heights of heegner cycles and derivatives of l series 0 introduction 99 1 intersections and heights 103 2 kuga sato varieties and cm cycles 117 3 heights of cm cycles 124 4 heights of heegner cycles 137 5 proof of the main identity and the consequences 147 references 151

borcherds products and arithmetic intersection theory on hilbert modular surfaces we prove an arithmetic version of a theorem of hirzebruch and zagier saying that hirzebruch zagier divisors on a hilbert modular surface are the coecients of an elliptic modular form of weight two moreover we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its petersson metric on a regular model of a hilbert modular surface and study faltings heights of arithmetic hirzebruch zagier divisors

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

theta lifts for lorentzian lattices and coefficients of mock theta functions we evaluate regularized theta lifts for lorentzian lattices in three different ways in particular we obtain formulas for their values at special points involving coefficients of mock theta functions by comparing the different evaluations we derive recurrences for the coefficients of mock theta functions such as hurwitz class numbers andrews spt function and ramanujan s mock theta functions

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

dynamics of neuronal waves first of all by studying the existence and stability of traveling wave fronts of the following nonlinear nonlocal model equation u t au x u alpha mathop int mathbb r k x y h u y t theta rm d y beta mathop int mathbb r k x y h u y t theta rm d y we derive relation between speed index function and stability index function for each of the waves this model was derived when studying working memory in synaptically coupled neuronal networks which incorporates low persistent activity rate and high saturating rate we will investigate dynamics of neuronal waves for this purpose we will be concerned with the equation for several different kinds of symmetric and asymmetric kernels and will compare speeds of the waves our analysis and results on the speed index functions facilitate our investigation on stability of the waves and the estimates of speeds secondly we are concerned with standing waves of the nonlinear nonhomogeneous system of integral differential equations begin array lll u t u w alpha mathop int limits mathbb r k x y h u y t theta rm d y beta mathop int limits mathbb r k x y h u y t theta rm d y mathcal i x t w t varepsilon u gamma w end array and the scalar equation begin array lll u t u alpha mathop int limits mathbb r k x y h u y t theta rm d y beta mathop int limits mathbb r k x y h u y t theta rm d y mathcal i x t end array

heights of kudla rapoport divisors and derivatives of l functions we study special cycles on integral models of shimura varieties associated with unitary similitude groups of signature n 1 1 we construct an arithmetic theta lift from harmonic maass forms of weight 2 n to the arithmetic chow group of the integral model of a unitary shimura variety by associating to a harmonic maass form f a linear combination of kudla rapoport divisors equipped with the green function given by the regularized theta lift of f our main result is an equality of two complex numbers 1 the height pairing of the arithmetic theta lift of f against a cm cycle and 2 the central derivative of the convolution l function of a weight n cusp form depending on f and the theta function of a positive definite hermitian lattice of rank n 1 when specialized to the case n 2 this result can be viewed as a variant of the gross zagier formula for shimura curves associated to unitary groups of signature 1 1 the proof relies on among other things a new method for computing improper arithmetic intersections

asymptotic stability of solitary waves we show that the family of solitary waves 1 solitons of the korteweg de vries equation partial t u u partial x u partial x 3 u 0 is asymptotically stable our methods also apply for the solitary waves of a class of generalized korteweg de vries equations partial t u partial x f u partial x 3 u 0 in particular we study the case wheref u up 1 p 1 p 1 2 3 and 3 0 withf c4 the same asymptotic stability result for kdv is also proved for the casep 2 the modified korteweg de vries equation we also prove asymptotic stability for the family of solitary waves for all but a finite number of values ofp between 3 and 4 the solitary waves are known to undergo a transition from stability to instability as the parameterp increases beyond the critical valuep 4 the solution is decomposed into a modulating solitary wave with time varying speedc t and phase t bound state part and an infinite dimensional perturbation radiating part the perturbation is shown to decay exponentially in time in a local sense relative to a frame moving with the solitary wave asp 4 the local decay or radiation rate decreases due to the presence of aresonance pole associated with the linearized evolution equation for solitary wave perturbations

channel coding rate in the finite blocklength regime this paper investigates the maximal channel coding rate achievable at a given blocklength and error probability for general classes of channels new achievability and converse bounds are given which are tighter than existing bounds for wide ranges of parameters of interest and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100 it is also shown analytically that the maximal rate achievable with error probability isclosely approximated by c v n q 1 where c is the capacity v is a characteristic of the channel referred to as channel dispersion and q is the complementary gaussian cumulative distribution function

restriction of coherent hilbert eisenstein series in pure appl math q 1 305 340 2005 yang constructed a family of coherent hilbert eisenstein series and asked several questions concerning the span of their diagonal restrictions in this paper we will answer one of them where the construction essentially comes from a biquadratic number field k surprisingly the span depends on the non vanishing of the central critical value of the l function associated to a weight 2 eigenform

on two geometric theta lifts the theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type in this paper we establish for the orthogonal group o p 2 an adjointness result between borcherds s singular theta lift and the kudla millson lift we extend this result to arbitrary signature by introducing a new singular theta lift for o p q on the geometric side this lift can be interpreted as a differential character in the sense of cheeger and simons for the cycles under consideration

on achievable rates of awgn energy harvesting channels with block energy arrival and non vanishing error probabilities this paper investigates the achievable rates of an additive white gaussian noise energy harvesting eh channel with an infinite battery the eh process is characterized by a sequence of blocks of harvested energy which is known causally at the source the harvested energy remains constant within a block while the harvested energy across different blocks is characterized by a sequence of independent and identically distributed random variables the blocks have length l which can be interpreted as the coherence time of the energy arrival process if l is a constant or grows sublinearly in the blocklength n we fully characterize the first order term in the asymptotic expansion of the maximum transmission rate subject to a fixed tolerable error probability varepsilon the first order term is known as the varepsilon capacity in addition we obtain lower and upper bounds on the second order term in the asymptotic expansion which reveal that the second order term is proportional to l n 1 2 for any varepsilon less than 1 2 the lower bound is obtained through analyzing the save and transmit strategy if l grows linearly in n we obtain lower and upper bounds on the varepsilon capacity which coincide whenever the cumulative distribution function of the eh random variable is continuous and strictly increasing in order to achieve the lower bound we have proposed a novel adaptive save and transmit strategy which chooses different save and transmit codes across different blocks according to the energy variation across the blocks

a nonsplit sum of coefficients of modular forms we introduce and study certain truncated sums of hecke eigenvalues of gl2 automorphic forms along quadratic polynomials a power saving estimate is established and new applications to moments of critical l values associated to quadratic fields are derived an application to the asymptotic behavior of the height of heegner points and singular moduli is discussed in detail

heegner divisors l functions and harmonic weak maass forms recent works mostly related to ramanujan s mock theta functions make use of the fact that harmonic weak maass forms can be combinatorial generating functions generalizing works of waldspurger kohnen and zagier we prove that such forms also serve as generating functions for central values and derivatives of quadratic twists of weight 2 modular l functions to obtain these results we construct differentials of the third kind with twisted heegner divisor by suitably generalizing the borcherds lift to harmonic weak maass forms the connection with periods fourier coefficients derivatives of l functions and points in the jacobian of modular curves is obtained by analyzing the properties of these differentials using works of scholl waldschmidt and gross and zagier

faltings heights of abelian varieties with complex multiplication let m be the shimura variety associated with the group of spinor similitudes of a quadratic space over q of signature n 2 we prove a conjecture of bruinier kudla yang relating the arithmetic intersection multiplicities of special divisors and big cm points on m to the central derivatives of certain l functions as an application of this result we prove an averaged version of colmez s conjecture on the faltings heights of cm abelian varieties

cm values of regularized theta lifts and harmonic weak maas forms of weight 1 we study special values of regularized theta lifts at complex multiplication cm points in particular we show that cm values of borcherds products can be expressed in terms of finitely many fourier coefficients of certain harmonic weak maas forms of weight 1 as it turns out these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve our results imply a conjecture of duke and li and give a new proof of the modularity of a certain arithmetic generating series of weight 1 studied by kudla rapoport and yang

spatially structured activity in synaptically coupled neuronal networks i traveling fronts and pulses we consider traveling front and pulse solutions to a system of integro differential equations used to describe the activity of synaptically coupled neuronal networks in a single spatial dimension our first goal is to establish a series of direct links between the abstract nature of the equations and their interpretation in terms of experimental findings in the cortex and other brain regions this is accomplished first by presenting a biophysically motivated derivation of the system and then by establishing a framework for comparison between numerical and experimental measures of activity propagation speed our second goal is to establish the existence of traveling pulse solutions using more rigorous methods two techniques are presented the first a shooting argument reduces the problem from finding a specific solution to an integro differential equation system to finding any solution to an ode system the second a singular perturbation argument provides a construction of traveling pulse solutions un

faltings heights of cm cycles and derivatives of l functions we study the faltings height pairing of arithmetic special divisors and cm cycles on shimura varieties associated to orthogonal groups we compute the archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a rankin l function we prove the conjecture in certain cases where the shimura variety has dimension 0 1 or 2 in particular we obtain a new proof of the gross zagier formula

an arithmetic intersection formula on hilbert modular surfaces in this paper we obtain an explicit arithmetic intersection formula on a hilbert modular surface between the diagonal embedding of the modular curve and a cm cycle associated to a non biquadratic cm quartic field this confirms a special case of the author s conjecture with j bruinier in by and is a generalization of the beautiful factorization formula of gross and zagier on singular moduli as an application we proved the first non trivial non abelian chowla selberg formula a special case of colmez conjecture

regularized theta lifts for orthogonal groups over totally real fields we define a regularized theta lift from sl2 to orthogonal groups over totally real fields it takes harmonic whittaker forms to automorphic green functions and weakly holomorphic whittaker forms to meromorphic modular forms on orthogonal groups with zeros and poles supported on special divisors generalizing borcherds work on automorphic products to prove our results we use the spectral expansion of the lift and study its relationship with the cohomological theta lift of kudla and millson

propagating activity patterns in thalamic neuronal networks singular perturbation methods are used to construct traveling waves for models of thalamic networks we first study a single layer of mutually inhibitory neurons each of which has the ability to rebound after hyperpolarization two types of waves are constructed smoothly propagating waves and lurching waves which propagate in a saltatory fashion we reduce the existence of these waves to simple boundary value problems or algebraic systems which are solved using the computational package auto the resulting calculations are compared to numerical simulations of the network finally some comments are made concerning two layer networks

information spectrum approach to second order coding rate in channel coding in this paper second order coding rate of channel coding is discussed for general sequence of channels the optimum second order transmission rate with a constant error constraint epsiv is obtained by using the information spectrum method we apply this result to the discrete memoryless case the discrete memoryless case with a cost constraint the additive markovian case and the gaussian channel case with an energy constraint we also clarify that the gallager bound does not give the optimum evaluation in the second order coding rate