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Richard Hill

Publications and source records attributed to Richard Hill.

29 records · Page 2Linked to original sources
arXiv2010

Emerton's Jacquet functors for non-Borel parabolic subgroups

This paper studies Emerton's Jacquet module functor for locally analytic representations of p-adic reductive groups. When P is a parabolic subgroup whose Levi factor M is not commutative, we show that passing to an isotypical subspace for the derived subgroup of M gives rise to essentially admissible locally analytic representations of the torus Z(M), which have a natural interpretation in terms of rigid geometry. We use this to extend Emerton's representation-theoretic construction of eigenvarieties by constructing eigenvarieties interpolating automorphic representations whose local components at p are not necessarily principal series.

math.NT↗
arXiv2007

On Emerton's $p$-adic Banach spaces

The purpose of the current paper is to introduce some new methods for studying the $p$-adic Banach spaces introduced by Emerton \cite{emerton}. We first relate these spaces to more familiar sheaf cohomology groups. As an application, we obtain a more general version of Emerton's spectral sequence. We also calculate the spaces in some easy cases. As a consequence, we obtain a number of vanishing theorems.

math.NT↗
arXiv2007

Geometric construction of metaplectic covers of $\GL_{n}$ in characteristic zero

This paper presents a new construction of the m-fold metaplectic cover of $\GL_{n}$ over an algebraic number field k, where k contains a primitive m-th root of unity. A 2-cocycle on $\GL_{n}(\A)$ representing this extension is given and the splitting of the cocycle on $\GL_{n}(k)$ is found explicitly. The cocycle is smooth at almost all places of k. As a consequence, a formula for the Kubota symbol on $\SL_{n}$ is obtained. The construction of the paper requires neither class field theory nor algebraic K-theory, but relies instead on naive techniques from the geometry of numbers introduced by W. Habicht and T. Kubota. The power reciprocity law for a number field is obtained as a corollary.

math.NT↗
arXiv2007

Shintani Cocycles on $\GL_{n}$

The aim of this paper is to define an n-1-cocycle $σ$ on $\GL_{n}(\Q)$ with values in a certain space $\hD$ of distributions on $\A_f^{n}\setminus\{0\}$. Here $\A_f$ denotes the ring of finite adèles of $\Q$, and the distributions take values in the Laurent series $\C((z_{1},...,z_{n}))$. This cocycle can be used to evaluate special values of Artin L-functions on number fields at negative integers. The construction generalizes that of Solomon in the case n=2.

math.NT↗
arXiv2007

A reinterpretation of Emerton's $p$-adic Banach spaces

It is shown that the $p$-adic Banach spaces introduced by Emerton are isomorphic to the cohomology groups of the sheaf of continuous $\Q_{p}$-valued functions on a certain space. Some applications of this result are discussed.

math.NT↗
arXiv2007

Construction of eigenvarieties in small cohomological dimensions for semi-simple, simply connected groups

We study low order terms of Emerton's spectral sequence for simply connected, simple groups. As a result, for real rank 1 groups, we show that Emerton's method for constructing eigenvarieties is successful in cohomological dimension 1. For real rank 2 groups, we show that a slight modification of Emerton's method allows one to construct eigenvarieties in cohomological dimension 2.

math.NT↗
arXiv2000

New Value of M_mu/M_e from Muonium Hyperfine Splitting

The complete contribution to the muonium hyperfine splitting of relative order alpha^3(m_e/m_mu)ln(alpha) is calculated. The result amounts to 0.013 kHZ, much smaller than suggested by a previous estimate, and leads to a 2-sigma shift of the most precise value for the muon-electron mass ratio, with the error reduced by approximately 30%. Analogous contributions are calculated for the positronium hyperfine splitting: (217/90-17*ln{2}/3)m_e(alpha^7/pi)ln(1/alpha) \approx -0.32 MHz; the remaining theoretical uncertainty should be well below experimental error, leaving discrepancies of 2.5-sigma and 3.5-sigma with the two most precise measurements.

hep-ph↗
arXiv2000

Nonperturbative Techniques for QED Bound States

Advantages of using a low-energy effective theory to study bound state properties are briefly discussed, and a nonperturbative implementation of such an effective theory is described within the context of nonrelativistic quantum mechanics. The hydrogen atom, in the approximation of a structureless, infinite-mass nucleus, but with the leading relativistic and radiative corrections included, is used to demonstrate the construction and solution of the effective theory. The resulting Hamiltonian incorporates a finite ultraviolet cutoff and can be solved nonperturbatively. An appendix lists explicit formulae for the various matrix elements necessary to diagonalize the Hamiltonian using gaussian basis sets.

hep-ph↗
arXiv1998

Trace Formulae, Zeta Functions, Congruences and Reidemeister Torsion in Nielsen Theory

In this paper we prove trace formulae for the Reidemeister number of a group endomorphism. This result implies the rationality of the Reidemeister zeta function in the following cases: the group is a direct product of a finite group and a finitely generated Abelian group; the group is finitely generated, nilpotent and torsion free. We continue to study analytical properties of the Nielsen zeta function. We connect the Reidemeister zeta function of an endomorphism of a direct product of a finite group and a finitely generated free Abelian group with the Lefschetz zeta function of the induced map on the unitary dual of the group. As a consequence we obtain a relation between a special value of the Reidemeister zeta function and a certain Reidemeister torsion. We also prove congruences for Reidemeister numbers of iterates of an endomorphism of a direct product of a finite group and a finitely generated free Abelian group which are the same as those found by Dold for Lefschetz numbers.

math.DG↗
arXiv1995

Trace formulas and dynamical zeta functions in the Nielsen theory

In this paper we prove the trace formulas for the Reidemeister numbers of group endomorphisms in the following cases:the group is finitely generated and an endomorphism is eventually commutative; the group is finite ; the group is a direct sum of a finite group and a finitely generated free abelian group; the group is finitely generated, nilpotent and torsion free . These results had previously been known only for the finitely generated free abelian group.As a consequence, we obtain under the same conditions on the fundamental group of a compact polyhedron,the trace formulas for the Reidemeister numbers of a continuous map and under suitable conditions the trace formulas for the Nielsen numbers of a continuous map.The trace formula for the Reidemeister numbers implies the rationality of the Reidemeister zeta function. We prove the rationality and functional equation for the Reidemeister zeta function of an endomorphisms of finitely generated torsion free nilpotent group and of a direct sum of a finite group and a finitely generated free abelian group.We give a new proof of the rationality of the Reidemeister zeta function in the case when the group is finitely generated and the endomorphism is eventually commutative and in the case when group is finite.We give also another proof for the positivity of the radius of convergence of the Nielsen zeta function and propose an exast algebraic lower bound estimation for the radius.We connect the Reidemeister zeta function of an endomorphism of a direct sum of a finite group and a finitely generated free abelian group with the Lefschetz zeta function of the unitary dual map,and as consequence obtain a connection of the Reidemeister zeta function with Reidemeister torsion.

chao-dyn↗