17,604 research outputs found
Curves, dynamical systems and weighted point counting
Suppose X is a (smooth projective irreducible algebraic) curve over a finite
field k. Counting the number of points on X over all finite field extensions of
k will not determine the curve uniquely. Actually, a famous theorem of Tate
implies that two such curves over k have the same zeta function (i.e., the same
number of points over all extensions of k) if and only if their corresponding
Jacobians are isogenous. We remedy this situation by showing that if, instead
of just the zeta function, all Dirichlet L-series of the two curves are equal
via an isomorphism of their Dirichlet character groups, then the curves are
isomorphic up to "Frobenius twists", i.e., up to automorphisms of the ground
field. Since L-series count points on a curve in a "weighted" way, we see that
weighted point counting determines a curve. In a sense, the result solves the
analogue of the isospectrality problem for curves over finite fields (also know
as the "arithmetic equivalence problem"): it says that a curve is determined by
"spectral" data, namely, eigenvalues of the Frobenius operator of k acting on
the cohomology groups of all l-adic sheaves corresponding to Dirichlet
characters. The method of proof is to shown that this is equivalent to the
respective class field theories of the curves being isomorphic as dynamical
systems, in a sense that we make precise.Comment: 11 page
Edge reconstruction of the Ihara zeta function
We show that if a graph has average degree , then the
Ihara zeta function of is edge-reconstructible. We prove some general
spectral properties of the edge adjacency operator : it is symmetric for an
indefinite form and has a "large" semi-simple part (but it can fail to be
semi-simple in general). We prove that this implies that if , one can
reconstruct the number of non-backtracking (closed or not) walks through a
given edge, the Perron-Frobenius eigenvector of (modulo a natural
symmetry), as well as the closed walks that pass through a given edge in both
directions at least once.
The appendix by Daniel MacDonald established the analogue for multigraphs of
some basic results in reconstruction theory of simple graphs that are used in
the main text.Comment: 19 pages, 2 pictures, in version 2 some minor changes and now
including an appendix by Daniel McDonal
Toroidal automorphic forms for some function fields
Zagier introduced toroidal automorphic forms to study the zeros of zeta
functions: an automorphic form on GL_2 is toroidal if all its right translates
integrate to zero over all nonsplit tori in GL_2, and an Eisenstein series is
toroidal if its weight is a zero of the zeta function of the corresponding
field. We compute the space of such forms for the global function fields of
class number one and genus g zero or one, and with a rational place. The space
has dimension g and is spanned by the expected Eisenstein series. We deduce an
"automorphic" proof for the Riemann hypothesis for the zeta function of those
curves.Comment: 7 pages, 2 figures; v2: minor correction
Unobserved individual and firm heterogeneity in wage and tenure functions: evidence from German linked employer-employee data
We estimate wage and job tenure functions that include individual and firm effects capturing time-invariant unobserved worker and firm heterogeneity using German linked employer-employee data (LIAB data set). We find that both types of heterogeneity are correlated to the observed characteristics and that it is therefore warranted to include individual and firm fixed effects in both the wage and the job tenure equation. We look into the correlation of the unobserved heterogeneity components with each other. We find that high-wage workers tend to be low-tenure workers, i.e. higher unobserved ability seems to be associated with higher job mobility. At firm level, there seems to be a trade-off between wages and job stability: High-wage firms tend to be low-tenure firms, which suggests that low job stability may be compensated by higher wages. High-wage workers seem to sort into low-wage/high-tenure firms. They seem to forgo some of their earnings potential in favour of higher job stability
Elliptic divisibility sequences and undecidable problems about rational points
Julia Robinson has given a first-order definition of the rational integers Z
in the rational numbers Q by a formula (\forall \exists \forall \exists)(F=0)
where the \forall-quantifiers run over a total of 8 variables, and where F is a
polynomial. This implies that the \Sigma_5-theory of Q is undecidable. We prove
that a conjecture about elliptic curves provides an interpretation of Z in Q
with quantifier complexity \forall \exists, involving only one universally
quantified variable. This improves the complexity of defining Z in Q in two
ways, and implies that the \Sigma_3-theory, and even the \Pi_2-theory, of Q is
undecidable (recall that Hilbert's Tenth Problem for Q is the question whether
the \Sigma_1-theory of Q is undecidable).
In short, granting the conjecture, there is a one-parameter family of
hypersurfaces over Q for which one cannot decide whether or not they all have a
rational point.
The conjecture is related to properties of elliptic divisibility sequences on
an elliptic curve and its image under rational 2-descent, namely existence of
primitive divisors in suitable residue classes, and we discuss how to prove
weaker-in-density versions of the conjecture and present some heuristics.Comment: 39 pages, uses calrsfs. 3rd version: many small changes, change of
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