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

deformations of galois representations and exceptional monodromy for any simple algebraic group g of exceptional type we construct geometric ell adic galois representations with algebraic monodromy group equal to g in particular producing the first such examples in types mathrm f 4 and mathrm e 6 to do this we extend to general reductive groups ravi ramakrishna s techniques for lifting odd two dimensional galois representations to geometric ell adic representations

on modular representations of bar q q arising from modular forms where g is the galois group gai i q and f is a finite field of characteristic i 3 suppose that p is modular of level n i e that it arises from a weight 2 newform of level dividing n and trivial nebentypus character then p is an odd representation the matrix p c where c is a complex conjugation in g has eigenvalues 1 1 since 1 and 1 are distinct in f p is absolutely irreducible and has a model over every subfield of f containing the set trace p we assume that f has been chosen so that it is generated by this set assume that p is a prime which exactly divides n we write p i n and restrict p to a decomposition group of g for the prime p view the restriction as a representation p p of gal t p qp we say that p is finite at p 34 or 35 w if there is a finite flat f vector space scheme h over zp for which the action of gal t p qp on the f vector space h 1 p gives pp i f p 9 is finite at p if and only if pp is unramified it is clear that p is finite at p whenever p is already modular of level n p in w of 34 cf 35 serre conjectured the converse i e that if p is finite then p is modular of level n p soon thereafter mazur 1 20 proved this conjecture in the case

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

mod l representations of arithmetic fundamental groups i an analog of serre s conjecture for function fields there is a well known conjecture of serre that any continuous irreducible representation gq gl2 f that is odd arises from a newform here gq is the absolute galois group of q and f is an algebraic closure of the finite field f of elements we formulate such a conjecture for n dimensional mod representations of 1 x for any positive integer n and where x is a geometrically irreducible smooth curve over a finite field k of characteristic p p and we prove this conjecture in a large number of cases in fact a proof of all cases of the conjecture for 2 follows from a result announced conditionally by gaitsgory in g the methods are different

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

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

even galois representations and the fontaine mazur conjecture we prove under mild hypotheses that there are no irreducible two dimensional ordinary even galois representations of mathrm gal overline mathbf q mathbf q with distinct hodge tate weights this is in accordance with the fontaine mazur conjecture if k q is an imaginary quadratic field we also prove again under certain hypotheses that mathrm gal overline mathbf q k does not admit irreducible two dimensional ordinary galois representations of non parallel weight

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

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

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

deformations of certain reducible galois representations ii we examine the deformation theory of two dimensional mod p reducible indecomposable galois representations removing or relaxing many of the hypotheses of part i we are able to prescribe local conditions on our deformations by allowing ramification at a set of primes congruent to 1 mod p not 1 mod p 2 and satisfying some other splitting conditions the new hypotheses admit irreducible ordinary characteristic zero lifts of many mod p representations which by skinner wiles allows us to conclude the modularity of such representations

on icosahedral artin representations ii we prove that some new infinite families of odd two dimensional icosahedral representations of the absolute galois group of inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i are modular and hence satsify the artin conjecture we also give an account of work of ramakrishna on lifting mod l galois representations to characteristic zero

modularity of certain potentially barsotti tate galois representations where is the reduction of p these results were subject to hypotheses on the local behavior of p at x i e the restriction of p to a decomposition group at x and to irreducibility hypotheses on 1p in this paper we build on the methods of 461 451 and 121 and relax the hypotheses on local behavior in particular we treat certain adic representations which are not semistable at x but potentially semistable we do this using results of 61 generalizing a theorem of ramakrishna 321 see fontaine mazur 22 131 for a slightly different point of view the results in 61 show that certain potentially barsotti tate deformation problems are smooth allowing us to define certain universal deformations for with the necessary galoistheoretic properties to apply wiles method to carry out the proof that these deformations are indeed realized in the cohomology of modular curves i e that the universal deformation rings are hecke algebras we need to identify the corresponding cohomology groups and prove they have the modular theoretic properties needed to apply wiles method as in 151 and 121 the identification is made by matching local behavior of automorphic representations and galois representations via the local langlands correspondence together with fontaine s theory at the prime we work directly with cohomology of modular curves instead of

deformations of galois representations and exceptional monodromy ii raising the level building on lifting results of ramakrishna khare and ramakrishna proved a purely galois theoretic level raising theorem for odd representations bar rho mathrm gal overline mathbb q mathbb q rightarrow mathrm gl 2 overline mathbb f ell in this paper we generalize these techniques from type a1 to general semi simple groups we then strengthen our previous results on constructing geometric galois representations with exceptional monodromy groups achieving such constructions for almost all ell rather than a density one set and achieving greater flexibility in the hodge numbers of the lifts the latter improvement requires the new level raising result

lower bounds on the energy of maps on demontre que pour la metrique standard sur rp n l application identite est minimisante de l energie dans sa classe d homotopie

information theoretical properties of tsallis entropies a chain rule and a subadditivity for the entropy of type which is one of the nonadditive entropies were derived by daroczy in this paper we study the further relations among tsallis type entropies which are typical nonadditive entropies the chain rule is generalized by showing it for tsallis relative entropy and the nonadditive entropy we show some inequalities related to tsallis entropies especially the strong subadditivity for tsallis type entropies and the subadditivity for the nonadditive entropies the subadditivity and the strong subadditivity naturally lead to define tsallis mutual entropy and tsallis conditional mutual entropy respectively and then we show again chain rules for tsallis mutual entropies we give properties of entropic distances in terms of tsallis entropies finally we show parametrically extended results based on information theory

finiteness theorems for nonnegatively curved vector bundles we prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds more generally we obtain finiteness results for open riemannian manifolds whose topology is concentrated on compact domains of bounded geometry