the integral monodromy of hyperelliptic and trielliptic curves we compute the mathbb z ell and mathbb z ell monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves in particular we provide a proof that the mathbb z ell monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group rm s p2g mathbb z ell we prove that the mathbb z ell monodromy of the moduli space of trielliptic curves with signature r s is the special unitary group rm su r s mathbb z ell otimes mathbb z zeta 3

the existence of an abelian variety over q isogenous to no jacobian we prove the existence of an abelian variety a of dimension g over q that is not isogenous to any jacobian subject to the necessary condition g 3 recently c chai and f oort gave such a proof assuming the andr eoort conjecture we modify their proof by constructing a special sequence of cm points for which we can avoid any unproven hypotheses we make use of various techniques from the recent work of klingler yafaev et al

minimizing neumann fundamental tones of triangles an optimal poincare inequality abstract the first nonzero eigenvalue of the neumann laplacian is shown to be minimal for the degenerate acute isosceles triangle among all triangles of given diameter hence an optimal poincare inequality for triangles is derived the proof relies on symmetry of the neumann fundamental mode for isosceles triangles with aperture less than 3 antisymmetry is proved for apertures greater than 3

walks on groups counting reducible matrices polynomials and surface and free group automorphisms we prove sharp limit theorems on random walks on graphs with values in finite groups we then apply these re sults together with some elementary algebraic geometry number theory and representation theory to finite quotients of lattices in semisimple lie groups specifically sl n z and sp 2n z to show that a random element in one of these lattices has irre ducible characteristic polynomials overz the term random can be defined in at least two ways in terms of height and also in terms of word length in terms of a generating set we show the result using both definitions we use the above results to show that a random in terms of word length element of the mapping class group of a surface is pseudo anosov and that a a random free group automorphism is irreducible with irreducible powers or strongly irreducible 1

asymptotic error probability analysis of quadratic receivers in rayleigh fading channels with applications to a unified analysis of coherent and noncoherent space time receivers a general asymptotic high signal to noise snr error analysis is introduced for quadratic receivers in frequency flat and multipath rayleigh fading channels with multiple transmit and receive antennas asymptotically tight expressions for the pairwise error probabilities are obtained for coherent noncoherent and differentially coherent space time receivers not only is our unified analysis applicable to more general modulation schemes and or channel models than previously considered but it also reveals a hitherto unrecognized eigenvalue structure that is common to all of these problems in addition to providing an easy recipe for computing the asymptotic pairwise error rates we make some conclusions regarding criteria for the design of signal constellations and codes such as a the same design criteria apply for both correlated and independent and identically distributed i i d fading processes and b for noncoherent communications unitary signals are optimal in the sense that they minimize the asymptotic union bound

hot spots conjecture for a class of acute triangles we show that the hot spots conjecture of j rauch holds for acute triangles if one of the angles is not larger than pi 6 more precisely we show that the second neumann eigenfunction on those acute triangles has no maximum or minimum inside the domain we first simplify the problem by showing that absence of critical points on two sides implies no critical points inside a triangle this result applies to any acute triangle and might help prove the conjecture for arbitrary acute triangles then we show that there are no critical points on two sides assuming one small angle we also establish simplicity for the smallest positive neumann eigenvalue for all non equilateral acute triangles this result was already known for obtuse triangles and it fails for the equilateral case

a counterexample to the hot spots conjecture we construct a counterexample to the hot spots conjecture there exists a bounded connected planar domain with two holes such that the second eigenvalue of the laplacian in that domain with neumann boundary conditions is simple and such that the corresponding eigenfunction attains its strict maximum at an interior point of that domain

eigenvalue inequalities for the laplacian with mixed boundary conditions abstract inequalities for the eigenvalues of the negative laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established the eigenvalues subject to a dirichlet boundary condition on a part of the boundary and a neumann boundary condition on the remainder of the boundary are estimated in terms of either dirichlet or neumann eigenvalues the results complement several classical inequalities between dirichlet and neumann eigenvalues due to polya payne levine and weinberger friedlander and others

on the hot spots conjecture of j rauch rodrigo banuelos research partially supported by nsf grant dms 9400854 krzysztof burdzy s research partially supported by nsf grant dms 9322689

maximizing neumann fundamental tones of triangles we prove sharp isoperimetric inequalities for neumann eigenvalues of the laplacian on triangular domains the first nonzero neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter and hence among all triangles of given area similar results are proven for the harmonic and arithmetic means of the first two nonzero eigenvalues

the hot spots conjecture for a certain class of planar convex domains we prove the hot spots conjecture of rauch five problems an introduction to the qualitative theory of partial differential equations partial differential equations and related topics program tulane univ new orleans la 1974 lecture notes in mathematics springer berlin 1975 vol 446 pp 355 369 for a certain class of planar convex domains specifically we show that an eigenfunction corresponding to the lowest nonzero eigenvalue of the neumann laplacian on attains its maximum minimum at points in one class of domains is the planar convex domain satisfying diam 2 1 378 when is a disk diam 2 1 273 hence this condition indicates that is a nearly circular planar convex domain however symmetries of the domain are not assumed we give other sufficient conditions for domains for which the conjecture holds we also give a new isoperimetric inequality

hat g local systems on smooth projective curves are potentially automorphic dfg grants fg 1920 and spp 1489 erc grant 290766 aamot nsf grant dms 1404769 nsf grant dms 1161671 humboldt research award clay mathematics institute erc grant no 714405 gmlp

scaling coupling of reflecting brownian motions and the hot spots problem we introduce a new type of coupling of reflecting brownian motions in smooth planar domains called scaling coupling we apply this to obtain monotonicity properties of antisymmetric second neumann eigenfunctions of convex planar domains with one line of symmetry in particular this gives the proof of the hot spots conjecture for some known types of domains and some new ones

location of hot spots in thin curved strips abstract the maxima and minima of neumann eigenfunctions of thin tubular neighbourhoods of curves on surfaces are located in terms of the maxima and minima of neumann eigenfunctions of the underlying curves in particular the hot spots conjecture for a new large class of domains possibly non convex and non euclidean is proved

exceptional splitting of reductions of abelian surfaces heuristics based on the sato tate conjecture and the lang trotter philosophy suggest that an abelian surface defined over a number field has infinitely many places of split reduction we prove this result for abelian surfaces with real multiplication as in previous work by charles and elkies this shows that a density 0 set of primes pertaining to the reduction of abelian varieties is infinite the proof relies on the arakelov intersection theory on hilbert modular surfaces

improved methods for calculating vectors of short length in a lattice including a complexity analysis the standard methods for calculating vectors of short length in a lattice use a reduction procedure followed by enumerating all vectors of z in a suitable box however it suffices to consider those x e z which lie in a suitable ellipsoid having a much smaller volume than the box we show in this paper that searching through that ellipsoid is in many cases much more efficient if combined with an appropriate reduction procedure our method allows to do computations in lattices of much higher dimensions several randomly constructed numerical examples illustrate the superiority of our new method over the known ones

the probability that the number of points on the jacobian of a genus 2 curve is prime in 2000 galbraith and mckee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points we show how their heuristics can be generalized to jacobians of curves of higher genus we then elaborate this in genus g 2 and study various related issues such as the probability of cyclicity and the probability of primality of the number of points on the curve itself finally we discuss the asymptotic behavior for g

analysis and design of short block ofdm spreading matrices for use on multipath fading channels we consider the use of block spreading in a multicarrier system to gain diversity advantage when employed over multipath fading channels the main idea is to split the full set of subcarriers into smaller blocks and spread the data symbols across these blocks via unitary spreading matrices in order to gain multipath diversity across each block at the receiver we pose the problem of designing the spreading matrix as a finite dimensional optimization problem in which the asymptotic error is minimized this formulation allows us to find explicit solutions for the optimal spreading matrices the performance is validated for the uncoded channel as well as for the coded channel employing turbo iterative decoding we further demonstrate that suboptimal linear complexity equalization strategies for spread orthogonal frequency division multiplexing ofdm do not gain any diversity advantage over traditional diagonal ofdm

on neumann eigenfunctions in lip domains atar s research partially supported by the fund for the promotion of research at the technion burdzy gratefull acknowledges the hospitality and financial support of technion israel and institut mittag leffler sweden this research was partially supported by nsf grant dms 0071486 and isf grant 12 98

a universal lattice code decoder for fading channels we present a maximum likelihood decoding algorithm for an arbitrary lattice code when used over an independent fading channel with perfect channel state information at the receiver the decoder is based on a bounded distance search among the lattice points falling inside a sphere centered at the received point by judicious choice of the decoding radius we show that this decoder can be practically used to decode lattice codes of dimension up to 32 in a fading environment

abelian varieties isogenous to a jacobian we dene a notion of weyl cm points in the moduli space ag 1 of g dimensional principally polarized abelian varieties and show that the andr e oort conjecture or the grh implies the following statement for any closed subvariety x ag 1 over q a there exists a weyl special point b 2 ag 1 q a such that b is not isogenous to the abelian variety a underlying any point a 2 x the title refers to the case when g 4 and x is the torelli locus in this case tsimerman has proved the statement unconditionally the notion of weyl special points is generalized to the context of shimura varieties and we prove a corresponding conditional statement with the ambient spaceag 1 replaced by a general shimura variety

the hot spots conjecture for domains with two axes of symmetry consider a convex planar domain with two axes of symmetry we show that the maximum and minimum of a neumann eigenfunction with lowest nonzero eigenvalue occur at points on the boundary only we deduce j rauch s hot spots conjecture in the following form if the initial temperature distribution is not orthogonal to the first nonzero eigenspace then the point at which the temperature achieves its maximum tends to the boundary in fact the maximum point reaches the boundary in finite time if the boundary has positive curvature results of this type have already been proved by bafiuelos and burdzy bb using the heat equation and probabilistic methods to deform initial conditions to eigenfunctions we introduce here a new technique based on deformation of the domain an advantage of our method is that it works even in the case of multiple eigenvalues on the way toward our results we prove monotonicity properties for neumann eigenfunctions for symmetric domains that need not be convex and deduce a sharp comparison of eigenvalues with the dirichlet problem of independent interest