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

goryachev chaplygin kovalevskaya and brdicka eardley nappi witten pp waves spacetimes with higher rank stackel killing tensors hidden symmetries of the goryachev chaplygin and kovalevskaya gyrostats spacetimes as well as the brdicka eardley nappi witten pp waves are studied we find out that these spacetimes possess higher rank stackel killing tensors and that in the case of the pp wave spacetimes the symmetry group of the stackel killing tensors is the well known newton hooke group

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

constructing semisimple p adic galois representations with prescribed properties consider a characteristic p representation of the absolute galois group of the rational numbers in this paper we show how to deform this representation to the p adics while guaranteeing that the characteristic polynomials of frobenius at a density one set of primes are algebraic and pure of specified weight the resulting representation is ramified at an infinite density zero set of primes as a consequence of the technique of proof we show that one can compatibly lift a mod pq representation again the resulting representation is ramified at infinitely many primes

counterexample to boundary regularity of a strongly pseudoconvex cr submanifold an addendum to the paper of harvey lawson clearly for any c f restricted on the line v c is an embedding outside the two points 0 c and 1 c f sends 0 t and 1 t to 0 t 0 for all t now take s which is the boundary of a ball b u v c2 u v 2 it is easy to see that the mapping f restricted on s is still an embedding the image of s under f is a strongly pseudoconvex cr manifold in c3 the variety that f s bounds is f b observe that f b has curve singularities along the line 0 t 0 we remark that f c2 is a hypersurface x y z c3 z2 zx x3 0 in c3

congruences between modular forms raising the level and dropping euler factors abstract we discuss the relationship among certain generalizations of results of hida ribet and wiles on congruences between modular forms hida s result accounts for congruences in terms of the value of an l function and ribet s result is related to the behavior of the period that appears there wiles theory leads to a class number formula relating the value of the l function to the size of a galois cohomology group the behavior of the period is used to deduce that a formula at nonminimal level is obtained from one at minimal level by dropping euler factors from the l function

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

infinitely ramified galois representations in this paper we show how to construct for most p 5 two types of surjective representations p gq gal q q gl2 zp that are ramified at an infinite number of primes the image of inertia at almost all of these primes will be torsion free the first construction is unconditional the catch is that we cannot say whether p igp gal qp q is crystalline or even potentially semistable the second construction assumes the generalized riemann hypothesis grh with this assumption we can further arrange that p igp is crystalline at p we remark that infinitely ramified reducible representations have been previously constructed by more elementary means

a rigid analytic gross zagier formula and arithmetic applications let f be a newform of weight 2 and squarefree level n its fourier coefficients generate a ring of whose fraction field kf has finite degree over q fix an imaginary quadratic field k of discriminant prime to n corresponding to a dirichlet character e the l series l f k s l f s l f 0 e s of f over k has an analytic continuation to the whole complex plane and a functional equation relating l f k s to l f k 2 s assume that the sign of this functional equation is 1 so that l f k s vanishes to even order at s 1 this is equivalent to saying that the number of prime factors of n which are inert in k is odd fix any such prime say p the field k determines a factorization n n nof n by taking n resp nto be the product of all the prime factors of n which are split resp

deformations and the rigidity method abstract in d rohrlich false division towers of elliptic curves j algebra 229 1 2000 249 279 d rohrlich a deformation of the tate module j algebra 229 1 2000 280 313 rohrlich proved rigidity for psl 2 z p t for p 5 obtained this group as a galois group over c t using modular function fields and derived from this interesting consequences for galois representations attached to the tate modules of elliptic curves furthermore in an unpublished preprint he established that the corresponding galois representation g c t gal c t alg c t psl 2 z p t is universal here we will turn things around we first provide a general framework for rigid deformations of projective representations of the absolute galois group of a function field in one variable over a separably closed base under natural rather general hypothesis we will determine the corresponding universal deformation ring if the residual representation is geometrically rigid which happens to be the case for many surjective representation to psl 2 f p p 2 which arise from belyi triples then certain universal deformations will be geometrically rigid too this will give new proofs for most of the results of rohrlich our method also applies to thompson tuples we then go on to give two further applications which are based on the example computed by rohrlich over f q t where q is a power of a prime l we construct infinite p adic galois extensions which have finite ramification and whose constant field is finite furthermore for p 5 and p 1 mod 4 we obtain a family of surjective galois representations gal q alg q 1 sl 2 z p 1 where the parameter runs over all p power roots of unity finally we exhibit a general class of rigid universal deformation rings which are finite flat over z p in particular this shows that the above examples of galois representations are not a singular event but a general phenomenon

on the density of modular points in universal deformation spaces based on comparison theorems for hecke algebras and universal deformation rings with strong restrictions at the critical prime l as provided by the results of wiles taylor diamond et al we prove under rather general conditions that the corresponding universal deformation spaces with no restrictions at l can be identified with certain hecke algebras of l adic modular forms as conjectured by gouvea thus generalizing previous work of gouvea and mazur along the way we show that the universal deformation spaces we consider are complete intersections flat over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i l of relative dimension three in which the modular points form a zariski dense subset furthermore the fibers above inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i l of these spaces are generically smooth

variation of iwasawa invariants in hida families let gq gl2 k be an absolutely irreducible modular galois representation over a finite field k of characteristic p assume further that is p ordinary and p distinguished in the sense that the restriction of to a decomposition group at p is reducible and non scalar the hida family h of is the set of all p ordinary p stabilized newforms f with mod p galois representation isomorphic to if is unramified at p then one must also fix an unramified line in and require that the ordinary line of f reduces to this fixed line these newforms are a dense set of points in a certain p adic analytic space of overconvergent eigenforms consisting of an intersecting system of branches i e irreducible components t a indexed by the minimal primes a of a certain hecke algebra to each modular form f h one may associate the iwasawa invariants an f an f alg f and alg f the analytic resp algebraic invariants are the number of zeroes of the p adic l function resp of the characteristic power series of the dual of the selmer group of f while the invariants are the exponents of the powers of p dividing the same objects in this paper we prove the following results on the behavior of these iwasawa invariants as f varies over h

kohn rossi cohomology and its application to the complex plateau problem ii let x be a compact connected strongly pseudoconvex cr manifold of real dimension 2n 1 in c n 1 tanaka introduced a spectral sequence e p q r x with e p q 1 x being the kohn rossi cohomology group and e k 0 2 x being the holomorphic de rham cohomology denoted by h k h x we study the holomorphic de rham cohomology in terms of the s invariant of the isolated singularities of the variety v bounded by x we give a characterization of the singularities with vanishing s invariants for n 3 yau used the kohn rossi cohomology groups to solve the classical complex plateau problem in 1981 for n 2 the problem has remained unsolved for over a quarter of a century in this paper we make progress in this direction by putting some conditions on x so that v will have very mild singularities specifically we prove that if dim x 3 and h 2 h x 0 then x is a boundary of complex variety v with only isolated quasi homogeneous singularities such that the dual graphs of the exceptional sets in the resolution are star shaped and all curves are rational

on higher dimensional complex plateau problem let x be a compact connected strongly pseudoconvex cr manifold of real dimension 2n 1 in mathbb c n it has been an interesting question to find an intrinsic smoothness criteria for the complex plateau problem for n ge 3 and n n 1 yau found a necessary and sufficient condition for the interior regularity of the harvey lawson solution to the complex plateau problem by means of kohn rossi cohomology groups on x in 1981 for n 2 and n ge n 1 the first and third authors introduced a new cr invariant g 1 1 x of x the vanishing of this invariant will give the interior regularity of the harvey lawson solution up to normalization for n ge 3 and n n 1 the problem still remains open in this paper we generalize the invariant g 1 1 x to higher dimension as g lambda n 1 x and show that if g lambda n 1 x 0 then the interior has at most finite number of rational singularities in particular if x is calabi yau of real dimension 5 then the vanishing of this invariant is equivalent to give the interior regularity up to normalization

eisenhart lift for euler s problem of two fixed centers abstract we apply the approach of eisenhart lift to the classical euler s bicentric system yielding a three dimensional geodesic system an extra potential is added to the geodesic hamiltonian to preserve the separability of the system we also make a generalization of the system by jacobi method the stackel separability of all of these systems are verified their hamilton jacobi equations are integrated in principle the explicit forms of the killing tensors on the associated riemannian manifolds are also shown

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

non hyperelliptic modular jacobians of dimension 3 we present a method to solve in an efficient way the problem of constructing the curves given by torelli s theorem in dimension 3 over the complex numbers for an absolutely simple principally polarized abelian threefold a over c given by its period matrix compute a model of the curve of genus three unique up to isomorphism whose jacobian equipped with its canonical polarization is isomorphic to a as a principally polarized abelian variety we use this method to describe the non hyperelliptic modular jacobians of dimension 3 we investigate all the non hyperelliptic new modular jacobians jac c f of dimension 3 which are isomorphic to a f where f s new 2 x 0 n n 4000

the iwasawa main conjectures for gl2 we prove the one two and three variable iwasawa greenberg main conjectures for a large class of modular forms that are ordinary with respect to an odd prime p the method of proof involves an analysis of an eisenstein ideal for ordinary hida families for gu 2 2

on serre s conjecture for 2 dimensional mod p representations of gal q q we prove the existence in many cases of minimally ramified p adic lifts of 2 dimensional continuous odd absolutely irreducible mod p representations of the absolute galois group of q it is predicted by serre s conjecture that such representations arise from newforms of optimal level and weight

exact solution of the goryachev chaplygin problem in quantum mechanics the quantum mechanical analogue of the classical integrable system originally founded by goryachev and chaplygin 1900 is considered in detail the problem is formulated in terms of the euclid e 3 group the euler poisson equations of motion and their integrals are derived the determination of the spectra of the integrals of motion is equivalent to construction of the special basis of representation of e 3 the eigenvalue problem admits separation in new variables which are closely connected with two boson creation annihilation operators they are the same as in the majorana representation of the lorentz group in the new variables the quantum mechanical equations of motion look similar to the classical ones the constants of motion are determined by the spectral problems for the two jacobi type tridiagonal infinite matrices some numerical results are given

universal deformation rings and fusion abstract we study in this paper the extent to which one can detect fusion in certain finite groups from information about the universal deformation rings r v of absolutely irreducible f p modules v the we consider are extensions of either abelian or dihedral groups g of order prime to p by an elementary abelian p group of rank 2

some remarks on the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication we establish several results towards the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields namely i the existence of an appropriate p adic l function building on works of hida and perrin riou ii the basic structure theory of the dual selmer group following works of coates hachimori venjakob et al and iii the implications of dihedral or anticyclotomic main conjectures with basechange the result of i is deduced from the construction of hida and perrin riou which in particular is seen to give a bounded distribution the result of ii allows us to deduce a corank formula for the p primary part of the tate shafarevich group of an elliptic curve in the z p 2 extension of an imaginary quadratic field finally iii allows us to deduce a criterion for one divisibility of the two variable main conjecture in terms of specializations to cyclotomic characters following a suggestion of greenberg as well as a refinement via basechange c 2011 elsevier inc all rights reserved

geometrical classification of killing tensors on bidimensional flat manifolds valence two killing tensors in the euclidean and minkowski planes are classified under the action of the group which preserves the type of the corresponding killing web the classification is based on an analysis of the system of determining partial differential equations for the group invariants and is entirely algebraic the approach allows one to classify both characteristic and noncharacteristic killing tensors

lattices in the cohomology of shimura curves we prove the main conjectures of breuil j reine angew math 2012 including a generalisation from the principal series to the cuspidal case and dembele j reine angew math 2012 subject to a mild global hypothesis that we make in order to apply certain r mathbb t theorems more precisely we prove a multiplicity one result for the mod p cohomology of a shimura curve at iwahori level and we show that certain apparently globally defined lattices in the cohomology of shimura curves are determined by the corresponding local p adic galois representations we also indicate a new proof of the buzzard diamond jarvis conjecture in generic cases our main tools are the geometric breuil mezard philosophy developed in emerton and gee j inst math jussieu 2012 and a new and more functorial perspective on the taylor wiles kisin patching method along the way we determine the tamely potentially barsotti tate deformation rings of generic two dimensional mod p representations generalising a result of breuil and mezard bull soc math de france 2012 in the principal series case

on meromorphic solutions of f2 g2 1 we show that meromorphic solutions f g of f2 g2 1 in c2 must be constant if fz2 and gz1 have the same zeros counting multiplicities we also apply the result to characterize meromorphic solutions of certain nonlinear partial differential equations

deformations of induced galois representations the object of this article is to study the behavior of certain deformation problems and hecke rings under base change to a real quadratic field these questions were the subject of conjectures by doi hida ishii see dhi98 and hid98 in this paper we will show that conjecture 2 2 of hid98 holds under a suitable hypothesis and that the isomorphism predicted by conjecture 3 8 of dhi98 holds up to a pseudo null cokernel the deformations studied in this paper are also interesting in light of the recent work by skinner and wiles see sw97 and sw99 on the deformation of reducible galois representations namely the ring r studied in hid98 and dhi98 turns out to be closely related to a pseudodeformation ring the explication of this connection is crucial to our results furthermore the central conjecture of hid98 turns out to be related to a certain class number condition which is exactly analogous to the condition that appears in sw97 finally we would like to point out that the deformations considered here give rise to a number of interesting examples specifically we are able to exhibit adic representations whose traces generate a nontrivial extension of and with the property that the specializations to certain arithmetic points of weight one are actually ramified over spec this is in contrast to the well known theorem of hida which asserts that arithmetic points of weight k 2 are smooth over spec to state the results more precisely we consider a real quadratic extension f q d of q with d 0 write for a generator of the group gal f q we fix a prime p 2 with d p 1 such that p pp in f with p p let fp denote the maximal algebraic extension of f unramified outside p and infinity let h gal fp f and let g gal fp q now consider a character h f

a local to global principle for deformations of galois representations given an absolutely irreducible galois representation ge gln k e a number eld k a nite eld of characteristic l 2 and a nite set of places q of e containing all places above l and1 and all where ramies there have been dened many functors representing strict equivalence classes of deformations of such a representation e g by mazur or wiles in 15 or 26 with various conditions on the behaviour of the deformations at the places in q and with the condition that the deformations are unramied outside q those functors are known to be representable for as above our goal is to present a rather general class of global deformation functors that satisfy local deformation conditions and to investigate for those under what conditions the global deformation functor is determined by the local deformation functors we will give precise conditions under which the local functors for all places in q are sucient to describe the global functor rst in a coarse form then in a rened form using auxiliary primes as done by taylor and wiles in 24 this has several consequences the strongest is that one can derive ring theoretic results for the universal deformation space by mazur if one uses results of diamond and wiles c f 11 and 26 and if one has a good understanding of all local situations furthermore it is easier to understand what happens under increasing the ramication

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

calculation of values of l functions associated to elliptic curves we calculated numerically the values of l functions of four typical elliptic curves in the critical strip in the range im s 0 which enables us to calculate for large im s furthermore we remark that a relation exists between sato tate conjecture and the generalized riemann hypothesis

various numerical invariants for isolated singularities definition 0 let v be a stein analytic space with x as its only singular point let 7r m v be a resolution of the singularity of v we shall denote dim h1 m 9 1 i n 1 by h i and dim hq m up for 1 p n 1 q n by hp q m so far as the classification problem is concerned h n i is one of the most important invariants in this paper we shall introduce a bunch of invariants cf definition 2 6 definition 4 1 and definition 5 1 which are naturally attached to isolated singularities these invariants are used to characterize the different notions of sheaves of germs of holomorphic differential forms on analytic spaces various formulae which relate all these invariants are proved we also show how to calculate these invariants explicitly our paper is organized as follows in section 2 we discuss the relationship between three different kinds of sheaves of germs of holomorphic forms introduced by grauert grothendieck noether and ferrari respectively and the dualizing sheaf on a complex analytic space which admits only isolated singularities we remark that the torsion sheaf of the sheaf of germs of holomorphic p forms in the sense of grauert grothendieck was studied by brieskorn 9 greuel 48 kantor 19 suzuki 38 vetter 39 and the author 43 in section 3 we relate the invariant g n l1 of a hypersurface isolated singularity with analytic invariants and topological invariants of any resolution of the singularity the noether s formula for the rank two bundle