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

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

platonic polyhedra topological constraints and periodic solutions of the classical n body problem we prove the existence of a number of smooth periodic motions u of the classical newtonian n body problem which up to a relabeling of the n particles are invariant under the rotation group mathcal r of one of the five platonic polyhedra the number n coincides with the order mathcal r of mathcal r and the particles have all the same mass our approach is variational and u is a minimizer of the lagrangian action mathcal a on a suitable subset mathcal k of the h1t periodic maps u 3n the set mathcal k is a cone and is determined by imposing on u both topological and symmetry constraints which are defined in terms of the rotation group mathcal r there exist infinitely many such cones mathcal k all with the property that mathcal a mathcal k is coercive for a certain number of them using level estimates and local deformations we show that minimizers are free of collisions and therefore classical solutions of the n body problem with a rich geometric kinematic structure

the number of limit cycles from a cubic center by the melnikov function of any order abstract in this paper we consider the system x y 1 x 2 p x y y x 1 x 2 q x y where p x y and q x y are arbitrary quadratic polynomials we study the maximum number of limit cycles bifurcating from the periodic orbits by using the melnikov function of any order we prove that the upper bound for the number of limit cycles is 3 and reached

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 planar minimizers and rotational symmetry in the n body problem abstract the newtonian n body problem admits uniformly rotating relative equilibrium solutions in the plane but not in r 3 when the bodies are allowed to interact in r 3 is it preferable in terms of action to leave the plane and follow a non planar trajectory we use the variational techniques of chenciner and venturelli celestial mech dyn astro 77 2000 139 to show that for an open set of masses there is a class of collision free action minimizing orbits of certain rotational symmetry in the four body problem which are non coplanar i e the planar relative equilibrium is not the least action solution among orbits in r 3 both periodic and quasi periodic solutions are constructed in this way we also discuss constructing collision free action minimizing solutions possessing d rotational symmetry along with various other symmetry constraints

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

highly eccentric hip hop solutions of the 2n body problem abstract we show the existence of families of hip hop solutions in the equal mass 2 n body problem which are close to highly eccentric planar elliptic homographic motions of 2 n bodies plus small perpendicular non harmonic oscillations by introducing a parameter the homographic motion and the small amplitude oscillations can be uncoupled into a purely keplerian homographic motion of fixed period and a vertical oscillation described by a hill type equation small changes in the eccentricity induce large variations in the period of the perpendicular oscillation and give rise via a bolzano argument to resonant periodic solutions of the uncoupled system in a rotating frame for small 0 the topological transversality persists and brouwer s fixed point theorem shows the existence of this kind of solutions in the full system

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

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

global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the n body problem abstract given n point masses turning in a plane at a constant speed this paper deals with the global bifurcation of periodic solutions for the masses in that plane and in space as a special case one has a complete study of n identical masses on a regular polygon and a central mass the symmetries of the problem are used in order to find the irreducible representations the linearization and with the help of the orthogonal degree theory all the symmetries of the bifurcating branches

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

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

on the existence of infinitely many periodic solutions to some problems of n body type we prove the existence of infinitely many periodic solutions with prescribed period to a class of problems of n body type

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

periodic orbits from second order perturbation via rational trigonometric integrals abstract the second order poincare pontryagin melnikov perturbation theory is used in this paper to study the number of bifurcated periodic orbits from certain centers this approach also allows us to give the shape and the period up to the first order we address these problems for some classes of abel differential equations and quadratic isochronous vector fields in the plane we prove that two is the maximum number of hyperbolic periodic orbits bifurcating from the isochronous quadratic centers with a birational linearization under quadratic perturbations of second order in particular the configurations 2 0 and 1 1 are realizable when two centers are perturbed simultaneously the required computations show that all the considered families share the same iterated rational trigonometric integrals

using melnikov functions of any order for studying limit cycles abstract in this work we study a class of planar hamiltonian systems perturbed with general quadratic polynomials in order to identify the number of limit cycles employing melnikov functions of any order we show that the class of systems can have at most three limit cycles and the limit is reached

symmetry groups and non planar collisionless action minimizing solutions of the three body problem in three dimensional space periodic and quasi periodic solutions of the n body problem can be found as minimizers of the lagrangian action functional restricted to suitable spaces of symmetric paths the main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three body problem in three dimensional space first we give a finite complete list of symmetry groups fitting to the minimization of the action with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives a second step is to prove that the resulting local and global symmetric action minimizers are always collisionless when they are not already bound to collisions furthermore we prove some results which address the question of whether minimizers are planar or non planar as a consequence of our theory we will give general criteria for a symmetry group to yield planar or homographic minimizers either homographic or not as in the chenciner montgomery eight solution on the other hand we will provide a rigorous proof of the existence of some interesting one parameter families of periodic and quasi periodic non planar orbits these include the choreographic marchal s p12 family with equal masses together with a less symmetric choreographic family which anyway probably coincides with the p12 family

binary decompositions for planar n body problems and symmetric periodic solutions a binary decomposition for a system of n masses is a way of treating the system as big n atop 2 big binaries with the total action exactly the same as that of the original system by considering binary decompositions we are able to provide effective lower bound estimates for the action of collision paths in several spaces of symmetric loops as applications we use our estimates to prove the existence of some new classes of symmetric periodic solutions for the n body problem

entire minimal parabolic trajectories the planar anisotropic kepler problem we continue the variational approach to parabolic trajectories introduced in our previous paper barutello et al entire parabolic trajectories as minimal phase transitions arxiv 1105 3358v1 2011 which sees parabolic orbits as minimal phase transitions we deepen and complete the analysis in the planar case for homogeneous singular potentials we characterize all parabolic orbits connecting two minimal central configurations as free time morse minimizers in a given homotopy class of paths these may occur for at most one value of the homogeneity exponent in addition we link this threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of fixed end problems and also with the existence of action minimizing periodic trajectories with nontrivial homotopy type

transitive decomposition of symmetry groups for the n body problem periodic and quasi periodic solutions of the n body problem are critical points of the action functional constrained to the sobolev space of symmetric loops variational methods yield collisionless orbits provided the group of symmetries fulfills certain conditions such as the rotating circle property here we generalize such conditions to more general group types and show how to constructively classify all groups satisfying such hypothesis by a decomposition into irreducible transitive components as examples we show approximate trajectories of some of the resulting symmetric minimizers

the third order melnikov function of a quadratic center under quadratic perturbations abstract we study quadratic perturbations of the integrable system 1 x d h where h x 2 y 2 2 we prove that the first three melnikov functions associated to the perturbed system give rise at most to three limit cycles

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

on the existence of collisionless equivariant minimizers for the classical n body problem we show that the minimization of the lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n body problem provided that some simple conditions on the symmetry group are satisfied more precisely we give a fairly general condition on symmetry groups g of the loop space for the n body problem with potential of homogeneous degree with 0 which ensures that the restriction of the lagrangian action mathcal a to the space g of g equivariant loops is coercive and its minimizers are collisionless without any strong force assumption in proving that local minima of g are free of collisions we develop an averaging technique based on marchal s idea of replacing some of the point masses with suitable shapes see 10 as an application several new orbits can be found with some appropriate choice of g furthermore the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits

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

a class of reversible quadratic systems with piecewise polynomial perturbations abstract this paper investigates a class of reversible quadratic systems perturbed inside piecewise polynomial differential systems of arbitrary degree n all possible phase portraits of the reversible quadratic systems on the plane are obtained by analytic techniques then we present the algebraic structure of its corresponding first order melnikov function which can be used to study the limit cycle bifurcation problem in this direction we use it to investigate the hopf bifurcation of the perturbed systems and obtain an upper bound for the hopf cyclicity meanwhile we employ it to discuss the maximal number of limit cycles emerging from the periodic orbits around the origin for a special case and also provide its upper bound

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

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

removing collision singularities from action minimizers for the n body problem with free boundaries for the planar and spatial n body problems it has been proved by marchal and chenciner that solutions for the minimizing problem with fixed ends are free from interior collisions this important result has been extended by ferrario terracini to newtonian type problems and equivariant problems it has also been used to construct many symmetric solutions for the n body problem

periodic solutions to some problems of n body type we prove the existence of at least one t periodic solution to a dynamical system of the type m i ddot u i sum limits j 1 j ne i n triangledown v ij u i u j text t 1 where the potentials v ij are t periodic in t and singular at the origin u i e r k i 1 n and k 3 we also provide estimates on the h 1 norm of this solution the proofs are based on a variant of the ljusternik schnirelman method the results here generalize to the n body problem some results obtained by bahri rabinowitz on the 3 body problem in 6