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Let be a finite group and be a field of characteristic . We investigate the homotopy category of the category of complexes of injective ( projective) -modules. If is a -group, this category is equivalent to the derived category of the cochains on the classifying space; if is not a -group, it has better properties than this derived category. The ordinary tensor product in with diagonal -action corresponds to the tensor product on .
We show that can be regarded as a slight enlargement of the stable module category . It has better formal properties inasmuch as the ordinary cohomology ring is better behaved than the Tate cohomology ring .
It is also better than the derived category , because the compact objects in form a copy of the bounded derived category , whereas the compact objects in consist of just the perfect complexes.
Finally, we develop the theory of support varieties and homotopy colimits in .
We investigate the joint moments of the -th power of the characteristic polynomial of random unitary matrices with the -th power of the derivative of this same polynomial. We prove that for a fixed , the moments are given by rational functions of , up to a well-known factor that already arises when .
We fully describe the denominator in those rational functions (this had already been done by Hughes experimentally), and define the numerators through various formulas, mostly sums over partitions.
We also use this to formulate conjectures on joint moments of the zeta function and its derivatives, or even the same questions for the Hardy function, if we use a “real” version of characteristic polynomials.
Our methods should easily be applied to other similar problems, for instance with higher derivatives of characteristic polynomials.
More data and computer programs are available as expanded content.
We prove that the natural map , where is an algebraic torus over a field of dimension at most , a smooth proper geometrically irreducible variety over containing as an open subset and is the group of classes of zero-dimensional cycles on of degree zero, is an isomorphism. In particular, the group is finite if is finitely generated over the prime subfield, over the complex field, or over a -adic field.
Soit . On étudie la catégorie des -paires où est un -module libre muni d’une action semi-linéaire et continue de et où est un -réseau stable par de . Cette catégorie contient celle des représentations -adiques, et est naturellement équivalente à la catégorie de tous les -modules sur l’anneau de Robba.
Let . We study the category of -pairs where is a free -module with a semilinear and continuous action of and where is a -stable -lattice in . This category contains the category of -adic representations and is naturally equivalent to the category of all -modules over the Robba ring.