Series Of Conjectures Research Articles

Mod pq Galois representations and Serre's conjecture

We consider linear representations of the Galois groups of number fields in two different characteristics and examine conditions under which they arise simultaneously from a motive.

The exact bounds for the degree of commutativity of a p-group of maximal class, I

The first major study of p-groups of maximal class was made by Blackburn in 1958. He showed that an important invariant of these groups is the ‘degree of commutativity.’ Recently (1995) Fernández-Alcober proved a best possible inequality for the degree of commutativity in terms of the order of the group. Recent computations for primes up to 43 show that sharper results can be obtained when an additional invariant is considered. A series of conjectures about this for all primes have been recorded in [A. Vera-López et al., preprint]. In this paper, we prove two of these conjectures.

Explicit Determination of the Images of the Galois Representations Attached to Abelian Surfaces with End(A) = Z

We give an effective version of a resuIt reported by Serre asserting that the images of the Galois representations attached to an abelian surface with End(A) = Z are as large as possible for almost every prime. Our algorithm depends on the truth of Serre's conjecture for two-dimensional odd irreducible Galois representations. Assuming this conjecture, we determine the finite set of primes with exceptional image. We also give infinite sets of primes for which we can prove (unconditionally) that the images of the corresponding Galois representations are large. We apply the results to a few examples of abelian surfaces.

A generalization of Serre's conjecture and some related issues

Several topics concerned with multivariate polynomial matrices like unimodular matrix completion, matrix determinantal or primitive factorization, matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original Serre problem (conjecture) solved nonconstructively in 1976 and constructively, more recently. This generalized Serre conjecture is proved to be equivalent to several other unsettled conjectures and, therfore, all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining.

On the modularity of ℚ-curves

A ℚ-curve is an elliptic curve over a number field K which is geometrically isogenous to each of its Galois conjugates. K. Ribet [17] asked whether every ℚ-curve is modular, and he showed that a positive answer would follow from J.-P. Serre's conjecture on mod p Galois representations. We answer Ribet's question in the affirmative, subject to certain local conditions at 3.

Multiplicities and a Dimension Inequality for Unmixed Modules

We prove the following result, which is motivated by the recent work of Kurano and Roberts on Serre's positivity conjecture. Assume that (R,m) is a local ring with finitely generated module M such that R/Ann(M) is quasi-unmixed and contains a field, and that p and q are prime ideals in the support of M such that p is analytically unramified, p+q=m and e(Mp)=e(M). ThendimR/p+dimR/q≤dimM.We also prove a similar theorem in a special case of mixed characteristic. Finally, we provide several examples to explain our assumptions as well as an example of a noncatenary local domain R with prime ideal p such that e(Rp)>e(R)=1.

Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique ≤2; Galois cohomology of quasi-split groups over fields of cohomological dimension ≤ 2

Let k be a perfect field with cohomological dimension ≤ 2. Serre's conjecture II claims that the Galois cohomology set H1(k,G) is trivial for any simply connected semi-simple algebraic G/k and this conjecture is known for groups of type 1A n after Merkurjev–Suslin and for classical groups and groups of type F4 and G2 after Bayer–Parimala. For any maximal torus T of G/k, we study the map H1(k, T) → H1(k, G) using an induction process on the type of the groups, and it yields conjecture II for all quasi-split simply connected absolutely almost k-simple groups with type distinct from E8. We also have partial results for E8 and for some twisted forms of simply connected quasi-split groups. In particular, this method gives a new proof of Hasse principle for quasi-split groups over number fields including the E8-case, which is based on the Galois cohomology of maximal tori of such groups.

An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod p$ cohomology of GL(n,ℤ)

An analogue of Serre's conjecture for Galois representations and Hecke eigenclasses in the mod p$ cohomology of GL(n,ℤ)

Essential dimension and the second serre conjecture for exceptional groups

We study the essential dimension of exceptional connected simply connected algebraic groups over algebraically closed fields. In the present paper, we find upper estimates for the essential dimensions of the groupsF4,E6, andE7. For the groupF4, the upper estimate, thus obtained coincides with the known lower estimate. We also prove the second Serre conjecture for the groupE6 and for a function field.

The Positivity of Intersection Multiplicities and Symbolic Powers of Prime Ideals

Serre's nonnegativity conjecture for intersection multiplicities has recently been proven by O. Gabber. In this paper we investigate Serre's positivity conjecture using the methods which he developed. We show in particular that the positivity conjecture has implications for properties of symbolic powers of prime ideals in regular local rings.

S4andS4Extensions of [formula omitted] Ramified at Only One Prime

For each primepin a certain family of odd primes, we construct anS4extension of Q unramified outsidep. We show that for allp≡3 (mod8) in our family, thisS4extension embeds in anS4extension, which is also unramified outsidep. Invoking Serre's conjecture (in a proven case) allows us to relate the splitting of primes in these extensions to certain modular forms of level 1.

Remarks on Methods of Fontaine and Faltings

In this note we prove the non existence of certain irreducible two dimensional representations of the the absolute Galois group. Such results are predicted by Serre's conjecture and we use Fontaine's methods to verify these predictions in a small number of cases. We also use a methods of Faltings to study certain finiteness conjectures for Galois representations. Key words: Serre's conjecture, Fontaine-Mazur conjectures, Galois representations.

Serre's Conjecture for Imaginary Quadratic Fields

We study an analog over an imaginary quadratic field K of Serre's conjecture for modular forms. Given a continuous irreducible representation ρ:Gal(Q/K) →GL2(Fl) we ask if ρ is modular. We give three examples of representations ρ obtained by restriction of even representations of Gal(Q/Q). These representations appear to be modular when viewed as representations over K, as shown by the computer calculations described at the end of the paper.

Hamiltonian Formalism in Quantum Mechanics

Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties are formulated in a series of Conjectures. To Vladimir Igorevich Arnol’d with admiration, on occasion of his 60th birthday.

Base Change, Lifting, and Serre's Conjecture

Base Change, Lifting, and Serre's Conjecture

Cayley-Bacharach theorems and conjectures

A theorem of Pappus of Alexandria, proved in the fourth century A.D., began a long development in algebraic geometry. In its changing expressions one can see reflected the changing concerns of the field, from synthetic geometry to projective plane curves to Riemann surfaces to the modern development of schemes and duality. We survey this development historically and use it to motivate a brief treatment of a part of duality theory. We then explain one of the modern developments arising from it, a series of conjectures about the linear conditions imposed by a set of points in projective space on the forms that vanish on them. We give a proof of the conjectures in a new special case.

Locational Avoidance by Nonmetropolitan Industry

The concept of locational avoidance is applied to the process of nonmetropolitan industrialisation by means of a periodisation involving two phases of locational avoidance. During the phase of Fordist mass production, certain labour-intensive industries decentralised to low-wage non-metropolitan areas to avoid locating close to other firms where there was a danger of wages subsequently being bid up. In the present phase, characterised by a tendency towards flexible accumulation, a new wave of more capital-intensive industries has sought out nonmetropolitan areas, again displaying a pattern of locational avoidance, but mainly in order to retain the human capital invested in their skilled labour force. This second dispersed arrangement of industry stands in stark contrast to the flexible production agglomerations that have been formed elsewhere in new industrial spaces during the same period, even though both were produced under regimes of flexible accumulation. A series of conjectures exploring these ideas is examined in light of the locational behaviour of firms locating in the Georgian Bay nonmetropolitan area north of Toronto.

Note on the Serre conjecture theorem of McGibbon and Neisendorfer

In this paper McGibbon-Neisendorfer's Serre Conjecture Theorem is improved. As corollaries, we apply our results to the homotopy groups of H-spaces.

Algorithmic aspects of Suslin's proof of Serre's conjecture

LetF be a unimodularr×s matrix with entries beingn-variate polynomials over an infinite fieldK. Denote by deg(F) the maximum of the degrees of the entries ofF and letd=1+deg(F). We describe an algorithm which computes a unimodulars×s matrixM with deg(M)=(rd) O(n) such thatFM=[I r ,O], where [I r ,O] denotes ther×s matrix obtained by adding to ther×r unit matrixI r s−r zero columns. We present the algorithm as an arithmetic network with inputs fromK, and we count field operations and comparisons as unit cost. The sequential complexity of our algorithm amounts to $$S^{O(r^2 )} r^{O(n^2 )} d^{O(n^2 + r^2 )} $$ field operations and comparisons inK whereas its parallel complexity isO(n 4 r 4log2(srd)). The complexity bounds and the degree bound for deg(M) mentioned above are optimal in order. Our algorithm is inspired by Suslin's proof of Serre's Conjecture.

PROVING PROPERTIES OF RULE-BASED SYSTEMS

Rule-based systems are being applied to tasks of increasing responsibility. Deductive methods are being applied to their validation, to detect flaws in these systems and to enable us to use them with more confidence. Each system of rules is encoded as a set of axioms that define the system theory. The operation of the rule language and information about the subject domain are also described in the system theory. Validation tasks, such as establishing termination, unreachability, or consistency, or verifying properties of the system, are all phrased as conjectures. If we succeed in establishing the validity of the conjecture in the system theory, we have carried out the corresponding validation task. If the proof is restricted to be sufficiently constructive, we may extract from it information other than a simple yes/no answer. For example, we may obtain a description of a situation in which an error or anomaly may occur. A method for the gradual formulation of specifications based on the attempted proof of a series of conjectures has been found to be suitable for rule-based systems. Such a specification can serve as the basis for a reengineering of the system using conventional software technology. Validation conjectures are proved or disproved by a new theorem-proving system, SNARK, which implements (nonclausal) resolution and paramodulation, an optional constructive restriction, and some facilities for proof by induction. The system has already been applied to prove properties of a number of simple rule-based systems.