almost all elliptic curves are serre curves using a multidimensional large sieve inequality we obtain a bound for the mean square error in the chebotarev theorem for division fields of elliptic curves that is as strong as what is implied by the generalized riemann hypothesis as an application we prove that according to height almost all elliptic curves are serre curves where a serre curve is an elliptic curve whose torsion subgroup roughly speaking has as much galois symmetry as possible

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

on the quantum filtration of the khovanov rozansky cohomology abstract we prove the quantum filtration on the khovanov rozansky link cohomology h p with a general degree n 1 monic potential polynomial p x is invariant under reidemeister moves and construct a spectral sequence converging to h p that is invariant under reidemeister moves whose e 1 term is isomorphic to the khovanov rozansky sl n cohomology h n then we define a generalization of the rasmussen invariant and study some of its properties we also discuss relations between upper bounds of the self linking number of transversal links in standard contact s 3

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

a slice genus lower bound from sl n khovanov rozansky homology we show that a generic perturbation of the doubly graded khovanov rozansky knot homology gives rise to a lower bound on the slice genus of a knot we prove a theorem about obtainable presentations of surfaces embedded in 4 space which we use to simplify significantly our algebraic computations

finite element analysis of non linear vibration of a circular plate with a concentric rigid mass abstract a finite element analysis of non linear vibration of an elastic circular plate with a concentric rigid mass is examined for different cases of boundary conditions the non linear partial differential equations and the associated boundary conditions are derived from an energy method by using the hamiltonian principle the applications of the finite element method to the dynamic problem rely on the use of a variational principle to derive the necessary element properties or equations the assembled equations for the plate are formed by summing each element equation obtained in consideration of a single element then the boundary conditions are imposed on the vector of nodal field variables so that the appropriate boundary conditions are satisfied the assembled equations form an eigenvalue problem and are solved for the eigenvalues as well as the unknown field variables the relations between the fundamental frequencies and the amplitudes of non linear vibrations of the circular isotropic elastic plate with a concentric rigid mass are obtained the results agree well with the solutions obtained by the kantorovich averaging method also the results yield the upper bounds of the solution to the stated problem

an endomorphism of the khovanov invariant abstract we construct an endomorphism of the khovanov invariant to prove h thinness and pairing phenomena of the invariants for alternating links as a consequence it follows that the khovanov invariant of an oriented nonsplit alternating link is determined by its jones polynomial signature and the linking numbers of its components

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

non linear vibration of a hinged circular plate with a concentric rigid mass abstract a computational analysis of the non linear vibration of a hinged elastic circular plate with a concentric rigid mass is examined for movable and immovable hinged cases the non linear partial differential equations are converted into the corresponding non linear ordinary differential equations by elimination of the time variable with a kantorovich time averaging method then by using a newton iteration technique and the concept of analytic continuation the solutions of the fundamental responses of the non linear vibration of the circular plate with a concentric rigid mass are obtained

geometrically nonlinear vibrations of rectangular plates carrying a concentrated mass abstract nonlinear forced vibrations of rectangular plates carrying a central concentrated mass are studied the plate is assumed to have immovable edges and rotational springs numerical results are presented for clamped plates the von karman nonlinear plate theory is used but in plane inertia in both the plate and the mass is retrained the problem is discretized into a multi degree of freedom dof system by using an energy approach and lagrange equations taking damping into account a pseudo arclength continuation method is used in order to obtain numerical solutions results are presented as both i frequency amplitude curves and ii time domain responses the effect of gravity and the effect of the consequent initial plate deflection are also investigated

elliptic curves with large rank over function fields we produce explicit elliptic curves over fp t whose mordell weil groups have arbitrarily large rank our method is to prove the conjecture of birch and swinnerton dyer for these curves or rather the tate conjecture for related elliptic surfaces and then use zeta functions to determine the rank in contrast to earlier examples of shafarevitch and tate our curves are not isotrivial asymptotically these curves have maximal rank for their conductor motivated by this fact we make a conjecture about the growth of ranks of elliptic

on the rank of an elliptic surface nagao has recently given a conjectural limit formula for the rank of an elliptic surface e in terms of a weighted average of fibral frobenius trace values we show that tate s conjecture on the order of vanishing of l 2 e s essentially implies nagao s formula in particular we prove nagao s formula for rational elliptic surfaces in the case that e is a twist we reduce nagao s and tate s conjectures to the case of products of curves and we verify the conjectures for many new classes of elliptic surfaces of kodaira dimension 0 and 1

a categorification of the jones polynomial author s khovanov mikhail abstract we construct a bigraded cohomology theory of links whose euler characteristic is the jones polynomial

rasmussen s spectral sequences and the sln concordance invariants combining known spectral sequences with a new spectral sequence relating reduced and unreduced sln homology yields a relationship between the homflypt homology of a knot and its sln concordance invariants as an application some of the sln concordance invariants are shown to be linearly independent

quasipositivity as an obstruction to sliceness for an oriented link l s 3 d 4 let s l be the greatest euler characteristic f of an oriented 2 manifold f without closed components smoothly embedded in d 4 with boundary l a knot k is slice if s k 1 realize d 4 in c 2 as z w z 2 w 2 1 it has been conjectured that if v is a nonsingular complex plane curve transverse to s 3 then s v s 3 v d 4 kronheimer and mrowka have proved this conjecture in the case that v d 4 is the milnor fiber of a singularity i explain how this seemingly special case implies both the general case and the slice bennequin inequality for braids as applications i show that various knots are not slice e g pretzel knots like p 3 5 7 all knots obtained from a positive trefoil o 2 3 by iterated untwisted positive doubling

new quantum obstructions to sliceness it is well known that generic perturbations of the complex frobenius algebra used to define khovanov cohomology each give rise to rasmussen s concordance invariant ss this gives a concordance homomorphism to the integers and a strong lower bound on the smooth slice genus of a knot similar behavior has been observed in sl n sl n khovanov rozansky cohomology where a perturbation gives rise to the concordance homomorphisms snsn for each n 2n 2 and where we have s2 ss2 s we demonstrate that snsn for n 3n 3 does not in fact arise generically and that varying the chosen perturbation gives rise both to new concordance homomorphisms and to new sliceness obstructions that are not equivalent to concordance homomorphisms

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

the effect of convolving families of l functions on the underlying group symmetries let fn and gm be families of primitive automorphic l functions for gln aq and glm aq respectively such that as n m the statistical behavior 1 level density of the low lying zeros of l functions in fn resp gm agrees with that of the eigenvalues near 1 of matrices in g1 resp g2 as the size of the matrices tend to infinity where each gi is one of the classical compact groups unitary u symplectic sp or orthogonal o so even so odd assuming that the convolved families of l functions fn gm are automorphic we study their 1 level density we also study convolved families of the form f gm for a fixed f under natural assumptions on the families which hold in many cases we can associate to each family l of l functions a symmetry constant cl equal to 0 resp 1 or 1 if the corresponding low lying zero statistics agree with those of the unitary resp symplectic or orthogonal group our main result is that cf g cf cg the symmetry type of the convolved family is the product of the symmetry types of the two families a similar statement holds for the convolved families f gm we provide examples built from dirichlet l functions and holomorphic modular forms and their symmetric powers an interesting special case is to convolve two families of elliptic curves with positive rank in this case the symmetry group of the convolution is independent of the ranks in accordance with the general principle of multiplicativity of the symmetry constants but the ranks persist before taking the limit n m as lower order terms

khovanov homology and the slice genus we use lee s work on the khovanov homology to define a knot invariant s we show that s k is a concordance invariant and that it provides a lower bound for the smooth slice genus of k as a corollary we give a purely combinatorial proof of the milnor conjecture

on the heegaard floer homology of branched double covers abstract let l s 3 be a link we study the heegaard floer homology of the branched double cover l of s 3 branched along l when l is an alternating link hf of its branched double cover has a particularly simple form determined entirely by the determinant of the link for the general case we derive a spectral sequence whose e 2 term is a suitable variant of khovanov s homology for the link l converging to the heegaard floer homology of l

the ozsvath szabo and rasmussen concordance invariants are not equal in this paper we present several counterexamples to rasmussen s conjecture that the concordance invariant coming from khovanov homology is equal to twice the invariant coming from ozsvath szabo floer homology the counterexamples are twisted whitehead doubles of 2 2n 1 torus knots