applications.