Let $p$ be an odd prime number and $E$ an elliptic curve defined over a number field $F$ with good reduction at every prime of $F$ above $p$. We compute the Euler characteristics of the signed Selmer groups of $E$ over the cyclotomic $\mathbb{Z}_{p}$-extension. The novelty of our result is that we allow the elliptic curve to have mixed reduction types for primes above $p$ and mixed signs in the definition of the signed Selmer groups.

We prove that for any prime number $p\geqslant 3$ , there exists a positive number $\unicode[STIX]{x1D705}_{p}$ such that $\unicode[STIX]{x1D712}({\mathcal{O}}_{X})\geqslant \unicode[STIX]{x1D705}_{p}c_{1}^{2}$ holds true for all algebraic surfaces $X$ of general type in characteristic $p$ . In particular, $\unicode[STIX]{x1D712}({\mathcal{O}}_{X})>0$ . This answers a question of Shepherd-Barron when $p\geqslant 3$ .

If $G$ is a pro-$p$, $p$-adic, Lie group containing no element of order $p$ and if $\Lambda (G)$ denotes the Iwasawa algebra of $G$ then we propose a number of invariants associated to finitely generated $\Lambda (G)$-modules, all given by various forms of Euler characteristic. The first turns out to be none other than the rank, and this gives a particularly convenient way of calculating the rank of Iwasawa modules. Others seem to play similar roles to the classical Iwasawa $\lambda $- and $\mu $-invariants. We explore some properties and give applications to the Iwasawa theory of elliptic curves.

2000 Mathematical Subject Classification: primary 16E10; seconday 11R23.

We compute the Euler characteristics of the individual connected components of the intersection of two opposed big cells in the real flag variety of type G2, verifying a conjecture of Rietsch [6].

2000 Mathematical Subject Classification: primary 14M15; secondary 20G20.

It is well known that every finite subgroup of GLd(Q[ell ]) is conjugate to a subgroup of GLd(Z[ell ]). However, this does not remain true if we replace general linear groups by symplectic groups. We say that G is a group of inertia type of G is a finite group which has a normal Sylow-p-subgroup with cyclic quotient. We show that if [ell ]>d+1, and G is a subgroup of Sp2d(Q[ell ]) of inertia type, then G is conjugate in GL2d(Q[ell ]) to a subgroup of Sp2d(Z[ell ]). We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime [ell ], isogeny classes of Abelian varieties all of whose polarizations have degree divisible by [ell ]2. We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over fields of positive characteristic.

We describe a general method for calculating equivariant Euler characteristics. The method exploits the fact that the γ-filtration on the Grothendieck group of vector bundles on a Noetherian quasi-projective scheme has finite length; it allows us to capture torsion information which is usually ignored by equivariant Riemann–Roch theorems. As applications, we study the G-module structure of the coherent cohomology of schemes with a free action by a finite group G and, under certain assumptions, we give an explicit formula for the equivariant Euler characteristic $\chi ({\cal O}_X)={\rm H}^0(X, {\cal O}_X)-{\rm H}^1(X, {\cal O}_X)$ in the Grothendieck group of finitely generated ${\bf Z}[G]$-modules, when X is a curve over ${\bf Z}$ and G has prime order.

We define a generalization of the Euler characteristic of a perfect complex of modules for the group ring of a finite group. This is combined with work of Lichtenbaum and Saito to define an equivariant Euler characteristic for G on regular projective surfaces over Z having a free action of a finite group. In positive characteristic we relate the Euler characteristic of G to the leading terms of the expansions of L-functions at s=1.