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Let be a complete discrete valuation field of equal characteristic . Using the tools of -adic differential modules, we define refined Artin and Swan conductors for a representation of the absolute Galois group with finite local monodromy; this leads to a description of the subquotients of the ramification filtration on . We prove that our definition of the refined Swan conductors coincides with that given by Saito, which uses étale cohomology. We also study its relation with the toroidal variation of Swan conductors.
For each positive-integer valued arithmetic function , let denote the image of , and put and . Recently Ford, Luca, and Pomerance showed that is infinite, where denotes Euler’s totient function and is the usual sum-of-divisors function. Work of Ford shows that as . Here we prove a result complementary to that of Ford et al. by showing that most -values are not -values, and vice versa. More precisely, we prove that, as ,
We show that a cuspidal automorphic representation of a unitary similitude group with archimedean component in a regular discrete series has an associated -dimensional -adic Galois representation with Frobenius eigenvalues given by the local base change parameters for all primes such that and are unramified.
The main goal of this article is to construct and study a family of Weil representations over an arbitrary locally noetherian scheme without restriction on characteristic. The key point is to recast the classical theory in the scheme-theoretic setting. As in work of Mumford, Moret-Bailly and others, a Heisenberg group (scheme) and its representation can be naturally constructed from a pair of an abelian scheme and a nondegenerate line bundle, replacing the role of a symplectic vector space. Once enough is understood about the Heisenberg group and its representations (e.g., the analogue of the Stone–von Neumann theorem), it is not difficult to produce the Weil representation of a metaplectic group (functor) from them. As an interesting consequence (when the base scheme is ), we obtain the new notion of mod Weil representations of -adic metaplectic groups on -vector spaces. The mod Weil representations admit an alternative construction starting from a -divisible group with a symplectic pairing.
We have been motivated by a few possible applications, including a conjectural mod theta correspondence for -adic reductive groups and a geometric approach to the (classical) theta correspondence.
We classify the irreducible projective representations of symmetric and alternating groups of minimal possible and second minimal possible dimensions, and get a lower bound for the third minimal dimension. On the way we obtain some new results on branching which might be of independent interest.
We prove that the ideal of the variety of secant lines to a Segre–Veronese variety is generated in degree three by minors of flattenings. In the special case of a Segre variety this was conjectured by Garcia, Stillman and Sturmfels, inspired by work on algebraic statistics, as well as by Pachter and Sturmfels, inspired by work on phylogenetic inference. In addition, we describe the decomposition of the coordinate ring of the secant line variety of a Segre–Veronese variety into a sum of irreducible representations under the natural action of a product of general linear groups.