2,162 research outputs found
Conjectures on the logarithmic derivatives of Artin L-functions II
We formulate a general conjecture relating Chern classes of subbundles of
Gauss-Manin bundles in Arakelov geometry to logarithmic derivatives of Artin
L-functions of number fields. This conjecture may be viewed as a far-reaching
generalisation of the (Lerch-)Chowla-Selberg formula computing logarithms of
periods of elliptic curves in terms of special values of the -function.
We prove several special cases of this conjecture in the situation where the
involved Artin characters are Dirichlet characters. This article contains the
computations promised in the article {\it Conjectures sur les d\'eriv\'ees
logarithmiques des fonctions L d'Artin aux entiers n\'egatifs}, where our
conjecture was announced. We also give a quick introduction to the
Grothendieck-Riemann-Roch theorem and to the geometric fixed point formula,
which form the geometric backbone of our conjecture.Comment: 54 page
On the determinant bundles of abelian schemes
Let \pi:\CA\ra S be an abelian scheme over a scheme which is
quasi-projective over an affine noetherian scheme and let \CL be a symmetric,
rigidified, relatively ample line bundle on \CA. We show that there is an
isomorphism
\det(\pi_*\CL)^{\o times 24}\simeq\big(\pi_*\omega_{\CA}^{\vee}\big)^{\o
times 12d}
of line bundles on , where is the rank of the (locally free) sheaf
\pi_*\CL. We also show that the numbers 24 and are sharp in the
following sense: if is a common divisor of 12 and 24, then there are data
as above such that
\det(\pi_*\CL)^{\o times
(24/N)}\not\simeq\big(\pi_*\omega_{\CA}^{\vee}\big)^{\o times (12d/N)}.Comment: 8 page
On a canonical class of Green currents for the unit sections of abelian schemes
We show that on any abelian scheme over a complex quasi-projective smooth
variety, there is a Green current for the zero-section, which is axiomatically
determined up to and -exact differential forms. This
current generalizes the Siegel functions defined on elliptic curves. We prove
generalizations of classical properties of Siegel functions, like distribution
relations, limit formulae and reciprocity laws.Comment: 42 page
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