We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose traces are weight 32 weakly holomorphic modular forms satisfying certain special properties. then use these to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups quadratic twists elliptic curves.

We consider two principal bundles of embeddings with total space Emb(M, N), with structure groups Di f f (M) and Di f f+(M), where Di f f+(M) is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: B(M, N) = Emb(M, N)/Di f f (M) and B+(M, N) = Emb(M, N)/Di f f+(M) from the various properti...

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 ...

Earlier we defined the class group ClX of Weil divisors for an algebraic varietyX and the Cartier class group CaClX of Cartier divisors (which is isomorphic to the Picard group of isomorphism classes of line bundles with tensor product). These groups are isomorphic when X is smooth. In general it is quite difficult to compute these groups. In this section we will give some classic examples with...

We give a proof that the Riemann hypothesis for hypersurfaces over finite fields implies the result for all smooth proper varieties, by a deformation argument which does not use the theory of Lefschetz pencils or the `-adic Fourier transform.

Arguments against scientific misconduct one finds in the literature generally fail to support current policies on research fraud: they may not prove wrong what is typically considered research misconduct and they tend to make wrong things that are not usually seen as scientific fraud, in particular honest errors. I argue that society cannot set a rule enjoining scientists to be honest, so any s...

In this paper, we introduce and study two new types of non-abelian zeta functions for curves over finite fields, which are defined by using (moduli spaces of) semi-stable vector bundles and non-stable bundles. A Riemann-Weil type hypothesis is formulated for zeta functions associated to semi-stable bundles, which we think is more canonical than the other one. All this is motivated by (and hence...

We introduce a notion of visibility for Mordell-Weil groups, make a conjecture about visibility, and support it with theoretical evidence and data. These results shed new light on relations between Mordell-Weil and Shafarevich-Tate groups. 11G05, 11G10, 11G18, 11Y40