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 $G$ has average degree $\bar d \geq 4$, then the Ihara zeta function of $G$ is edge-reconstructible. We prove some general spectral properties of the edge adjacency operator $T$: 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 $\bar d>4$, one can reconstruct the number of non-backtracking (closed or not) walks through a given edge, the Perron-Frobenius eigenvector of $T$ (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

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

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

Hecke algebra isomorphisms and adelic points on algebraic groups

Let $G$ denote a linear algebraic group over $\mathbf{Q}$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism of points on $G$ over the finite adeles of $K$ and $L$, respectively. We establish conditions on the group $G$, related to the structure of its Borel groups, under which $K$ and $L$ have isomorphic adele rings. Under these conditions, if $K$ or $L$ is a Galois extension of $\mathbf{Q}$ and $G(\mathbf{A}_{K,f})$ and $G(\mathbf{A}_{L,f})$ are isomorphic, then $K$ and $L$ are isomorphic as fields. We use this result to show that if for two number fields $K$ and $L$ that are Galois over $\mathbf{Q}$, the finite Hecke algebras for $\mathrm{GL}(n)$ (for fixed $n > 1$) are isomorphic by an isometry for the $L^1$-norm, then the fields $K$ and $L$ are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field if it is Galois over $\mathbf{Q}$.Comment: 19 pages - completely rewritte

Mumford curves with maximal automorphism group II: Lame type groups in genus 5-8

A Mumford curve of genus g=5,6,7 or 8 over a non-archimedean field of characteristic p (such that if p=0, the residue field characteristic exceeds 5) has at most 12(g-1) automorphisms. In this paper, all curves that attain this bound and their automorphism groups (called of Lame type) are explicitly determined.Comment: 14 pages, contains three eps-picture