Kirti Joshi

On vector bundles destabilized by Frobenius pull-back

Let X be a smooth projective curve of genus g>1 over an algebraically closed field of positive characteristic. This paper is a study of a natural stratification, defined by the absolute Frobenius morphism of X, on the moduli space of vector bundles. In characteristic two, there is a complete classification of semi-stable bundles of rank 2 which are destabilized by Frobenius pull-back. We also show that these strata are irreducible and obtain their respective dimensions. In particular, the dimension of the locus of bundles of rank two which are destabilized by Frobenius is 3g-4. These Frobenius destabilized bundles also exist in characteristics two, three and five with ranks 4, 3 and 5, respectively. Finally, there is a connection between (pre)-opers and Frobenius destabilized bundles. This allows an interpretation of some of the above results in terms of pre-opers and provides a mechanism for constructing Frobenius destabilized bundles in large characteristics.

Frobenius splitting and ordinarity

We examine the relationship between the notion of Frobe- nius splitting and ordinarity for varieties. We show the following: a) The de Rham-Witt cohomology groups H i (X, W(OX)) of a smooth projec- tive Frobenius split variety are finitely generated over W(k). b) we provide counterexamples to a question of V. B. Mehta that Frobenius split varieties are ordinary or even Hodge-Witt. c) a Kummer K3 sur- face associated to an Abelian surface is F-split (ordinary) if and only if the associated Abelian surface is F-split (ordinary). d) for a K3-surface defined over a number field, there is a set of primes of density one in some finite extension of the base field, over which the surface acquires ordinary reduction.

Moduli of Vector Bundles on Curves in Positive Characteristics

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.

Exotic torsion, frobenius splitting and the slope spectral sequence

In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic p > 0 is Hodge?Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N. Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications

Remarks on Methods of Fontaine and Faltings

In this note we prove the non existence of certain irreducible two dimensional representations of the the absolute Galois group. Such results are predicted by Serres conjecture and we use Fontaines methods to verify these predictions in a small number of cases. We also use a methods of Faltings to study certain finiteness conjectures for Galois representations. Key words: Serres conjecture, Fontaine-Mazur conjectures, Galois representations.

Kodaira–Akizuki–Nakano Vanishing: a Variant

In this note we wish to prove a purely characteristic p > 0 variant of the Kodaira–Akizuki–Nakano vanishing for smooth complete intersections of dimension at least two in projective space. This has some interesting applications; in particular, we show that all Frobenius pull-backs of the tangent bundle of any complete intersection of general type and of dimension at least three in Pn are stable. We also show (see Remark 3.4) that a small modification of the techniques of [5] and a theorem of Mehta and Ramanathan (see [3]) together allow us to extend this stability result to smooth projective hypersurfaces of degree d, where (n+1)/2<d<n+1 (that is, to some Fano hypersurfaces). It is well known that behaviour of stability under Frobenius pull-backs is a subtle problem of the theory of vector bundles in characteristic p>0, and hence this result is not without interest. We end with an obvious conjectural form of our variant for a general class of varieties. 1991 Mathematics Subject Classification 14J760.

TWO REMARKS ON SUBVARIETIES OF MODULI SPACES

We study two natural questions about subvarieties of moduli spaces. In the first section, we study the locus of curves equipped with F-nilpotent bundles and its relationship to the p-rank zero locus of the moduli space of curves of genus g. In the second section, we study subvarieties of moduli spaces of vector bundles on curves. We prove an analogue of a result of F. Oort about proper subvarieties of moduli of abelian varieties.

On the Coleman-Chabauty bound

Abstract The Coleman-Chabauty bound is an upper bound for the number of rational points on a curve of genus g ≥ 2 whose Jacobian has Mordell-Weil rank r less than g . The bound is given in terms of the genus of the curve and the number of F p -points on the reduced curve, for all primes p of good reduction such that p > 2 g . In this Note we show that the hypothesis on the Mordell-Weil rank is essential. We do so by exhibiting, for each prime p ≥ 5, an explicit family of curves of genus ( p − 1)/2 (and rank at least ( p − 1)/2) for which the bound in question does not hold. Our examples show that the difference between the number of rational points and the bound in question can in fact be linear in the genus. Under mild assumptions, our curves have rank at least twice their genus.

On the action of Hecke operators on Drinfeld modular forms

Abstract In this paper we determine the explicit structure of the semisimple part of the Hecke algebra that acts on Drinfeld modular forms of full level modulo T . We show that modulo T the Hecke algebra has a non-zero semisimple part. In contrast, a well-known theorem of Serre asserts that for classical modular forms the action of T l for any odd prime l is nilpotent modulo 2. After proving the result for Drinfeld modular forms modulo T , we use computations of the Hecke action modulo T to show that certain powers of the Drinfeld modular form h cannot be eigenforms. Finally, we pose a question a positive answer to which will mean that the Hecke algebra that acts on Drinfeld modular forms of full level is not smooth for large weights, which again contrasts the classical situation.

Infinite hilbert class field towers from galois representations

We investigate class field towers of number fields obtained as fixed fields of modular representations of the absolute Galois group of the rational numbers. First, for each