Torsion Zero-Cycles on a Product of Canonical Lifts of Elliptic Curves

Let X be a surface over a p-adic field with good reduction and let Y be its special fiber. We write T(X) and T(Y) for the kernels of the Albanese maps of X and Y, respectively. Then, F(X) = T(X)/T(X)_div is conjectured to be finite, where T(X)_div is the maximal divisible subgroup of T(X). Furthermore, F(X) is conjectured to be isomorphic to T(Y) modulo p-primary torsion. We show that the p-primary torsion subgroup of F(X) can be arbitrary large even though we fix the special fiber Y.     

Cohomology Algebras of Blocks of Finite Groups and Brauer Correspondence

M. Linckelmann defined the cohomology algebras of blocks of finite groups. This note is an attempt to analyze an inclusion of cohomology algebras of blocks that corresponds under Brauer correspondence through transfer maps between the Hochschild cohomology algebras of the blocks.Presented by Jon Carlson.     

A Combinatorial Characterization of Hermitian Curves

A unital U with parameter q is a 2 – (q ³ + 1, q + 1, 1) design. If a point set U in PG(2, q ²) together with its (q + 1)-secants forms a unital, then U is called a Hermitian arc. Through each point p of a Hermitian arc H there is exactly one line L having with H only the point p in common; this line L is called the tangent of H at p. For any prime power q, the absolute points and nonabsolute lines of a unitary polarity of PG(2, q ²) form a unital that is called the classical unital. The points of a classical unital are the points of a Hermitian curve in PG(2, q ²).Let H be a Hermitian arc in the projective plane PG(2, q ²). If tangents of H at collinear points of H are concurrent, then H is a Hermitian curve. This result proves a well known conjecture on Hermitian arcs.     

Embedding of a Maximal Curve in a Hermitian Variety

Let X be a projective, geometrically irreducible, non-singular, algebraic curve defined over a finite field F q ² of order q ². If the number of F q ²-rational points of X satisfies the Hasse–Weil upper bound, then X is said to be F q ²-maximal. For a point P ₀ X(F q ²), let be the morphism arising from the linear series D: = |(q + 1)P ₀|, and let N: = dim(D). It is known that N 2 and that is independent of P ₀ whenever X is F q ²-maximal.     

Curves with finite turn

In this paper we study the notions of finite turn of a curve and finite turn of tangents of a curve. We generalize the theory (previously developed by Alexandrov, Pogorelov, and Reshetnyak) of angular turn in Euclidean spaces to curves with values in arbitrary Banach spaces. In particular, we manage to prove the equality of angular turn and angular turn of tangents in Hilbert spaces. One of the implications was only proved in the finite dimensional context previously, and equivalence of finiteness of turn with finiteness of turn of tangents in arbitrary Banach spaces. We also develop an auxiliary theory of one-sidedly smooth curves with values in Banach spaces. We use analytic language and methods to provide analogues of angular theorems. In some cases our approach yields stronger results (for example Corollary 5.12 concerning the permanent properties of curves with finite turn) than those that were proved previously with geometric methods in Euclidean spaces. The author was partially supported by the grant GAČR 201/03/0931 and by the NSF grant DMS-0244515.     

Generic Idempotent Modules for a Finite Group

Let G be a finite group and k an algebraically closed field of characteristic p. Let F _U be the Rickard idempotent k G-module corresponding to the set U of subvarieties of the cohomology variety V _G which are not irreducible components. We show that F _U is a finite sum of generic modules corresponding to the irreducible components of V _G . In this context, a generic module is an indecomposable module of infinite length over k G but finite length as a module over its endomorphism ring.     

On the crystalline cohomology of Deligne–Lusztig varieties

Let $X\to Y^0$ be an abelian prime-to-p Galois covering of smooth schemes over a perfect field k of characteristic $p>0$. Let Y be a smooth compactification of $Y^0$ such that $Y-Y^0$ is a normal crossings divisor on Y. We describe a logarithmic F-crystal on Y whose rational crystalline cohomology is the rigid cohomology of X, in particular provides a natural $W[F]$-lattice inside the latter; here W is the Witt vector ring of k. If a finite group G acts compatibly on X, $Y^0$ and Y then our construction is G-equivariant. As an example we apply it to Deligne–Lusztig varieties. For a finite field k, if $\mathbb{G}$ is a connected reductive algebraic group defined over k and $\mathbb{L}$ a k-rational torus satisfying a certain standard condition, we obtain a meaningful equivariant $W[F]$-lattice in the cohomology (ℓ-adic or rigid) of the corresponding Deligne–Lusztig variety and an expression of its reduction modulo p in terms of equivariant Hodge cohomology groups.     

Dirac cohomology of unitary representations of equal rank exceptional groups

In this paper, we consider the unitary representations of equal rank exceptional groups of type E with a regular lambda-lowest K-type and classify those unitary representations with the nonzero Dirac cohomology.     

Deforming Curves in Jacobians to Non-Jacobians I: Curves in <Emphasis Type="Italic">C</Emphasis><Superscript>(2)</Superscript>

We introduce deformation theoretic methods for determining when a curve X in a nonhyperelliptic Jacobian JC will deform with JC to a non-Jacobian. We apply these methods to a particular class of curves in the second symmetric power of C. More precisely, given a pencil of degree d on C, let X be the curve parametrizing pairs of points in divisors of (see the paper for the precise scheme-theoretical definition). We prove that if X deforms infinitesimally out of the Jacobian locus with JC then either d=4 or d=5, dim H° and C has genus 4 This material is based upon work partially supported by the National Security Agency under Grant No. MDA904-98-1-0014 and the National Science Foundation under Grant No. DMS-0071795. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF) or the National Security Agency (NSA)     

Visibility of Shafarevich-Tate Groups of Abelian Varieties

We investigate Mazur's notion of visibility of elements of Shafarevich-Tate groups of abelian varieties. We give a proof that every cohomology class is visible in a suitable abelian variety, discuss the visibility dimension, and describe a construction of visible elements of certain Shafarevich-Tate groups. This construction can be used to give some of the first evidence for the Birch and Swinnerton-Dyer conjecture for abelian varieties of large dimension. We then give examples of visible and invisible Shafarevich-Tate groups.     

Zonal Functions for the Unitary Groups and Applications to Hermitian Lattices

We study the decomposition of the space L²(Sⁿ⁻¹) under the actions of the complex and quaternionic unitary groups. We give an explicit basis for the space of zonal functions, which in the second case takes account of the action of the group of quaternions of norm 1. We derive applications to hermitian lattices.     

Explicit Classification for Torsion Subgroups of Rational Points of Elliptic Curves

We study the classification of elliptic curves E over the rationals ℚ according to the torsion sugroups E _tors(ℚ). More precisely, we classify those elliptic curves with E _tors(ℚ) being cyclic with even orders. We also give explicit formulas for generators of E _tors(ℚ). These results, together with the recent results of K. Ono for the non-cyclic E _tors(ℚ), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2. Received July 29, 1999, Revised March 9, 2001, Accepted July 20, 2001     

Enumerating Rational Curves: The Rational Fibration Method

A new method to approach enumerative questions about rational curves on algebraic varieties is described. The idea is to reduce the counting problems to computations on the Néron-Severi group of a ruled surface. Applications include a short proof of Kontsevich's formula for plane curves and the solution of the analogous problem for the Hirzebruch surface F₃.     

The Higher K-Theory of Real Curves

If X is a smooth curve defined over the real numbers , we show that K _n (X) is the sum of a divisible group and a finite elementary Abelian 2-group when n 2. We determine the torsion subgroup of K _n (X), which is a finite sum of copies of and 2, only depending on the topological invariants of X() and X(), and show that (for n 2) these torsion subgroups are periodic of order 8.     

The Complete Determination of the Minimum Distance of Two-Point Codes on a Hermitian Curve

This is a continuation of the previous papers [3, 4, 5]. We finish determining the minimum distance of two-point codes on a Hermitian curve. Masaaki Homma: Partially supported by Grant-in-Aid for Scientific Research (15500017), JSPS. Seon Jeong Kim: Partially supported by Korea Research Foundation Grant (KRF-2004-041-C00016)     

Generalising a characterisation of Hermitian curves

This article proves a characterisation of the classical unital that is a generalisation of a characterisation proved in 1982 by Lefèvre-Percsy. It is shown that if is a Buekenhout-Metz unital with respect to a line in such that a line of not through meets in a Baer subline, then is classical. An immediate corollary is that if is a unital in PG such that is Buekenhout-Metz with respect to two distinct lines, then is classical. Received 5 August 1999; revised 15 February 2000.     

The Selmer Groups of Elliptic Curves

Let E/K be an elliptic curve with K-rational p-torsion points.The p-Selmer group of E is described by the image of a map λk and hence an upper bound of its order is given in terms of the class numbers of the S-ideal class group of K and the p-division field of E.