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The obstruction to the local-global principle for a hermitian lattice can be quantified by computing the mass of . The mass formula expresses the mass of as a product of local factors, called the local densities of . The local density formula is known except in the case of a ramified hermitian lattice of residue characteristic 2.
Let be a finite unramified field extension of . Ramified quadratic extensions fall into two cases that we call Case 1 and Case 2. In this paper, we obtain the local density formula for a ramified hermitian lattice in Case 1, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with the paper of W. T. Gan and J.-K. Yu (Duke Math. J. 105 (2000), 497–524), allows the computation of the mass formula for a hermitian lattice in Case 1.
Tits has defined Steinberg groups and Kac–Moody groups for any root system and any commutative ring . We establish a Curtis–Tits-style presentation for the Steinberg group of any irreducible affine root system with rank , for any . Namely, is the direct limit of the Steinberg groups coming from the - and -node subdiagrams of the Dynkin diagram. In fact, we give a completely explicit presentation. Using this we show that is finitely presented if the rank is and is finitely generated as a ring, or if the rank is and is finitely generated as a module over a subring generated by finitely many units. Similar results hold for the corresponding Kac–Moody groups when is a Dedekind domain of arithmetic type.
The discriminant is a classical invariant associated to algebras which are finite over their centers. It was shown recently by several authors that if the discriminant of is “sufficiently nontrivial” then it can be used to answer some difficult questions about . Two such questions are: What is the automorphism group of ? Is Zariski cancellative?
We use the discriminant to study these questions for a class of (generalized) quantum Weyl algebras. Along the way, we give criteria for when such an algebra is finite over its center and prove two conjectures of Ceken, Wang, Palmieri and Zhang.
We introduce a regularized theta lift for reductive dual pairs , for a quadratic vector space over a totally real number field. The lift takes values in the space of (1,1)-currents on the Shimura variety attached to ; we show its values are cohomologous to currents given by integration on special divisors against automorphic Green functions. A later paper will show how to evaluate the new lift on differential forms obtained as usual (nonregularized) theta lifts.
We give a version of the Eichler–Shimura isomorphism with a nonabelian in group cohomology. Manin has given a map from vectors of cusp forms to a noncommutative cohomology set by means of iterated integrals. We show that Manin’s map is injective but far from surjective. By extending Manin’s map we are able to construct a bijective map and remarkably this establishes the existence of a nonabelian version of the Eichler–Shimura map.
We study the existential (and parts of the universal-existential) theory of equicharacteristic henselian valued fields. We prove, among other things, an existential Ax–Kochen–Ershov principle, which roughly says that the existential theory of an equicharacteristic henselian valued field (of arbitrary characteristic) is determined by the existential theory of the residue field; in particular, it is independent of the value group. As an immediate corollary, we get an unconditional proof of the decidability of the existential theory of .
It is well known that, for any finitely generated torsion module over the Iwasawa algebra , where is isomorphic to , there exists a continuous -adic character of such that, for every open subgroup of , the group of -coinvariants is finite; here denotes the twist of by . This twisting lemma was already used to study various arithmetic properties of Selmer groups and Galois cohomologies over a cyclotomic tower by Greenberg and Perrin-Riou. We prove a noncommutative generalization of this twisting lemma, replacing torsion modules over by certain torsion modules over with more general -adic Lie group . In a forthcoming article, this noncommutative twisting lemma will be used to prove the functional equation of Selmer groups of general -adic representations over certain -adic Lie extensions.