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

hypergeometric functions over finite fields in this paper the analogy between the character sum expansion of a complex valued function over gf q and the power series expansion of an analytic function is exploited in order to develop an analogue for hypergeometric series over finite fields it is shown that such functions satisfy many summation and transformation formulas analogous to their classical counterparts

modularity of the rankin selberg l series and multiplicity one for sl 2 let f g be primitive cusp forms holomorphic or otherwise on the upper half plane h of levels n m respectively with unitarily normalized l functions l s f equation and l s g equation when p does not divide n resp m the inverse roots p p resp p p are nonzero with sum ap resp bp for every p prime to nm set lp s f g 1 p pp s 1 p pp s 1 p pp s 1 p pp s 1 let l s f g denote the incomplete euler product of lp s f g over all p not dividing nm this is closely related to the convolution l series sum over n 1 a sub n b sub n n s whose miraculous properties were first studied by rankin and selberg

galois representations with quaternion multiplication associated to noncongruence modular forms in this paper we study the compatible family of degree 4 scholl representations associated with a space s of weight 2 noncongruence cusp forms satisfying quaternion mul tiplications over a biquadratic extension of q it is shown that is automorphic that is its associated l function has the same euler factors as the l function of an automorphic form for gl4 over q further it yields a relation between the fourier coefficients of noncongruence cusp forms in s and those of certain automorphic forms via the three term atkin and swinnerton dyer congruences

extensions of the multiplicity conjecture the multiplicity conjecture of herzog huneke and srinivasan states an upper bound for the multiplicity of any graded k algebra as well as a lower bound for cohen macaulay algebras in this note we extend this conjecture in several directions we discuss when these bounds are sharp find a sharp lower bound in the case of not necessarily arithmetically cohen macaulay one dimensional schemes of 3 space and propose an upper bound for finitely generated graded torsion modules we establish this bound for torsion modules whose codimension is at most two

on l independence of algebraic monodromy groups in compatible systems of representations consider a profinite group g and a collection of continuous representations g gln q indexed by a set l of rational primes suppose that g is endowed with a dense subset of frobenius elements f a the system is called a compatible system of adic representations if for every a the characteristic polynomial of f has coefficients in the field of rational numbers and does not depend on in 6 5 we will give a more precise and less restrictive definition which allows us to throw out some bad pairs in order to accomodate ramification our notion slightly generalizes serre s original definition 14 to recover serre s definition we take g to be the galois group of a number field k and f to be frobenius representatives for primes of k let g be the zariski closure of g in gln q this is the algebraic monodromy group at our question is the following how does g vary with one hopes for some kind of independence at best there can exist a global algebraic group g gln q such that every g is conjugate to g q q unfortunately this does not always happen in the abstract setting in which we work so we must settle for weaker independence results we first recall what is already known in this direction the compatibility condition bears only on the semisimple part of the elements f so we lose no information by assuming all to be semisimple representations alternatively we could take arbitrary representations and define g as the quotient of the zariski closure of g by its unipotent radical thus g is a family of reductive groups serre has proved

some global generalizations of the birkhoff kellogg theorem and applications on etudie certaines generalisations globales du theoreme de birkhoff kellogg et on donne des applications

maximality of galois actions for compatible systems in the sense of serre 19 of course the image of is not generally zariski dense in gln q so the maximality condition must be formulated relative to the zariski closure g of gal k k one might hope that the image of is a maximal compact subgroup of g q but this is too optimistic the center causes problems and we do not even know a priori that g is reductive to formulate a maximality conjecture which avoids such problems we introduce maps g g g

potential automorphy and change of weight we prove an automorphy lifting theorem for l adic representations where we impose a new condition at l which we call potentential diagonalizability this result allows for change of weight and seems to be substantially more exible than previous theorems along the same lines we derive several applications for instance we show that any irreducible totally odd essentially self dual regular weakly compatible system of l adic representations of the absolute galois group of a totally real eld is potentially automorphic and hence is pure and its l function has meromorphic continuation to the whole complex plane and satises the expected functional equation

on atkin and swinnerton dyer congruence relations 2 in this paper we give an example of a noncongruence subgroup whose three dimensional space of cusp forms of weight 3 has the following properties for each of the four residue classes of odd primes modulo 8 there is a basis whose fourier coefficients at infinity satisfy a three term atkin and swinnerton dyer congruence relation which is the p adic analogue of the three term recursion satisfied by the coefficients of classical hecke eigenforms we also show that there is an automorphic l function over mathbb q whose local factors agree with those of the l adic scholl representations attached to the space of noncongruence cusp forms

boundary conditions for condensing mappings on etudie la condition limite f x x 0 pour des applications condensantes on considere la situation 0 1

new approaches to bounding the multiplicity of an ideal abstract we use the theory of resolutions for a given hilbert function to investigate the multiplicity conjectures of huneke and srinivasan and herzog and srinivasan to prove the conjectures for all modules with a particular hilbert function we show that it is enough to prove the statements only for elements at the bottom of the partially ordered set of resolutions with that hilbert function this enables us to test the conjectured upper bound for the multiplicity efficiently with the computer algebra system macaulay 2 and we verify the upper bound for many artinian modules in three variables with small socle degree moreover with this approach we show that though numerical techniques have been sufficient in several of the known special cases they are insufficient to prove the conjectures in general finally we apply a result of herzog and srinivasan on ideals with a quasipure resolution to prove the upper bound for cohen macaulay quotients by ideals with generators in high degrees relative to the regularity

modularity of 2 adic barsotti tate representations we prove a modularity lifting theorem for two dimensional 2 adic potentially barsotti tate representations this proves hypothesis h of khare wintenberger and completes the proof of serre s conjecture the main new ingredient is a classification of connected finite flat group schemes over rings of integers of finite extensions of 2

a generalization of guo s theorem and applications as an application of this theorem the guo s fixed point theorem on the compression and expansion of a domain can be deduced see 2 in this paper we extend this theorem to the case when a is a strict setcontraction mapping and give some applications it should be noticed that the guo s theorem mentioned above was repeatedly proved by the same method in 3 theorem 1 and 3 also tried to extend this theorem to strict set contractions and condensing mappings but as petryshyn pointed out in 4 this try was unsuccessful in fact theorems 2 and 3 in 3 which are the main results of 3 are incorrect

modular forms and k3 surfaces abstract for every known hecke eigenform of weight 3 with rational eigenvalues we exhibit a k3 surface over q associated to the form this answers a question asked independently by mazur and van straten the proof builds on a classification of cm forms by the second author

an extension of guo s theorem via k ψ contractive retractions abstract let x be a infinite dimensional banach space we generalize guo s theorem d j guo eigenvalues and eigenvectors of nonlinear operators chinese ann math 2 1981 65 80 english to k contractions and condensing mappings under a condition which depends on the infimum k of all k 1 for which there exists a k contractive retraction of the closed unit ball of the space x onto its boundary