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We introduce and analyze a general class of not necessarily bounded multiplicative functions, examples of which include the function , where and where counts the number of distinct prime factors of , as well as the function , where denotes the Fourier coefficients of a primitive holomorphic cusp form.
For this class of functions we show that after applying a -trick, their elements become orthogonal to polynomial nilsequences. The resulting functions therefore have small uniformity norms of all orders by the Green–Tao–Ziegler inverse theorem, a consequence that will be used in a separate paper in order to asymptotically evaluate linear correlations of multiplicative functions from our class. Our result generalizes work of Green and Tao on the Möbius function.
It is shown that the Orlik–Terao algebra is graded isomorphic to the special fiber of the ideal generated by the -fold products of the members of a central arrangement of size . This momentum is carried over to the Rees algebra (blowup) of and it is shown that this algebra is of fiber-type and Cohen–Macaulay. It follows by a result of Simis and Vasconcelos that the special fiber of is Cohen–Macaulay, thus giving another proof of a result of Proudfoot and Speyer about the Cohen–Macaulayness of the Orlik–Terao algebra.
We show, for a smooth projective variety over an algebraically closed field with an effective Cartier divisor , that the torsion subgroup can be described in terms of a relative étale cohomology for any prime . This extends a classical result of Bloch, on the torsion in the ordinary Chow group, to the modulus setting. We prove the Roitman torsion theorem (including -torsion) for when is reduced. We deduce applications to the problem of invariance of the prime-to- torsion in under an infinitesimal extension of .
We investigate the local descents for special orthogonal groups over -adic local fields of characteristic zero, and obtain explicit spectral decomposition of the local descents at the first occurrence index in terms of the local Langlands data via the explicit local Langlands correspondence and explicit calculations of relevant local root numbers. The main result can be regarded as a refinement of the local Gan–Gross–Prasad conjecture (2012).
Let be a vector space of dimension over , a finite field of elements, and let be a linear group. A base for is a set of vectors whose pointwise stabilizer in is trivial. We prove that if is a quasisimple group (i.e., is perfect and is simple) acting irreducibly on , then excluding two natural families, has a base of size at most 6. The two families consist of alternating groups acting on the natural module of dimension or , and classical groups with natural module of dimension over subfields of .
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