Issue 4, Volume 37, December 2012

2012 · Open access

1. HYPERBOLIC ALGEBRAIC VARIETIES AND HOLOMORPHIC DIFFERENTIAL EQUATIONS JEAN-PIERRE DEMAILLY PDF

   441-512
2. TITS ALTERNATIVE FOR AUTOMORPHISM GROUPS OF COMPACT KAHLER MANIFOLDS TIEN-CUONG DINH PDF

   ▸ Abstract

   We survey some properties of the automorphism groups of compact Kähler manifolds. In particular, we present recent results by Keum, Oguiso and Zhang on the structure of these groups from the Tits alternative point of view. Several other related results will be also discussed.

   513-529
3. A FINITENESS THEOREM FOR GALOIS REPRESENTATIONS OF FUNCTION FIELDS OVER FINITE FIELDS (AFTER DELIGNE) HÉLÈNE ESNAULT, MORITZ KERZ PDF

   ▸ Abstract

   We give a detailed account of Deligne’s letter [13] to Drinfeld dated June 18, 2011, in which he shows that there are finitely many irreducible lisse $\overline{\mathbb Q}_{\ell}$-sheaves with bounded ramification, up to isomorphism and up to twist, on a smooth variety defined over a finite field. The proof relies on Lafforgue’s Langlands correspondence over curves [27]. In addition, Deligne shows the existence of affine moduli of finite type over $\mathbb Q$. A corollary of Deligne’s finiteness theorem is the existence of a number field which contains all traces of the Frobenii at closed points, which was the main result of [12] and which answers positively his own conjecture [9, Conj. 1.2.10 (ii)].

   531-562
4. REALISING UNSTABLE MODULES AS THE COHOMOLOGY OF SPACES AND MAPPING SPACES GÉRALD GAUDENS, LIONEL SCHWARTZ PDF

   ▸ Abstract

   This report discusses the question whether or not a given unstable module is the mod-$p$ cohomology of a space. One first discusses shortly the Hopf invariant 1 problem and the Kervaire invariant 1 problem and gives their relations to homotopy theory and geometric topology. Then one describes some more qualitative results, emphasizing the use of the space map$(B\mathbb Z/p, X)$ and of the structure of the category of unstable modules.

   563-577
5. HANOI LECTURES ON THE ARITHMETIC OF HYPERELLIPTIC CURVES BENEDICT H. GROSS PDF

   579-588