L-adic Cohomology Research Articles

Exponential sums and ℓ-adic cohomology: A survey

This is the text of two lectures given at the Exponential Sums Conference held at The Hebrew University of Jerusalem in January 1998. In these lectures I reviewed some important results on exponential sums, which have been obtained by Deligne, Katz and others as consequences of the Grothendieck trace formula for the l-adic cohomology and Deligne’s fundamental result on the Weil conjecture.

POLES OF ZETA FUNCTIONS OF COMPLETE INTERSECTIONS

A vanishing theorem is proved for l-adic cohomology with compact support on an affine (singular) complete intersection. As an application, it is shown that for an affine complete intersection defined over a finite field of q elements, the reciprocal "poles" of the zeta function are always divisible by q as algebraic integers. A p-adic proof is also given, which leads to further q-divisibility of the poles or equivalently an improvement of the polar part of the Ax-Katz theorem for an affine complete intersection. Similar results hold for a projective complete intersection.

A comparison theorem for l-adic cohomology

HTML view is not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button. We show that, for certain types of rigid analytic varieties X and constructible l-adic sheaves (F$_n$)$_n$ on, one has H$^p_c$ (X,(F$_n$)$_n$) ${\stackrel{\sim}{\longrightarrow}} {\displaystyle\lim_{\stackrel{\scriptstyle{\longleftarrow}} {\scriptstyle{n}}}}$ H$^p_c$ (X,F$_n$). As an application we obtain that, for an algebraic variety X and associated rigid analytic variety X$^rig$, the l-adic cohomology of X and X$^rig$ agree.

Weights of the l-adic cohomology of a proper toric variety

We describe the weights of the l-adic cohomology groups of a toric variety over a finite field k in terms of the Ishida complexes of Z-modules. As a consequence, we conclude that, for an r-dimensional proper toric variety X, the m-th cohomology group Hm (X⊗k[kbar],Ql) is of pure weight if m = 0,1,2,3,2r - 3,2r - 2,2r - 1,2r.Furthermore, we show that, for any m such that 3 < m < 2r - 3,there exists an r-dimensional proper toric variety whose m-th cohomology group Hm (X⊗k[kbar],Q l ) is not pure.

Subspace Arrangements over Finite Fields: Cohomological and Enumerative Aspects

The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic relations between these two points of view. Counting points is also related to the ℓ-adic cohomology of the arrangement (as a variety). We describe the eigenvalues of the Frobenius map acting on this cohomology, which corresponds to a finer decomposition of the zeta function. The ℓ-adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data. Finally, it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases.

Green Functions and Character Sheaves

purely combinatorial way, using earlier ideas of P. Hall. In the general case, they were introduced [3] in terms of l-adic cohomology, as values of certain virtual representations of R'(O)(T is a maximal torus defined over Fq) at unipotent elements; the characters of RG( 0) at arbitrary elements were then expressible in a simple form in terms of Green functions of G and of smaller groups. One drawback of the definition of [3] was that it did not allow one to compute the Green functions except in some simple cases. Another definition of Green functions was proposed by Springer, first in terms of trigonometric sums on the Lie algebra and later in geometrical terms [17]. This definition was applicable only in large characteristic and in that case it was identified with the definition in [3] by Springer [17] and Kazhdan [10]. In [12], I observed that Green functions for GLn can be redefined in terms of intersection cohomology. This led me to a new definition of Green functions