1,456 research outputs found
A combinatorial Li-Yau inequality and rational points on curves
We present a method to control gonality of nonarchimedean curves based on graph theory. Let k denote a complete nonarchimedean valued field.We first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally,we apply the results to give a lower bound for the gonality of arbitraryDrinfeld modular curves over finite fields and for general congruence subgroups Γ of Γ (1) that is linear in the index [Γ (1) : Γ ], with a constant that only depends on the residue field degree and the degree of the chosen “infinite” place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian
Rational discrete cohomology for totally disconnected locally compact groups
Rational discrete cohomology and homology for a totally disconnected locally
compact group is introduced and studied. The -
identities associated to the rational discrete bimodule allow
to introduce the notion of rational duality groups in analogy to the discrete
case. It is shown that semi-simple groups defined over a non-discrete,
non-archimedean local field are rational t.d.l.c. duality groups, and the same
is true for certain topological Kac-Moody groups. However, Y. Neretin's group
of spheromorphisms of a locally finite regular tree is not even of finite
rational discrete cohomological dimension. For a unimodular t.d.l.c. group
of type it is possible to define an Euler-Poincar\'e
characteristic which is a rational multiple of a Haar measure. This
value is calculated explicitly for Chevalley groups defined over a
non-discrete, non-archimedean local field and some other examples
The boundary of the outer space of a free product
Let be a countable group that splits as a free product of groups of the
form , where is a finitely generated free
group. We identify the closure of the outer space
for the axes topology with the space of
projective minimal, \emph{very small} -trees, i.e. trees
whose arc stabilizers are either trivial, or cyclic, closed under taking roots,
and not conjugate into any of the 's, and whose tripod stabilizers are
trivial. Its topological dimension is equal to , and the boundary has
dimension . We also prove that any very small
-tree has at most orbits of branch points.Comment: v3: Final version, to appear in the Israel Journal of Mathematics.
Section 3, regarding the definition and properties of geometric trees, has
been rewritten to improve the exposition, following a referee's suggestio
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