Let $\mathbb{F}$ be a finite real abelian extension of $\mathbb{Q}$. Let $M$ be an odd positive integer. For every squarefree positive integer $r$ the prime factors of which are congruent to 1 modulo $M$ and split completely in $\mathbb{F}$, the corresponding Kolyvagin class ${{k}_{r}}\in {{\mathbb{F}}^{\times }}/{{\mathbb{F}}^{\times M}}$ satisfies a remarkable and crucial recursion which for each prime number $\ell $ dividing $r$ determines the order of vanishing of ${{k}_{r}}$ at each place of $\mathbb{F}$ above $\ell $ in terms of ${{k}_{r/\ell }}$. In this note we give the recursion a new and universal interpretation with the help of the double complex method introduced by Anderson and further developed by Das and Ouyang. Namely, we show that the recursion satisfied by Kolyvagin classes is the specialization of a universal recursion independent of $\mathbb{F}$ satisfied by universal Kolyvagin classes in the group cohomology of the universal ordinary distribution à la Kubert tensored with $\mathbb{Z}/M\mathbb{Z}$. Further, we show by a method involving a variant of the diagonal shift operation introduced by Das that certain group cohomology classes belonging (up to sign) to a basis previously constructed by Ouyang also satisfy the universal recursion.

Soit $G$ un groupe fini agissant sur une courbe algébrique projective et lisse $X$ sur un corps algébriquement clos $k$. Dans cet article, on donne une formule de Riemann-Roch pour la caractéristique d'Euler équivariante d'un $G$-faisceau inversible $\mathcal{L}$, à valeurs dans l'anneau ${{R}_{k}}\left( G \right)$ des caractères du groupe $G$. La formule donnée a un bon comportement fonctoriel en ce sens qu'elle relève la formule classique le long du morphisme dim: ${{R}_{k}}\left( G \right)\to \mathbb{Z}$, et est valable même pour une action sauvage. En guise d'application, on montre comment calculer explicitement le caractère de l'espace des sections globales d'une large classe de $G$-faisceaux inversibles, en s'attardant sur le cas particulier délicat du faisceau des différentielles sur la courbe.

A topology on $\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on $\mathbb{Z}$, which includes the $p$-adics, and one in which the set of rational primes $\mathbb{P}$ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and $k$-free numbers.

For a given Sturm-Liouville equation whose leading coefficient function changes sign, we establish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues (unbounded from both below and above) for a separated self-adjoint boundary condition can be numbered in terms of the Prüfer angle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also relate this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.

An $E$-ring is a unital ring $R$ such that every endomorphism of the underlying abelian group ${{R}^{+}}$ is multiplication by some ring element. The existence of almost-free $E$-rings of cardinality greater than ${{2}^{{{\aleph }_{0}}}}$ is undecidable in ZFC. While they exist in Gödel's universe, they do not exist in other models of set theory. For a regular cardinal ${{\aleph }_{1}}\le \text{ }\!\!\lambda\!\!\text{ }\le {{2}^{{{\aleph }_{0}}}}$ we construct $E$-rings of cardinality $\lambda $ in ZFC which have ${{\aleph }_{1}}$-free additive structure. For $\text{ }\!\!\lambda\!\!\text{ }={{\aleph }_{1}}$ we therefore obtain the existence of almost-free $E$-rings of cardinality ${{\aleph }_{1}}$ in ZFC.

We develop an explicit skein-theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the homology of $U\left( 1 \right)$-representation varieties on the one side and the combinatorially constructed Hennings TQFT based on the quasitriangular Hopf algebra $\mathcal{N=}\mathbb{Z}\text{/2}\ltimes {{\wedge }^{*}}{{\mathbb{R}}^{2}}$ on the other side. We find that both TQFT's are $\text{SL}\left( 2,\,\mathbb{R} \right)$-equivariant functors and, as such, are isomorphic. The $\text{SL}\left( 2,\,\mathbb{R} \right)$-action in the Hennings construction comes from the natural action on $\mathcal{N}$ and in the case of the Frohman–Nicas theory from the Hard–Lefschetz decomposition of the $U\left( 1 \right)$-moduli spaces given that they are naturally Kähler. The irreducible components of this TQFT, corresponding to simple representations of $\text{SL}\left( 2,\,\mathbb{Z} \right)\,\text{and}\,\text{Sp}\left( 2g,\,\mathbb{Z} \right)$, thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg–Witten theories, Casson type theories for homology circles à la Donaldson, higher rank gauge theories following Frohman and Nicas, and the $\mathbb{Z}/p\mathbb{Z}$ reductions of Reshetikhin–Turaev theories over the cyclotomic integers $\mathbb{Z}[{{\zeta }_{p}}]$. We also conjecture that the Hennings TQFT for quantum-$\mathfrak{s}{{\mathfrak{l}}_{2}}$ is the product of the Reshetikhin–Turaev TQFT and such a homological TQFT.

We define a total ordering of the pure braid groups which is invariant under multiplication on both sides. This ordering is natural in several respects. Moreover, it well-orders the pure braids which are positive in the sense of Garside. The ordering is defined using a combination of Artin's combing technique and the Magnus expansion of free groups, and is explicit and algorithmic.

By contrast, the full braid groups (on 3 or more strings) can be ordered in such a way as to be invariant on one side or the other, but not both simultaneously. Finally, we remark that the same type of ordering can be applied to the fundamental groups of certain complex hyperplane arrangements, a direct generalization of the pure braid groups.

Equivariant holomorphic maps of Hermitian symmetric domains into Siegel upper half spaces can be used to construct families of abelian varieties parametrized by locally symmetric spaces, which can be regarded as complex torus bundles over the parameter spaces. We extend the construction of such torus bundles using 2-cocycles of discrete subgroups of the semisimple Lie groups associated to the given symmetric domains and investigate some of their properties. In particular, we determine their cohomology along the fibers.

Xu introduced a class of nongraded Hamiltonian Lie algebras. These Lie algebras have a Poisson bracket structure. In this paper, the isomorphism classes of these Lie algebras are determined by employing a “sandwich” method and by studying some features of these Lie algebras. It is obtained that two Hamiltonian Lie algebras are isomorphic if and only if their corresponding Poisson algebras are isomorphic. Furthermore, the derivation algebras and the second cohomology groups are determined.