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Let be a complete discretely valued field of equal characteristic with possibly imperfect residue field, and let be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on defined by Abbes and Saito (Amer. J. Math 124:5, 879–920) coincide with the differential Artin conductors and Swan conductors of Galois representations of defined by Kedlaya (Algebra Number Theory 1:3, 269–300). As a consequence, we obtain a Hasse–Arf theorem for arithmetic ramification filtrations in this case. As applications, we obtain a Hasse–Arf theorem for finite flat group schemes; we also give a comparison theorem between the differential Artin conductors and Borger’s conductors (Math. Ann. 329:1, 1–30).
Given a fixed integer , we consider closed subgroups of , where is sufficiently large in terms of . Assuming that the identity component of the Zariski closure of in does not admit any nontrivial torus as quotient group, we give a condition on the () reduction of which guarantees that is of bounded index in .
It is well-known that for a large class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we exploit this property to study the structure of such rings. One of our results states that the Picard group of the punctured spectrum of such a ring cannot have -torsion. When is a local complete intersection, this recovers (with a purely local algebra proof) an analogous statement for complete intersections in projective spaces first given by Deligne in SGA and also a special case of a conjecture by Gabber. Our method also leads to many simply constructed examples where rigidity for the Frobenius endomorphism does not hold, even when the rings are Gorenstein with isolated singularity. This is in stark contrast to the situation for complete intersection rings. A related length criterion for modules of finite length and finite projective dimension is discussed towards the end.
Let be a prime integer and a field of characteristic different from . We prove that the essential -dimension ed of the class of central simple algebras of degree is at least . The integer measures complexity of the class of central simple algebras of degree over field extensions of .
It is now known that for any prime and any finite semiabelian -group , there exists a (tame) realization of as a Galois group over the rationals with exactly ramified primes, where is the minimal number of generators of , which solves the minimal ramification problem for finite semiabelian -groups. We generalize this result to obtain a theorem on finite semiabelian groups and derive the solution to the minimal ramification problem for a certain family of semiabelian groups that includes all finite nilpotent semiabelian groups . Finally, we give some indication of the depth of the minimal ramification problem for semiabelian groups not covered by our theorem.
In this paper I introduce modular symbols for Maass wave cusp forms. They appear in the guise of finitely additive functions on the boolean algebra generated by intervals with nonpositive rational ends, with values in analytic functions (pseudomeasures in the sense of Manin and Marcolli). We explain the basic issues and draw an analogy with the -adic case. We then construct the new modular symbols, followed by the related Lévy–Mellin transforms. This work builds on the fundamental study of Lewis and Zagier (2001).