Characteristic cycle and wild ramification for nearby cycles of étale sheaves

In this article, we give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes and Saito’s logarithmic ramification filtration. This provides a positive answer to the main conjecture in [24] for smooth morphisms in equal characteristic. We also study the ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait.

WILD RAMIFICATION IN TRINOMIAL EXTENSIONS AND GALOIS GROUPS

It is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.

Wild ramification and restrictions to curves

We prove that wild ramification of a constructible sheaf on a surface is determined by that of the restrictions to all curves. We deduce from this result that the Euler–Poincaré characteristic of a constructible sheaf on a variety of arbitrary dimension over an algebraically closed field is determined by wild ramification of the restrictions to all curves. We similarly deduce from it that so is the alternating sum of the Swan conductors of the cohomology groups, for a constructible sheaf on a variety over a local field.

Explicit Serre weights for two-dimensional Galois representations

We prove the explicit version of the Buzzard–Diamond–Jarvis conjecture formulated by Dembeleet al.(Serre weights and wild ramification in two-dimensional Galois representations, Preprint (2016),arXiv:1603.07708[math.NT]). More precisely, we prove that it is equivalent to the original Buzzard–Diamond–Jarvis conjecture, which was proved for odd primes (under a mild Taylor–Wiles hypothesis) in earlier work of the third author and coauthors.