Volume 2 (2002), Number 2. Abstracts

V. Beresnevich, V. Bernik, D. Kleinbock and G. Margulis. Metric Diophantine Approximation: The Khintchine—Groshev Theorem for Nondegenerate Manifolds [PDF]

The main objective of this paper is to prove a Khintchine type theorem on divergence of linear Diophantine approximation on nondegenerate manifolds, which completes earlier results for convergence.

Keywords. Diophantine approximation, Khintchine type theorems, metric theory of Diophantine approximation.

2000 Mathematics Subject Classification. Primary 11J83; Secondary 11K60.

P. Deligne. Catégories Tensorielles [PDF]

We give a super mathematics analogue to the theorem that, over an algebraically closed field of characteristic zero, categories of representations of affine group schemes, with their associative, commutative and unital tensor product, are characterized by the property that for any object large enough exterior powers vanish. Exterior powers are replaced by arbitrary Schur functors.

Keywords. Tensor category, Tannaka duality, fiber functor, super group.

2000 Mathematics Subject Classification. Primary: 18D99; Secondary: 20C99.

G. Faltings. Group Schemes with Strict $\mathcal{O}$-action [PDF]

Let $\mathcal{O}$ denote the ring of integers in a p-adic local field. Recall that $\mathcal{O}$-modules are formal groups with an $\mathcal{O}$-action such that the induced action on the Lie algebra is via scalars. In the paper this notion is generalised to finite flat group schemes. It is shown that the usual properties carry over. For example, Cartier duality holds with the multiplicative group replaced by the Lubin—Tate group. We also show that liftings over $\mathcal{O}$-divided powers are controlled by Dieudonné modules or, better, by complexes. For these facts new proofs have to be invented, because the classical recipe of embedding into abelian varieties is not available.

Keywords. Finite flat group schemes, Lubin—Tate groups, $\mathcal{O}$-modules.

2000 Mathematics Subject Classification. 14L15.

J. de Jong. Counting Elliptic Surfaces over Finite Fields [PDF]

We count the number of isomorphism classes of elliptic curves of given height d over the field of rational functions in one variable over the finite field of q elements. We also estimate the number of isomorphism classes of elliptic surfaces over the projective line, which have a polarization of relative degree 3. This leads to an upper bound for the average 3-Selmer rank of the aforementionned curves. Finally, we deduce a new upper bound for the average rank of elliptic curves in the large d limit, namely the average rank is asymptotically bounded by 1.5+O(1/q).

Keywords. Elliptic curves, elliptic surfaces, rank, average rank, Selmer group.

2000 Mathematics Subject Classification. 14G, 11G, 14H25, 1452.

A. Panchishkin. A New Method of Constructing p-adic L-functions Associated with Modular Forms [PDF]

We give a new method of constructing admissible p-adic measures associated with modular cusp eigenforms, starting from distributions with values in spaces of modular forms. A canonical projection operator is used onto the characteristic subspace of an eigenvalue $\alpha$ of the Atkin—Lehner operator Up. An algebraic version of nearly holomorphic modular forms is given and used in constructing p-adic measures.

Keywords. Modular forms, Eisenstein series, p-adic L-functions, special values.

2000 Mathematics Subject Classification. 11F33, 11F67, 11F30.

M. Tsfasman and S. Vlăduț. Infinite Global Fields and the Generalized Brauer—Siegel Theorem [PDF]

The paper has two purposes. First, we start to develop a theory of infinite global fields, i.e., of infinite algebraic extensions either of $\mathbb{Q}$ or of $\mathbb{F}_r(t)$. We produce a series of invariants of such fields, and we introduce and study a kind of zeta-function for them. Second, for sequences of number fields with growing discriminant, we prove generalizations of the Odlyzko—Serre bounds and of the Brauer—Siegel theorem, taking into account non-archimedean places. This leads to asymptotic bounds on the ratio $\log hR/\log\sqrt{|D|}$ valid without the standard assumption $n/\log\sqrt{|D|}\rightarrow 0$, thus including, in particular, the case of unramified towers. Then we produce examples of class field towers, showing that this assumption is indeed necessary for the Brauer—Siegel theorem to hold. As an easy consequence we ameliorate existing bounds for regulators.

Keywords. Global field, number field, curve over a finite field, class number, regulator, discriminant bound, explicit formulae, infinite global field, Brauer—Siegel theorem.

2000 Mathematics Subject Classification. 11G20, 11R37, 11R42, 14G05, 14G15, 14H05.

Yu. Zarhin. Very Simple 2-adic Representations and Hyperelliptic Jacobians [PDF]

Let $K$ be a field of characteristic zero, $n \ge 5$ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $\mathrm{S}_n$ or the alternating group $\mathrm{A}_n$. Let $C\colon y^2 = f(x)$ be the corresponding hyperelliptic curve and $X = J(C)$ its Jacobian defined over~$K$. For each prime $\ell$ we write $V_{\ell}(X)$ for the $\mathbf{Q}_{\ell}$-Tate module of $X$ and $e_{\lambda}$ for the Riemann form on $V_{\ell}(X)$ attached to the theta divisor. Let $\mathfrak{sp}(V_{\ell}(X),e_{\lambda})$ be the $\mathbf{Q}_{\ell}$-Lie algebra of the symplectic group of $e_{\lambda}$. Let $\mathfrak{g}_{\ell,X}$ be the $\mathbf{Q}_{\ell}$-Lie algebra of the image of the Galois group $\mathrm{Gal}(K)$ of $K$ in $\mathrm{Aut}(V_{\ell}(X))$. Assuming that $K$ is finitely generated over $\Q$, we prove that $\mathfrak{g}_{\ell,X} = \mathbf{Q}_{\ell}\operatorname{Id} \oplus \mathfrak{sp}(V_{\ell}(X),e_{\lambda})$ where $\operatorname{Id}$ is the identity operator.

Keywords. Abelian varieties, $\ell$-adic representations, hyperelliptic Jacobians, very simple representations.

2000 Mathematics Subject Classification. Primary 14H40; Secondary 14K05, 11G30, 11G10.

MMJ Cover

Moscow Mathematical Journal
is distributed by the
American Mathematical Society
for the
Independent University of Moscow

Online ISSN 1609-4514
© 2002, Independent University of Moscow

AMS Logo Medium