Those $(2m-1)$-edge-colourings of a spanning subgraph of $K_{2m}$, consisting of $K_{r}$ and independent edges, that can be embedded in a $(2m-1)$-edge-colouring of $K_{2m}$ are characterised.

Weierstrass points are special points on a Riemann surface that carry a lot of information. Ogg studied such points on $X_0(pM)$ (for $M$ such that $X_0(M)$ has genus zero and $p$ prime with $p\nmid M$), and he proved that if $Q$ is a $\mathbb{Q}$-rational Weierstrass point on $X_0(pM)$, then its reduction modulo $p$ is supersingular. The paper shows that, for square-free $M$ on the list, all supersingular $j$-invariants are obtained in this way. Furthermore, for most cases where $M$ is prime, the explicit correspondence between Weierstrass points and supersingular $j$-invariants in characteristic $p$ is described. Along the way, a useful formula of Rohrlich for computing a certain Wronskian of modular forms modulo $p$ is generalized.

Let $F(n,r,k)$ denote the maximum possible number of distinct edge-colorings of a simple graph on $n$ vertices with $r$ colors which contain no monochromatic copy of $K_k$. It is shown that for every fixed $k$ and all $n\,{>}\,n_0(k)$, $F(n,2,k)\,{=}\,2^{t_{k-1}(n)}$ and $F(n,3,k)\,{=}\,3^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the maximum possible number of edges of a graph on $n$ vertices with no $K_k$ (determined by Turán's theorem). The case $r\,{=}\,2$ settles an old conjecture of Erdős and Rothschild, which was also independently raised later by Yuster. On the other hand, for every fixed $r\,{>}\,3$ and $k\,{>}\,2$, the function $F(n,r,k)$ is exponentially bigger than $r^{t_{k-1}(n)}.$ The proofs are based on Szemerédi's regularity lemma together with some additional tools in extremal graph theory, and provide one of the rare examples of a precise result proved by applying this lemma.

The non-degeneracy of the canonical $p$-adic height pairing defined by Perrin-Riou and Schneider on an elliptic curve over a number field with good, ordinary reduction is still unknown.

Following the work done for the real-valued pairing, the behaviour of the $p$-adic height is analysed as a point varies on a section of a family of elliptic curves, and so new information is obtained about this pairing. In particular, the variation is $p$-adically continuous and the non-degeneracy of a set of sections can be checked simultaneously for almost all elements of the family. The paper contains some conjectures about the valuation of the $p$-adic regulator and its consequences for the growth of the Mordell–Weil group in cyclotomic $\mathbb{Z}_p$-extensions.

Waldspurger's involution for the genuine irreducible supercuspidal representations of $\widetilde{\mathrm{SL}_{2}(F)}$ is parametrized in terms of types in the case $F$$p$-adic with $p$ odd. In particular, it is shown that the in-volution is given by conjugating by an element of $\widetilde{\mathrm{GL}_{2}(F)}$ and twisting one of the defining parameters of an associated type by a quadratic character, the relevant parameter being a character on the norm one elements of a quadratic extension.

A conjecture is formulated which relates the equivariant local epsilon constant of a Galois extension of $p$-adic fields to a natural algebraic invariant coming from étale cohomology. Some evidence for the conjecture is provided and its relation to a conjecture for the equivariant global epsilon constant of an extension of number fields formulated by Bley and Burns is established.

The paper gives fundamental results on the universal abelian covers of rational surface singularities. Let $(X,o)$ be a normal complex surface singularity germ with a rational homology sphere link. Then $(X,o)$ has the universal abelian cover $(Y,o) \longrightarrow (X,o)$. It is shown that if $(X,o)$ is rational or minimally elliptic, and if it has a star-shaped resolution graph, then $(Y,o)$ is a complete intersection (a partial answer to the conjecture of Neumann and Wahl). A way is given to compute the multiplicity and the embedding dimension of $(Y,o)$ from the resolution graph of $(X,o)$ in the case when $(X,o)$ is rational.

Let $E$ be an elliptic curve defined over a number field $F$. The paper concerns the structure of the $p^{\backslash}\infty$-Selmer group of $E$ over $p$-adic Lie extensions $F_{\backslash}\infty$ of $F$ which are obtained by adjoining to $F$ the $p$-division points of an abelian variety $A$ defined over $F$. The main focus of the paper is the calculation of the Gal$(F_{\backslash}\infty/F)$-Euler characteristic of the $p^{\backslash}\infty$-Selmer group of $E$. The main theory is illustrated with the example of an elliptic curve of conductor 294.

The depth of the associated graded ring of the powers of an ideal $I$ of a local ring $R$ is studied. It is proved that the depth of the associated graded ring of $I^n$ is asymptotically constant when $n$ tends to infinity, and this value is characterized in terms of Valabrega–Valla conditions of $I^m$ for some large integer $m\ge 0$. As a corollary, a generalization is obtained of the $2$-dimensional algebraic version of the Grauert–Riemenschneider vanishing theorem (due to Huckaba and Huneke) to ideals satisfying the second Valabrega–Valla condition. The positiveness of Hilbert coefficients is also studied, and Valabrega–Valla conditions are linked to the vanishing of the cohomology groups of the closed fiber of the blowing up of ${\bf Spec}(R)$ along the closed sub-scheme defined by $I$.

Let $M$ be a module over the ring $R$. Extensive use is made of Krull codimension to study further the Artinian-finitary automorphism group $\[ F_1\hbox{\rm Aut}_RM \,{=}\, \{g\,{\in}\hbox{ Aut}_RM \,{:}\, M(g - 1) \hbox{ is } R\hbox{-Artinian}\} \]$ of $M$ over $R$. Substantial progress is made where either $M$ is residually Noetherian or $R$ is commutative. There are some group-theoretic consequences of the two main structure theorems.

Several authors have proved Lefschetz type formulas for the local Euler obstruction. In particular, a result of this type has been proved that turns out to be equivalent to saying that the local Euler obstruction, as a constructible function, satisfies the local Euler condition (in bivariant theory) with respect to general linear forms. The purpose of the paper is to determine what prevents the local Euler obstruction from satisfying the local Euler condition with respect to functions which are singular at the considered point. This is measured by an invariant (or ‘defect’) of such functions. An interpretation of this defect is given in terms of vanishing cycles, which allows it to be calculated algebraically. When the function has an isolated singularity, the invariant can be defined geometrically, via obstruction theory. This invariant unifies the usual concepts of the Milnor number of a function and the local Euler obstruction of an analytic set.

Patchworking of singular hypersurfaces is used to construct projective hypersurfaces with prescribed singularities. For all $n\,{\geq}\, 2$, an asymptotically proper existence result is deduced for hypersurfaces in $\P^n$ with singularities of corank at most 2 prescribed up to analytical or topological equivalence. In the case of $T$-smooth hypersurfaces with only simple singularities, the result is even asymptotically optimal, that is, the leading coefficient in the sufficient existence condition cannot be improved, which is new even in the case of plane curves. Furthermore, an asymptotically proper existence result is proved for hypersurfaces in $\P^n$ with quasihomogeneous singularities. The estimates substantially improve all known (general) existence results for hypersurfaces with these singularities.

The theory of order $p$ linear differential equations with univalued solutions can be presented in terms of the Schubert calculus in the Grassmannian of $p$-dimensional subspaces of the vector space of complex polynomials in one variable. The regular singular points and the exponents of an equation determine an intersection of Schubert varieties.

Heine and Stieltjes in their studies of order $p = 2$ linear differential equations with polynomial coefficients having a polynomial solution of a prescribed degree discovered that such a solution was given by a critical point of a certain remarkable symmetric function. In terms of the Schubert calculus, the critical points determine the elements of the intersection of Schubert varieties.

The case $p > 2$ is studied. A function whose critical points determine the non-degenerate elements in the intersection of Schubert varieties is presented. The number of Fuchsian equations with univalued solutions and prescribed exponents at singular points is estimated from above by the intersection number of the corresponding Schubert classes. Conjecturally, for generic disposition of singular points, these numbers coincide, that is, the intersection of Schubert varieties is transversal and consists of non-degenerate elements only. For $p = 2$ the conjecture was proved earlier by the author.

Leaning on a remarkable paper of Pryce, the paper studies two independent classes of topological Abelian groups which are strictly angelic when endowed with their Bohr topology. Some extensions are given of the Eberlein–šmulyan theorem for the class of topological Abelian groups, and finally, for a large subclass of the latter, Bohr angelicity is related to the Schur property.

Localization and dévissage theorems are proved for the hermitian $K$-theory of rings that are analogous to well-known theorems in algebraic $K$-theory. The proofs rely on, among other things, a study of derived categories, a generalization of a theorem of Pedersen and Weibel to the hermitian setting, and a cofinality result for triangular Witt groups. Applications include a proof of a conjecture of Karoubi and algebraic Bott periodicity.

Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type parameters is constructed in nonelementary abelian groups, the groups $\Z_4^{2k}\times \Z_2^{4 \ell-4k}$ for all $k$ when $\ell$ is odd and for all $k < \ell$ when $\ell$ is even. Similarly, partial difference sets with Latin square type parameters are constructed in the same groups for all $k$ when $\ell$ is even and for all $k<\ell$ when $\ell$ is odd. These constructions provide the first example where the non-homomorphic bijection approach outlined by Hagita and Schmidt can produce difference sets in groups that previously had no known constructions. Computer computations indicate that the strongly regular graphs associated to the partial difference sets are not isomorphic to the known graphs, and it is conjectured that the family of strongly regular graphs will be new.

Graded versions of the principal series modules of the category $\cO$ of a semisimple complex Lie algebra $\mg$ are defined. Their combinatorial descriptions are given by some Kazhdan–Lusztig polynomials. A graded version of the Duflo–Zhelobenko four-term exact sequence is proved. This gives results about composition factors of quotients of the universal enveloping algebra of $\mg$ by primitive ideals; in particular an upper bound is obtained for the multiplicities of such composition factors. Explicit descriptions are given of principal series modules for Lie algebras of rank $2$. It can be seen that these graded versions of principal series representations are neither rigid nor Koszul modules.

Let $G$ be a reductive $p$-adic group. Consider the category of smooth (complex) representations of $G$ in which a (fixed) closed cocompact subgroup of the centre acts by a (fixed) character. It is well known that the supercuspidal representations in this category are both injective and projective. It is shown that, conversely, an admissible injective or projective object is necessarily supercuspidal.

A locally projective amalgam is formed by the stabilizer $G(x)$ of a vertex $x$ and the global stabilizer $G\{x,y\}$ of an edge containing $x$ in a group $G$, acting faithfully and locally finitely on a connected graph $\Gm$ of valency $2^n\,{-}\,1$ so that (i) the action is 2-arc-transitive, (ii) the sub-constituent $G(x)^{\Gm(x)}$ is the linear group ${\rm SL}_n(2) \,{\cong}\, {\rm L}_n(2)$ in its natural doubly transitive action, and (iii) $[t,G\{x,y\}] \,{\le}\, O_2(G(x) \,{\cap}\, G\{x,y\})$ for some $t \,{\in}\, G\{x,y\} \setminus G(x)$. Djoković and Miller used the classical Tutte theorem to show that there are seven locally projective amalgams for $n\,{=}\,2$. Trofimov's theorem was used by the first author and Shpectorov to extend the classification to the case $n \,{\ge}\, 3$. It turned out that for $n\,{\geq}\,3$, besides two infinite series of locally projective amalgams (embedded into the groups ${\rm AGL}_n(2)$ and $O_{2n}^+(2)$), there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups $M_{22}$, $M_{23}$, $Co_2$, $J_4$ and $\hbox{\it BM}$. For a locally projective amalgam $\cA$, the minimal degree $m\,{=}\,m(\cA)$ of its complex representation (which is a faithful completion into ${\rm GL}_m(\CC)$) is calculated. The minimal representations are analysed and three open questions on exceptional locally projective amalgams are answered. It is shown that

$\cA_4^{(1)}$ possesses ${\rm SL}_{20}(13)$ as a faithful completion in which the third geometric subgroup is improper;

$\cA_4^{(2)}$ possesses the alternating group ${\rm Alt}_{64}$ as a completion constrained at levels 2 and 3;

$\cA_4^{(5)}$ possesses ${\rm Alt}_{256}$ as a completion which is constrained at level 2 but not at level 3.

A functorial and categorical defined cyclotomic trace is given, extending the usual one for rings to ring spectra. There are two ingredients to this: first a cyclotomic trace is needed that accepts a categorical input with few restrictive assumptions. This is important in its own right, since this allows one to transport rich structures through the cyclotomic trace. Secondly, a sufficiently nice model is needed for the category of finitely generated free modules which is functorial in the ring spectrum.