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

Coulomb analogy for nonhermitian degeneracies near quantum phase transitions

Degeneracies near the real axis in a complex-extended parameter space of a hermitian Hamiltonian are studied. We present a method to measure distributions of such degeneracies on the Riemann sheet of a selected level and apply it in classification of quantum phase transitions. The degeneracies are shown to behave similarly as complex zeros of a partition function.Comment: 4 page

Norm kernels and the closeness relation for Pauli-allowed basis functions

The norm kernel of the generator-coordinate method is shown to be a symmetric kernel of an integral equation with eigenfunctions defined in the Fock--Bargmann space and forming a complete set of orthonormalized states (classified with the use of SU(3) symmetry indices) satisfying the Pauli exclusion principle. This interpretation allows to develop a method which, even in the presence of the SU(3) degeneracy, provides for a consistent way to introduce additional quantum numbers for the classification of the basis states. In order to set the asymptotic boundary conditions for the expansion coefficients of a wave function in the SU(3) basis, a complementary basis of functions with partial angular momenta as good quantum numbers is needed. Norm kernels of the binary systems 6He+p, 6He+n, 6He+4He, and 8He+4He are considered in detail.Comment: 25 pages; submitted to Few-Body System

In-the-Gap SU UMa-Type Dwarf Nova, Var73 Dra with a Supercycle of about 60 Days

An intensive photometric-observation campaign of the recently discovered SU UMa-type dwarf nova, Var73 Dra was conducted from 2002 August to 2003 February. We caught three superoutbursts in 2002 October, December and 2003 February. The recurrence cycle of the superoutburst (supercycle) is indicated to be $\sim$60 d, the shortest among the values known so far in SU UMa stars and close to those of ER UMa stars. The superhump periods measured during the first two superoutbursts were 0.104885(93) d, and 0.10623(16) d, respectively. A 0.10424(3)-d periodicity was detected in quiescence. The change rate of the superhump period during the second superoutburst was $1.7\times10^{-3}$, which is an order of magnitude larger than the largest value ever known. Outburst activity has changed from a phase of frequent normal outbursts and infrequent superoutbursts in 2001 to a phase of infrequent normal outbursts and frequent superoutbursts in 2002. Our observations are negative to an idea that this star is an related object to ER UMa stars in terms of the duty cycle of the superoutburst and the recurrence cycle of the normal outburst. However, to trace the superhump evolution throughout a superoutburst, and from quiescence more effectively, may give a fruitful result on this matter.Comment: 9 pages, 8 figures, submitted to A&