878 research outputs found
The Berry-Keating Hamiltonian and the Local Riemann Hypothesis
The local Riemann hypothesis states that the zeros of the Mellin transform of
a harmonic-oscillator eigenfunction (on a real or p-adic configuration space)
have real part 1/2. For the real case, we show that the imaginary parts of
these zeros are the eigenvalues of the Berry-Keating hamiltonian H=(xp+px)/2
projected onto the subspace of oscillator eigenfunctions of lower level. This
gives a spectral proof of the local Riemann hypothesis for the reals, in the
spirit of the Hilbert-Polya conjecture. The p-adic case is also discussed.Comment: 9 pages, no figures; v2 included more mathematical background, v3 has
minor edits for clarit
From the function-sheaf dictionary to quasicharacters of -adic tori
We consider the rigid monoidal category of character sheaves on a smooth
commutative group scheme over a finite field and expand the scope of
the function-sheaf dictionary from connected commutative algebraic groups to
this setting. We find the group of isomorphism classes of character sheaves on
and show that it is an extension of the group of characters of by a
cohomology group determined by the component group scheme of . We also
classify all morphisms in the category character sheaves on . As an
application, we study character sheaves on Greenberg transforms of locally
finite type N\'eron models of algebraic tori over local fields. This provides a
geometrization of quasicharacters of -adic tori.Comment: Added examples and incorporated referee's suggestions. To be
published in Journal of the Institute of Mathematics of Jussie
Old and new results on normality
We present a partial survey on normal numbers, including Keane's
contributions, and with recent developments in different directions.Comment: Published at http://dx.doi.org/10.1214/074921706000000248 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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