Index formulae for integral Galois modules

We prove very general index formulae for integral Galois modules, specifically for units in rings of integers of number fields, for higher K-groups of rings of integers, and for Mordell-Weil groups of elliptic curves over number fields. These formulae link the respective Galois module structure to other arithmetic invariants, such as class numbers, or Tamagawa numbers and Tate-Shafarevich groups. This is a generalisation of known results on units to other Galois modules and to many more Galois groups, and at the same time a unification of the approaches hitherto developed in the case of units.Comment: 14 pages; final versio

Factor equivalence of Galois modules and regulator constants

We compare two approaches to the study of Galois module structures: on the one hand factor equivalence, a technique that has been used by Fr\"ohlich and others to investigate the Galois module structure of rings of integers of number fields and of their unit groups, and on the other hand regulator constants, a set of invariants attached to integral group representations by Dokchitser and Dokchitser, and used by the author, among others, to study Galois module structures. We show that the two approaches are in fact closely related, and interpret results arising from these two approaches in terms of each other. We also use this comparison to derive a factorisability result on higher $K$-groups of rings of integers, which is a direct analogue of a theorem of de Smit on $S$-units.Comment: Minor corrections and some more details added in proofs; 11 pages. Final version to appear in Int. J. Number Theor

Extended equivariant Picard complexes and homogeneous spaces

Let k be a field of characteristic 0 and let X be a smooth geometrically integral k-variety. In our previous paper we defined the extended Picard complex UPic(X) as a certain complex of Galois modules in degrees 0 and 1. We computed the isomorphism class of UPic(G) in the derived category of Galois modules for a connected linear k-group G. In this paper we assume that X is a homogeneous space of a connected linear k-group G with geometric stabilizer H. We compute the isomorphism class of UPic(X) in the derived category of Galois modules in terms of the character groups of G and H. The proof is based on the notion of the extended equivariant Picard complex UPic_G(X) of a G-variety X.Comment: 32 pages. Final version, to appear in Transformation Group

Algebraic tori revisited

Let $K/k$ be a finite Galois extension and \pi = \fn{Gal}(K/k). An algebraic torus $T$ defined over $k$ is called a $\pi$-torus if T\times_{\fn{Spec}(k)} \fn{Spec}(K)\simeq \bm{G}_{m,K}^n for some integer $n$. The set of all algebraic $\pi$-tori defined over $k$ under the stably isomorphism form a semigroup, denoted by $T(\pi)$. We will give a complete proof of the following theorem due to Endo and Miyata \cite{EM5}. Theorem. Let $\pi$ be a finite group. Then $T(\pi)\simeq C(\Omega_{\bm{Z}\pi})$ where $\Omega_{\bm{Z}\pi}$ is a maximal $\bm{Z}$-order in $\bm{Q}\pi$ containing $\bm{Z}\pi$ and $C(\Omega_{\bm{Z}\pi})$ is the locally free class group of $\Omega_{\bm{Z}\pi}$, provided that $\pi$ is isomorphic to the following four types of groups : $C_n$ ($n$ is any positive integer), $D_m$ ($m$ is any odd integer $\ge 3$), $C_{q^f}\times D_m$ ($m$ is any odd integer $\ge 3$, $q$ is an odd prime number not dividing $m$, $f\ge 1$, and $(\bm{Z}/q^f\bm{Z})^{\times}=\langle \bar{p}\rangle$ for any prime divisor $p$ of $m$), $Q_{4m}$ ($m$ is any odd integer $\ge 3$, $p\equiv 3 \pmod{4}$ for any prime divisor $p$ of $m$).Comment: To appear in Asian J. Math. ; the title is change

On the dihedral Euler characteristics of Selmer groups of abelian varieties

This note shows how to use the framework of Euler characteristic formulae to study Selmer groups of abelian varieties in certain dihedral or anticyclotomic extensions of CM fields via Iwasawa main conjectures, and in particular how to verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this setting. When the Selmer group is cotorsion with respect to the associated Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with respect to the associated Iwasawa algebra, we give a conjectural description of the Euler characteristic of the cotorsion submodule, and explain how to deduce inequalities from the associated main conjecture divisibilities of Perrin-Riou and Howard.Comment: 26 pages. Previous discussion of two-variable setting removed, and discussion of the indefinite setting modified accordingly. To appear in the HIM "Arithmetic and Geometry" conference proceeding

On anticyclotomic mu-invariants of modular forms

Let f be a modular form of weight 2 and trivial character. Fix also an imaginary quadratic field K. We use work of Bertolini-Darmon and Vatsal to study the mu-invariant of the p-adic Selmer group of f over the anticyclotomic Zp-extension of K. In particular, we verify the mu-part of the main conjecture in this context. The proof of this result is based on an analysis of congruences of modular forms, leading to a conjectural quantitative version of level-lowering (which we verify in the case that Mazur's principle applies)