2 Faltings’ theorem 15 2.1 Prelude: the Shafarevich problem . . . . . . . . . . . . . . . . 15 2.2 First reduction: the Kodaira–Parshin trick . . . . . . . . . . . 17 2.3 Second reduction: passing to the jacobian . . . . . . . . . . . 19 2.4 Third reduction: passing to isogeny classes . . . . . . . . . . . 19 2.5 Fourth reduction: from isogeny classes to `-adic representations 21 2.6 The isogen...

1. A complex Banach algebra A is a compact (weakly compact) algebra if its left and right regular representations consist of compact (weakly compact) operators. Let E be any subset of A and denote by Ei and Er the left and right annihilators of E. A is an annihilator algebra if A¡= (0) —Ar, Ir^{fS) for each proper closed left ideal / and Ji t¿ (0) for each proper closed right ideal /. In [6, Th...

We investigate integer solutions of the superelliptic equation (1) z = F (x, y), where F is a homogenous polynomial with integer coefficients, and of the generalized Fermat equation (2) Ax +By = Cz, where A,B and C are non-zero integers. Call an integer solution (x, y, z) to such an equation proper if gcd(x, y, z) = 1. Using Faltings’ Theorem, we shall show that, other than in certain exception...

We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. Our methods come from the theory of Hopf algebra crossed products. For primitive ideals we obtain an analogue of Duflo’s Theorem, which says that every primitive ideal is the annihilator of a simple highest weight...

Theorem A. Let A be an abelian variety over a number field k and X a subvariety of A. Then there are a finite number of translated abelian subvarieties B1, . . . , Bn over k such that Bi ⊂ X and the closure X(k) of X(k) in X is contained in ⋃ i Bi. In this short note, as applications of the above Faltings’ theorem, we will give several remarks on rational points of varieties whose cotangent bun...

A well known theorem of Duflo claims that the annihilator of a Verma module in the enveloping algebra of a complex semisimple Lie algebra is generated by its intersection with the centre. For a Lie superalgebra this result fails to be true. For instance, in the case of the orthosymplectic Lie superalgebra osp(1, 2), Pinczon gave in [Pi] an example of a Verma module whose annihilator is not gene...

A keystone in the classical theory of diophantine approximation is the construction of an auxilliary polynomial. The polynomial is constructed so that it is forced (for arithmetic reasons) to vanish at certain approximating points and this contradicts an upper bound on the order of vanishing obtained by other (usually geometric) techniques; the contradiction then allows one to prove finiteness ...

In 1980, Faltings proved, by deep local algebra methods, a local result regarding formal functions which has the following global geometric fact as a consequence. Theorem. − Let k be an algebraically closed field (of any characteristic). Let Y be a closed subvariety of a projective irreducible variety X defined over k. Assume that X ⊂ Pn, dim(X) = d > 2 and Y is the intersection of X with r hyp...

In [15], Kaplansky introduced Baer rings as rings in which every right (left) annihilator ideal is generated by an idempotent. According to Clark [9], a ring R is called quasi-Baer if the right annihilator of every right ideal is generated (as a right ideal) by an idempotent. Further works on quasi-Baer rings appear in [4, 6, 17]. Recently, Birkenmeier et al. [8] called a ring R to be a right (...