To get a feeling for our level of ignorance in the face of such questions, consider that, before Faltings, there was not a single curve X (of genus > 1) for which we knew this statement to be true for all number fields K over which X is defined! Already in the twenties, Weil and Siegel made serious attempts to attack the problem. Siegel, influenced by Weil's thesis, used methods of diophantine ...

We survey and compare invariants of modules over the polynomial ring and the exterior algebra. In our considerations, we focus on the depth. The exterior analogue of depth was first introduced by Aramova, Avramov and Herzog. We state similarities between the two notion of depth and exhibit their relation in the case of squarefree modules. Work of Conca, Herzog and Hibi and Trung, respectively, ...

We describe the two-sided ideals in the universal enveloping algebras of the Lie algebras of vector fields on the line and the circle which annihilate the tensor density modules. Both of these Lie algebras contain the projective subalgebra, a copy of sl2. The restrictions of the tensor density modules to this subalgebra are duals of Verma modules (of sl2) for Vec(R) and principal series modules...

The Stickelberger elements attached to an abelian extension of number fields conjecturally participate, under certain conditions, in annihilator relations involving higher algebraic K-groups. In [13], Snaith introduces canonical Galois modules hoped to appear in annihilator relations generalising and improving those involving Stickelberger elements. In this paper we study the first of these mod...

A right ideal A of a ring R is called annihilator-small if A+ T = R; T a right ideal, implies that l(T ) = 0; where l( ) indicates the left annihilator. The sum Ar of all such right ideals turns out to be a two-sided ideal that contains the Jacobson radical and the left singular ideal, and is contained in the ideal generated by the total of the ring. The ideal Ar is studied, conditions when it ...

Many scenarios where participants hold private information require payments to encourage truthful revelation. Some of these scenarios have no natural residual claimant who would absorb the budget surplus or cover the deficit. Faltings [7] proposed the idea of excluding one agent uniformly at random and making him the residual claimant. Based on this idea, we propose two classes of public good m...

Abstract We build on the recent techniques of Codogni and Patakfalvi (2021, Inventiones Mathematicae 223, 811–894), which were used to establish theorems about semi-positivity Chow Mumford line bundles for families $\mathrm {K}$ -semistable Fano varieties. Here, we apply Central Limit Theorem ascertain asymptotic probabilistic nature vertices Harder Narasimhan polygons . As an application our m...

A conjecture of Mordell, recently proven by Faltings [7], states that a curve of genus at least 2 has only finitely many rational points. Faltings' proof is not effective, although a careful reworking of his proof, combined with some further ideas of Faltings, Mumford, Parshin, and Raynaud, allows one to give an upper bound for the number of rational points. (See [19, XI, §2].) Unfortunately, t...