zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

how porous is the graph of brownian motion we prove that the graph of brownian motion is almost surely porous and determine the hausdorff dimension of sets with a given porosity index in particular we show that the porosity index of the graph is yo 0 6948

backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

torsion points on cm elliptic curves over real number fields we study torsion subgroups of elliptic curves with complex multiplication cm defined over number fields which admit a real embedding we give a complete classification of the groups which arise up to isomorphism as the torsion subgroup of a cm elliptic curve defined over a number field of odd degree there are infinitely many restricting to the case of prime degree we show that there are only finitely many isomorphism classes more precisely there are six olson groups which arise as torsion subgroups of cm elliptic curves over number fields of every degree and there are precisely 17 non olson cm elliptic curves defined over a prime degree number field

non parametric sets of regular realizations over number fields abstract given a number field k we show that for many finite groups g all the galois extensions of k with galois group g cannot be obtained by specializing any given finitely many galois extensions e k t with galois group g and e k regular our examples include abelian groups dihedral groups symmetric groups general linear groups over finite fields etc we also provide a similar conclusion while specializing any given infinitely many galois extensions e k t with galois group g and e k regular of a certain type under a conjectural uniform faltings theorem

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

fourier coefficients of cusp forms of half integral weight let mathfrak f be a cusp form of half integral weight whose fourier coefficients mathfrak a mathfrak f n are all real we study the sign change problem of mathfrak a mathfrak f n when n runs over some specific sets of integers lower bounds of the best possible order of magnitude are established for the number of those coefficients that have the same signs these give an improvement on some recent results of bruinier and kohnen modular forms on schiermonnikoong cambridge university press cambridge 57 66 2008 and kohnen int j number theory 6 1255 1259 2010

selmer companion curves we say that two elliptic curves e1 e2 over a number field k are n selmer companions for a positive integer n if for every quadratic character of k there is an isomorphism seln e 1 k seln e 2 k between the nselmer groups of the quadratic twists e 1 e 2 we give sufficient conditions for two elliptic curves to be n selmer companions and give a number of examples of non isogenous pairs of companions

the spin of prime ideals fixing a nontrivial automorphism of a number field k we associate to ideals in k an invariant with values in 0 1 which we call the spin and for which the associated l function does not possess euler products we are nevertheless able using the techniques of bilinear forms to handle spin value distribution over primes obtaining stronger results than the analogous ones which follow from the technology of l functions in its current state the initial application of our theorem is to the arithmetic statistics of selmer groups of elliptic curves

selmer groups of quadratic twists of elliptic curves 1 1 e y a1xy a3y x 3 a2x 2 a4x a6 where a1 a2 a3 a4 a6 z let n e denote the conductor of e j e the j invariant of e and l e s n 1 a n n s the hasse weil l function of e if e is modular then let fe z n 1 ae n q n s2 n e 1 be the associated weight 2 cusp form here 1 denotes the trivial dirichlet character throughout d will denote a square free integer and d shall denote the kronecker character for the field q d let h d denote the order of cl d the ideal class group of q d if l is prime then define h d l by

root numbers and ranks in positive characteristic abstract for a global field k and an elliptic curve e over k t silverman s specialization theorem implies rank e k t rank e t k for all but finitely many t p 1 k if this inequality is strict for all but finitely many t the elliptic curve e is said to have elevated rank all known examples of elevated rank for k q rest on the parity conjecture for elliptic curves over q and the examples are all isotrivial some additional standard conjectures over q imply that there does not exist a non isotrivial elliptic curve over q t with elevated rank in positive characteristic an analogue of one of these additional conjectures is false inspired by this for the rational function field k u over any finite field with characteristic 2 we construct an explicit 2 parameter family e c d of non isotrivial elliptic curves over k t depending on arbitrary c d such that under the parity conjecture each e c d has elevated rank

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

the circle method and bounds for l functions i let f be a hecke maass or holomorphic primitive cusp form of arbitrary level and nebentypus and let chi be a primitive character of conductor m for the twisted l function l s f otimes chi we establish the hybrid subconvex bound begin aligned l left frac 1 2 it f otimes chi right ll m 3 t frac 1 2 frac 1 18 varepsilon end aligned for t in mathbb r the implied constant depends only on the form f and varepsilon

small gaps between prime numbers the work of goldston pintz yildirim in early 2005 dan goldston janos pintz and cem yildirim 12 made a spectacular breakthrough in the study of prime numbers resolving a long standing open problem they proved that there are infinitely many primes for which the gap to the next prime is as small as we want compared to the average gap between consecutive primes before their work it was known only that there were infinitely many gaps which were about a quarter the size of the average gap the new result may be viewed as a step towards the famous twin prime conjecture that there are infinitely many prime pairs p and p 2 the gap here being 2 the smallest possible gap between primes perhaps most excitingly their work reveals a connection between the distribution of primes in arithmetic progressions and small gaps between primes assuming certain admittedly difficult conjectures on the distribution of primes in arithmetic progressions they are able to prove the existence of infinitely many prime pairs that differ by at most 16 the aim of this article is to explain some of the ideas involved in their work let us begin by explaining the main question in a little more detail the number of primes up to x denoted by x is roughly x log x for large values of x this is the celebrated prime number theorem therefore if we randomly choose an integer near x then it has about a 1 in log x chance of being prime in other words as we look at primes around size x the average gap between consecutive primes is about log x as x increases the primes get sparser and the gap between consecutive primes tends to increase here are some natural questions about these gaps between prime numbers do the gaps always remain roughly about size log x or do we sometimes get unexpectedly large gaps and sometimes surprisingly small gaps can we say something about the statistical distribution of these gaps that is can we quantify how often the gap is between say log x and log x given 0 except for the primes 2 and 3 clearly the gap between consecutive primes must be even does every even number occur infinitely often as a gap between consecutive primes for example the twin prime conjecture says that the

trilinear forms with kloosterman fractions abstract we give new bounds for a m n m n a e a m n where m n and a are arbitrary coefficients improving upon a result of duke friedlander and iwaniec 7 we also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations

le probleme de tchebychev pour le douzieme polynome cyclotomique let p n denote the largest prime factor of the integer n and 12 n n 4 n 2 1 we prove that for x large enough we have p x x 1 c with c 10 26531

brownian motion at a slow point on rappelle des resultats de greenwood et perkins 1983 84 et on etudie les proprietes probabilistes et de trajectoire du processus x t b t t b t on montre en particulier que x est un processus de markov non homogene satisfaisant une certaine equation differentielle stochastique

an additive problem in the fourier coefficients of cusp forms we establish an estimate on sums of shifted products of fourier coefficients coming from holomorphic or maass cusp forms of arbitrary level and nebentypus these sums are analogous to the binary additive divisor sum which has been studied extensively as an application we derive extending work of duke friedlander and iwaniec a subconvex estimate on the critical line for l functions associated to character twists of these cusp forms

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

average rank in families of quadratic twists a geometric point of view we investigate the average rank in the family of quadratic twists of a given elliptic curve defined over mathbb q when the curves are ordered using the canonical height of their lowest non torsion rational point

on multi symplectic partitioned runge kutta methods for hamiltonian wave equations many conservative pdes such as various wave equations schrodinger equations kdv equations and so on allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes in this note we show the discretization to hamiltonian wave equations in space and time using two symplectic partitioned runge kutta methods respectively leads to multi symplectic integrators which preserve a symplectic conservation law under some conditions we discuss the energy and momentum conservative property of partitioned runge kutta methods for the wave equations with a quadratic potential

rankin selberg l functions in the level aspect keywords moments rankin selberg convolution level aspect convexity breaking reference tan article 2002 003doi 10 1215 s0012 7094 02 11416 1 record created on 2008 11 14 modified on 2017 05 12

on bombieri s asymptotic sieve if a sequence an of non negative real numbers has best possible distribution in arithmetic progressions bombieri showed that one can deduce an asymptotic formula for the sum n x a n k n for k 2 by constructing appropriate sequences we show that any weakening of the well distribution property is not sufficient to deduce the same conclusion

on brownian slow points it is shown that for a wiener process x t both the quantities mathop inf limits t mathop overline lim limits h text to text 0 x t text text h x t sqrt h text and mathop text sup limits t mathop underline lim limits h text to text 0 text x t text text h x t sqrt h are almost surely equal to 1

ranks of twists of elliptic curves and hilbert s tenth problem in this paper we investigate the 2 selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields we give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2 selmer rank and we give lower bounds for the number of twists with bounded conductor that have a given 2 selmer rank as a consequence under appropriate hypotheses we can find many twists with trivial mordell weil group and assuming the shafarevich tate conjecture many others with infinite cyclic mordell weil group using work of poonen and shlapentokh it follows from our results that if the shafarevich tate conjecture holds then hilbert s tenth problem has a negative answer over the ring of integers of every number field

bertini theorems over finite fields let x be a smooth quasiprojective subscheme of pn of dimension m i 0 over fq then there exist homogeneous polynomials f over fq for which the intersection of x and the hypersurface f 0 is smooth in fact the set of such f has a positive density equal to aex m 1 1 where aex s zx q s is the zeta function of x an analogue for regular quasiprojective schemes over z is proved assuming the abc conjecture and another conjecture

elements of class groups and shafarevich tate groups of elliptic curves the problem of estimating the number of imaginary quadratic fields whose ideal class group has an element of order l 2 is classical in number theory analogous questions for quadratic twists of elliptic curves have been the focus of recent interest whereas works of c stewart and j top st and of f gouvea and b mazur gm address the nontriviality of mordell weil groups less is known about the nontriviality of shafarevich tate groups here we obtain a new type of result for the nontriviality of class groups of imaginary quadratic fields using the circle method and then we combine it with works of g frey f v kolyvagin k and k ono o2 to obtain results on the nontriviality of shafarevich tate groups of certain elliptic curves for e x0 11 these results imply that x d 0 d fundamental and e d 5 0 x3 5log 2x

the size of selmer groups for the congruent number problem where np is the number of solutions of the congruence y x d2x mod p then ld s has an analytic continuation as an entire function on the complex plane the conjecture of birch and swinnerton dyer then states in particular that the rank r d of ed is equal to the order r d of ld s at s 1 this being the so called analytic rank while we cannot at present find r d in all cases we can at least determine whether or not ld 1 0 and hence conjecturally whether or not r d 0 moreover one has a functional equation for ld s which relates its values at s and 2 s via a sign change ed 1 one may deduce that 1 r d ed it would then follow from the conjecture of birch and swinnerton dyer that the rank is positive whenever ed 1 according to the calculations of birch and stephens 2 one has

prime values of a 2 p 4 a 2 p 4 we prove an asymptotic formula for the number of primes of the shape a 2 p 4 thereby refining the well known work of friedlander and iwaniec ann math 2 148 3 945 1040 1998 along the way we prove a result on equidistribution of primes up to x in which the moduli may be almost as large as x 2

heegner zeros of theta functions heegner divisors play an important role in number theory however little is known on whether a modular form has heegner zeros in this paper we start to study this question for a family of classical theta functions and prove a quantitative result which roughly says that many of these theta functions have a heegner zero of discriminant 7 this leads to some interesting questions on the arithmetic of certain elliptic curves which we also address here

on ranks of twists of elliptic curves and power free values of binary forms let e be an elliptic curve over q by the rank of e we shall mean the rank of the group of rational points of e mestre 31 improving on the work of neron 34 cf 13 39 and 46 has shown that there is an infinite family of elliptic curves over q with rank at least 12 however computational work see 3 4 7 and 47 suggests that a typical elliptic curve will have much smaller rank with curves of rank 0 or 1 being predominant indeed brumer 6 has proved subject to the birch swinnerton dyer conjecture the shimura taniyama weil conjecture and the generalized riemann hypothesis that the average rank of an elliptic curve ordered according to its faltings height is at most 2 3 in this article we shall study the behaviour of the rank as we run over twists of a given elliptic curve over q that is we shall restrict our attention to families of elliptic curves defined over q which are isomorphic over c there are families of quadratic cubic quartic and sextic twists see for example proposition 5 4 of chapter x of 42 let e be an elliptic curve over q with weierstrass equation y2 x ax b and for any non zero integer d let ed 2 3 denote a quadratic twist of e given by the equation dy2 x ax b let r d denote the rank of ed note that if d1 and d2 are non zero integers then ed is isomorphic to ed over q if and only if d1 d2 is the square of a rational number subject to the conjectures of birch and swinnerton dyer and of shimura taniyama and weil goldfeld 14 conjectured in 1979 that

the multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this article we consider partitioned runge kutta prk methods for hamiltonian partial differential equations pdes and present some sufficient conditions for multi symplecticity of prk methods of hamiltonian pdes

a conditioned limit theorem for random walk and brownian local time on square root boundaries we count the number of times a random walk exits from a square root boundary and show that the normalized counting process and the normalized random walk converge jointly in law to a local time whose inverse is a stable subordinator of a known index and a brownian motion the study of this limit process leads to some precise sample path properties of brownian motion these properties improve earlier results of dvoretsky and kahane on the existence of small oscillations in the brownian path

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points ii we prove a general subconvex bound in the level aspect for rankin selberg l functions associated with two primitive holomorphic or maass cusp forms over q we use this bound to establish the equidistribution of incomplete galois orbits of heegner points on shimura curves associated with indefinite quaternion algebras over q

the square free sieve and the rank of elliptic curves let e be an elliptic curve over q a celebrated theorem of mordell asserts that e q the abelian group of rational points of e is finitely generated by the rank of e we mean the rank of e q thus the rank of e is positive if and only if e possesses an infinity of rational points relatively few general qualitative assertions can be made about the rank as e varies how large can the rank get although we expect that there are elliptic curves over q with arbitrarily high ranks this is presently unknown what is the average size of the rank recently brumer and mcguinness have reported on their study of 310 716 elliptic curves of prime conductor less than 108 where they have found that 20 06 of those curves have even rank 2 cf bm also see forthcoming work of brumer where subject to a number of standard conjectures he shows that 2 3 is an upper bound for the average rank for all elliptic curves over q ordered in terms of their faltings height what is the behavior of the rank over the family of twists of a given elliptic curve here one has three natural kinds of families of twists 1 quadratic twists one can take any elliptic curve and systematically twist it by all quadratic characters this is the type of family of elliptic curves we are concerned with in this paper specifically if e is an elliptic curve over q given by the weierstrass equation y2 x a x b and d is any squarefree integer the quadratic twist of e by d ed is given by the equation

on multisymplectic integrators based on runge kutta nystrom methods for hamiltonian wave equations abstract many conservative partial differential equations pdes such as wave equations schrodinger equations kdv equations maxwell equations and so on allow for a multisymplectic formulation which can be regarded as a generalization of the symplectic structure of hamiltonian ordinary differential equations odes in this note for hamiltonian wave equations we show the discretization in space and time using two symplectic runge kutta nystrom srkn methods respectively leads to a multisymplectic integrator which can preserve a discrete multisymplectic conservation law moreover we discuss the energy and momentum conservative properties of the multisymplectic integrator for the wave equations with a quadratic potential

breaking the 12 barrier for the twisted second moment of dirichlet l functions abstract we study the second moment of dirichlet l functions to a large prime modulus q twisted by the square of an arbitrary dirichlet polynomial we break the 1 2 barrier in this problem and obtain an asymptotic formula provided that the length of the dirichlet polynomial is less than q 51 101 q 1 2 1 202 as an application we obtain an upper bound of the correct order of magnitude for the third moment of dirichlet l functions we give further results when the coefficients of the dirichlet polynomial are more specialized

on the hausdorff dimension of the brownian slow points for each c1 c2 consider the set of one sided slow points of the brownian path b defined by s c 1 c 2 omega t exists delta 0 text b t h b t in c 1 h 1 2 c 2 h 1 2 forall 0 underset raise0 3em hbox smash scriptscriptstyle leqslant h underset raise0 3em hbox smash scriptscriptstyle leqslant delta we find the hausdorff dimension of s c1 c2 as well as s c1 c2 a where a a progressively measurable set satisfying certain hypotheses in particular the hausdorff dimension of the two sided slow points s c1 c2 and the one sided and two sided slow points in the zero set of b are obtained by making an appropriate choice of a these results show for example that s c c o or o according as c is greater than or less than the smallest positive zero of m 1 2 1 2 x2 2 1 3069 where m is the confluent hypergeometric function this confirms a conjecture of burgess davis whose methods play a major role in the proofs the results are also extended to higher dimensions

siegel cusp forms of degree 2 are determined by their fundamental fourier coefficients we prove that a siegel cusp form of degree 2 for the full modular group is determined by its set of fourier coefficients a s with 4 det s ranging over odd squarefree integers as a key step to our result we also prove that a classical cusp form of half integral weight and level 4n with n odd and squarefree is determined by its set of fourier coefficients a d with d ranging over odd squarefree integers a result that was previously known only for hecke eigenforms

a derived category approach to generic vanishing we prove a generic vanishing theorem for coherent sheaves on an abelian variety over an algebraically closed field k when k c this implies a conjecture of green and lazarsfeld

galois representations attached to q curves and the generalized fermat equation a 4 b 2 c p we prove that the equation a 4 b 2 c p has no solutions in coprime positive integers when p 211 the main step is to show that for all sufficiently large primes p every q curve over an imaginary quadratic field k with a prime of potentially multiplicative reduction greater than 6 has a surjective mod p galois representation the bound on p depends on k and the degree of the isogeny between e and its galois conjugate but is independent of the choice of e the proof of this theorem combines geometric arguments due to mazur momose darmon and merel with an analytic estimate of the average special values of certain l functions

bombieri vinogradov for multiplicative functions and beyond the x1 2 barrier abstract part and parcel of the study of multiplicative number theory is the study of the distribution of multiplicative functions in arithmetic progressions although appropriate analogies to the bombieri vingradov theorem have been proved for particular examples of multiplicative functions there has not previously been headway on a general theory seemingly none of the different proofs of the bombieri vingradov theorem for primes adapt well to this situation in this article we find out why such a result has been so elusive and discover what can be proved along these lines and develop some limitations for a fixed residue class a we extend such averages out to moduli x 20 39

equidistribution of eisenstein series on geodesic segments abstract we show an asymptotic formula for the l 2 norm of the eisenstein series e z 1 2 i t restricted to a segment of a geodesic connecting infinity and an arbitrary real for generic geodesics of this form the asymptotic formula shows that the eisenstein series satisfies restricted que on the other hand for rational geodesics the eisenstein series does not satisfy restricted quantum ergodicity as an application we show that the zero set of the eisenstein series intersects every such geodesic segment provided t is large we also make analogous conjectures for the maass cusp forms in particular we predict that cusp forms do not satisfy restricted quantum ergodicity for rational geodesics

on the negative pell equation we give asymptotic upper and lower bounds for the number of squarefree d 0 d x such that the equation x 2 dy 2 1 is solvable these estimates as usual can equivalently be interpreted in terms of real quadratic fields with a fundamental unit with norm 1 and give strong evidence in the direction of a conjecture due to p stevenhagen