20,996 research outputs found
Toroidal automorphic forms, Waldspurger periods and double Dirichlet series
The space of toroidal automorphic forms was introduced by Zagier in the
1970s: a GL_2-automorphic form is toroidal if it has vanishing constant Fourier
coefficients along all embedded non-split tori. The interest in this space
stems (amongst others) from the fact that an Eisenstein series of weight s is
toroidal for a given torus precisely if s is a non-trivial zero of the zeta
function of the quadratic field corresponding to the torus.
In this paper, we study the structure of the space of toroidal automorphic
forms for an arbitrary number field F. We prove that it decomposes into a space
spanned by all derivatives up to order n-1 of an Eisenstein series of weight s
and class group character omega precisely if s is a zero of order n of the
L-series corresponding to omega at s, and a space consisting of exactly those
cusp forms the central value of whose L-series is zero.
The proofs are based on an identity of Hecke for toroidal integrals of
Eisenstein series and a result of Waldspurger about toroidal integrals of cusp
forms combined with non-vanishing results for twists of L-series proven by the
method of double Dirichlet series.Comment: 14 page
A multi-variable version of the completed Riemann zeta function and other -functions
We define a generalisation of the completed Riemann zeta function in several
complex variables. It satisfies a functional equation, shuffle product
identities, and has simple poles along finitely many hyperplanes, with a
recursive structure on its residues. The special case of two variables can be
written as a partial Mellin transform of a real analytic Eisenstein series,
which enables us to relate its values at pairs of positive even points to
periods of (simple extensions of symmetric powers of the cohomology of) the CM
elliptic curve corresponding to the Gaussian integers. In general, the totally
even values of these functions are related to new quantities which we call
multiple quadratic sums.
More generally, we cautiously define multiple-variable versions of motivic
-functions and ask whether there is a relation between their special values
and periods of general mixed motives. We show that all periods of mixed Tate
motives over the integers, and all periods of motivic fundamental groups (or
relative completions) of modular groups, are indeed special values of the
multiple motivic -values defined here.Comment: This is the second half of a talk given in honour of Ihara's 80th
birthday, and will appear in the proceedings thereo
Effective action and heat kernel in a toy model of brane-induced gravity
We apply a recently suggested technique of the Neumann-Dirichlet reduction to
a toy model of brane-induced gravity for the calculation of its quantum
one-loop effective action. This model is represented by a massive scalar field
in the -dimensional flat bulk supplied with the -dimensional kinetic
term localized on a flat brane and mimicking the brane Einstein term of the
Dvali-Gabadadze-Porrati (DGP) model. We obtain the inverse mass expansion of
the effective action and its ultraviolet divergences which turn out to be
non-vanishing for both even and odd spacetime dimensionality . For the
massless case, which corresponds to a limit of the toy DGP model, we obtain the
Coleman-Weinberg type effective potential of the system. We also obtain the
proper time expansion of the heat kernel in this model associated with the
generalized Neumann boundary conditions containing second order tangential
derivatives. We show that in addition to the usual integer and half-integer
powers of the proper time this expansion exhibits, depending on the dimension
, either logarithmic terms or powers multiple of one quarter. This property
is considered in the context of strong ellipticity of the boundary value
problem, which can be violated when the Euclidean action of the theory is not
positive definite.Comment: LaTeX, 20 pages, new references added, typos correcte
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