tunneling through a singular potential barrier quantum tunneling of a nonrelativistic particle through a singular potential barrier v is studied on the line the hamiltonian is a self adjoint extension of the operator h1 d2 dx2 v x if h1 is essentially self adjoint on its natural domain the tunneling is forbidden for a class of potentials that includes all semiclassically impenetrable barriers if h1 is not essentially self adjoint the friedrichs extension of h1 yields no tunneling for another class of potentials which again includes the semiclassically impenetrable ones in general the occurrence of tunneling is not excluded and depends on the self adjoint extension we choose as the hamiltonian of our problem as an example we evaluate the transmission coefficient for all self adjoint extensions of the operator h1 referring to v x gx 2 with 0 g 3 4

mixing automorphisms of compact groups and a theorem of schlickewei we prove that every mixing d by automorphisms of a compact connected abelian group is mixing of all orders

rank one drinfeld modules on elliptic curves the sgn normalized rank one drinfeld modules 0 associated with all elliptic curves e over fq for 4 q 13 are computed in explicit form such 0 for q 4 were computed previously these computations verify a conjecture of dorman on the norm of j q aq l and also suggest some interesting new properties of 0 we prove dorman s conjecture in the ramified case we also prove the formula deg n a q hk 1 q where n a is the norm of a and hk is the class number of k fq e we describe a remarkable conjectural property of the trace of a in even characteristic that holds in all the examples in his recent paper 1 on the factorization of norms of j invariants of ranktwo drinfeld modules with complex multiplications d dorman conjectured that such norms are monic elements of fq x dorman computed the jinvariants in several examples and found his conjecture valid in all of them for this purpose he used the rank one examples from 6 the first extensive computational test of the conjecture was carried out by one of us dummit who computed the rank one drinfeld modules associated with all elliptic curves over fq with q 13 see 3 below the results of these computations may be found on the microfiche card included at the end of this issue inspired by these computations one of us hayes proved dorman s conjecture when the infinite place is ramified see 2 below in 3 we describe an algorithm for computing the rank one drinfeld modules associated with any hyperelliptic curve over fqq and we prove some basic attributes of the algorithm in 4 we prove formulas for the degrees of the norm and trace of the j invariants of drinfeld modules associated with elliptic curves these formulas were first observed computationally they suggest a number of interesting questions for elliptic curves with complex multiplications in characteristic zero in 5 we state some conjectures about the remarkable form of the trace term in characteristic two these conjectures are supported by all our computations

isomorphism rigidity of irreducible algebraic ℤd actions an irreducible algebraic d action on a compact abelian group x is a d action by automorphisms of x such that every closed invariant subgroup y x is finite we prove the following result if d 2 then every measurable conjugacy between irreducible and mixing algebraic d actions on compact zero dimensional abelian groups is affine for irreducible expansive and mixing algebraic d actions on compact connected abelian groups the analogous statement follows essentially from a result by katok and spatzier on invariant measures of such actions cf 4 and 3 by combining these two theorems one obtains isomorphism rigidity of all irreducible expansive and mixing algebraic d actions with d 2

complex potential description of the damped harmonic oscillator the multidimensional damped harmonic oscillator is treated by means of a non self adjoint hamiltonian with complex potential the propagator referring to the evolution semigroup is evaluated from the lie trotter formula the one dimensional case is discussed in detail with the following results a the nondamped limit gives the correct propagator including the maslov phase factor b for some initial conditions the classical limit of the solution can differ from the behavior of the classical damped oscillator the difference being negligible in the case of weak damping and c the point spectrum of the considered pseudo hamiltonian is found

quantum motion on a half line connected to a plane in this paper the free motion of a particle on a manifold that consists of a one dimensional and a two dimensional part connected in one point is discussed the class of admissible hamiltonians is found using the theory of self adjoint extensions particular attention is paid to those hamiltonians that allow the particle to pass through the point singularity the reflection coefficient and other quantities characterizing scattering on the connection point are calculated a possible application is also discussed

algebraic ℤ actions of entropy rank one we investigate algebraic z d actions of entropy rank one namely those for which each element has finite entropy such actions can be completely described in terms of diagonal actions on products of local fields using standard adelic machinery this leads to numerous alternative characterizations of entropy rank one both geometric and algebraic we then compute the measure entropy of a class of skew products where the fiber maps are elements from an algebraic z d action of entropy rank one this leads via the relative variational principle to a formula for the topological entropy of continuous skew products as the maximum of a finite number of topological pressures we use this to settle a conjecture concerning the relational entropy of commuting toral automorphisms

mahler measure and entropy for commuting automorphisms of compact groups we compute the joint entropy ofd commuting automorphisms of a compact metrizable group letr d u 1 1 d 1 1 be the ring of laurent polynomials ind commuting variables andm be anr d module then the dual groupx m ofm is compact and multiplication onm by each of thed variables corresponds to an action m of d by automorphisms ofx m every action of d by automorphisms of a compact abelian group arises this way iff r d our main formula shows that the topological entropy of alpha r d left langle f right rangle is given by h left alpha r d left langle f right rangle right log m left f right int limits 0 1 ldots int limits 0 1 log left f left e 2 pi it 1 ldots e 2 pi it d right right dt 1 ldots dt d where m f is the mahler measure off this reduces to the classical result for toral automorphisms via jensen s formula while the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7 3 4 2 occur if p is a non principal prime ideal we show h left alpha r d p right 0 using an analogue of the yuzvinskii thomas addition formula we computeh m for arbitraryr d modulesm and then the joint entropy for an action of d on a not necessarily abelian compact group using a result of boyd we characterize those m which have completely positive entropy in terms of the prime ideals associated tom and show this condition implies that m is mixing of all orders we also establish an analogue of berg s theorem proving that if m has finite entropy then haar measure is the unique measure of maximal entropy if and only if m has completely positive entropy finally we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy

isomorphism rigidity of commuting automorphisms r let d 1 and let x and y be two zero entropy z d actions on compact abelian groups by d commuting automorphisms we show that if all lower rank subactions of and have completely positive entropy then any measurable equivariant map from x to y is an affine map in particular two such actions are measurably conjugate if and only if they are algebraically conjugate

compact group automorphisms addition formulas and fuglede kadison determinants for a countable amenable group and an element f in the integral group ring z being invertible in the group von neumann algebra of we show that the entropy of the shift action of on the pontryagin dual of the quotient of z by its left ideal generated by f is the logarithm of the fuglede kadison determinant of f for the proof we establish an p version of rufus bowen s denition of topological entropy addition formulas for group extensions of countable amenable group actions and an approximation formula for the fuglede kadison determinant of f in terms of the determinants of perturbations of the compressions of f

invariant measures on g γ for split simple lie groups g we study the left action of a cartan subgroup on the space x g where is a lattice in a simple split connected lie group g of rank n 1 let be an invariant measure on x we give several conditions using entropy and conditional measures each of which characterizes the haar measure on x furthermore we show that the conditional measure on the foliation of unstable manifolds has the structure of a product measure the main new element compared to the previous work on this subject is the use of noncommutativity of root foliations to establish rigidity of invariant measures 2003 wiley periodicals inc

entropy geometry and disjointness for zero dimensional algebraic actions we show that many algebraic actions of higher rank abelian groups on zero dimensional compact abelian groups are mutually disjoint the proofs exploit differences in the entropy geometry arising from subdynamics and a form of abramov rokhlin formula for half space entropies

on the uniform boundedness conjecture for drinfeld modules in analogy with the famous theorems of mazur and merel on the torsion subgroups of elliptic curves one can formulate similar conjectures for the torsion points of drinfeld modules we prove some partial results for rank 2 drinfeld q t modules for example the uniform boundedness of the primary torsion

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 arbitrary drinfeld modular curves over finite fields and for general congruence subgroups vargamma of vargamma 1 that is linear in the index vargamma 1 vargamma 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

group theoretical approach to reflectionless potentials we examine the general form of potentials with zero reflection coefficient in one dimensional hamiltonians connected with casimir invariants of non compact groups

the andre oort conjecture for products of drinfeld modular curves let z x1 xn be a product of drinfeld modular curves we characterize those algebraic subvarieties x z containing a zariski dense set of cm points i e points corresponding to n tuples of drinfeld modules with complex multiplication and suitable level structure this is a characteristic p analogue of a special case of the andre oort conjecture

elliptic curves with abelian division fields let e be an elliptic curve over mathbb q and let n ge 1 the central object of study of this article is the division field mathbb q e n that results by adjoining to mathbb q the coordinates of all n torsion points on e overline mathbb q in particular we classify all curves e mathbb q such that mathbb q e n is as small as possible that is when mathbb q e n mathbb q zeta n and we prove that this is only possible for n 2 3 4 or 5 more generally we classify all curves such that mathbb q e n is contained in a cyclotomic extension of mathbb q or equivalently by the kronecker weber theorem when mathbb q e n mathbb q is an abelian extension in particular we prove that this only happens for n 2 3 4 5 6 or 8 and we classify the possible galois groups that occur for each value of n

the mordell lang conjecture for function fields we give a proof of the geometric mordell lang conjecture in any characteristic our method involves a model theoretic analysis of the kernel of manin s homomorphism and of a certain analog in characteristic p department of mathematics massachusetts institute of technology 2 277 cambridge massachusetts 02139 current address department of mathematics hebrew university jerusalem israel e mail address ehud math mit edu

sur la nature non cyclotomique des points d ordre fini des courbes elliptiques appendice par e kowalski et p michel

cycles of quadratic polynomials and rational points on a genus 2 curve it has been conjectured that for n sufficiently large there are no quadratic polynomials in q z with rational periodic points of period n morton proved there were none with n 4 by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to x 1 16 whose rational points had been previously computed we prove there are none with n 5 here the relevant curve has genus 14 but it has a genus 2 quotient whose rational points we compute by performing a 2 descent on its jacobian and applying a refinement of the method of chabauty and coleman we hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves we also describe the three possible galois stable 5 cycles and show that there exist galois stable n cycles for infinitely many n furthermore we answer a question of morton by showing that the genus 14 curve and its quotient are not modular finally we mention some partial results for n 6

connection conditions and the spectral family under singular potentials to describe a quantum system whose potential is divergent at one point one must provide proper connection conditions for the wavefunctions at the singularity generalizing the scheme used for point interactions in one dimension we present a set of connection conditions which are well defined even if the wavefunctions and or their derivatives are divergent at the singularity our generalized scheme covers the entire u 2 family of quantizations self adjoint hamiltonians admitted for the singular system we use this scheme to examine the spectra of the coulomb potential v x e2 x and the harmonic oscillator with square inverse potential v x m 2 2 x2 g x2 and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra we further show that for any parity invariant singular potential v x v x the spectrum is determined solely by the eigenvalues of the characteristic matrix u u 2

homoclinic groups ie groups and expansive algebraic actions we give algebraic characterizations for expansiveness of algebraic actions of countable groups the notion of p expansiveness is introduced for algebraic actions and we show that for countable amenable groups a finitely presented algebraic action is 1 expansive exactly when it has finite entropy we also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms and show that the ie group determines the pinsker factor for such actions for an expansive algebraic action of a polycyclic by finite group on x it is shown that the entropy of the action is equal to the entropy of the induced action on the pontryagin dual of the homoclinic group the homoclinic group is a dense subgroup of the ie group the homoclinic group is nontrivial exactly when the action has positive entropy and the homoclinic group is dense in x exactly when the action has completely positive entropy