We use generating functions to relate the expected values of polynomial factorization statistics over @math to the cohomology of ordered configurations in @math as a representation of the symmetric group. Our methods lead to a new proof of the twisted Grothendieck-Lefschetz formula for squarefree polynomial factorization statistics of Church, Ellenberg, and Farb.
The first moment formula is derived from the Grothendieck-Lefschetz trace formula for ' e tale cohomology with twisted coefficients'' from which the name is borrowed. The author and Lagarias @cite_2 use Theorem to establish a representation theoretic interpretation of the @math , where @math is the probability of a random squarefree polynomial having factorization type @math .
@cite_2
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We use generating functions to relate the expected values of polynomial factorization statistics over @math to the cohomology of ordered configurations in @math as a representation of the symmetric group. Our methods lead to a new proof of the twisted Grothendieck-Lefschetz formula for squarefree polynomial factorization statistics of Church, Ellenberg, and Farb. We study the distribution in short intervals of certain arithmetic functions, including the von Mangoldt function and the M "obius function, on polynomials over a finite field @math . Using the Grothendieck-Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the @math -adic cohomology of these varieties, yielding an asymptotic bound on each moment for fixed degree @math in the limit as @math . In this paper we compute the cohomology with compact supports of a Siegelthreefold as a virtual module over the product of the Galois group of ( Q ^ ) over ( Q ) and the Hecke algebra. We use a method which has been developed by Ihara, Langlands and Kottwitz: comparison of the Grothendieck--Lefschetz formula and the Arthur--Selberg trace formula. We prove a conjecture of Knutson asserting that the Schubert structure constants of the cohomology ring of a two-step flag variety are equal to the number of puzzles with specified border labels that can be created using a list of eight puzzle pieces. As a consequence, we obtain a puzzle formula for the Gromov-Witten invariants defining the small quantum cohomology ring of a Grassmann variety of type A. The proof of the conjecture proceeds by showing that the puzzle formula defines an associative product on the cohomology ring of the two-step flag variety. It is based on an explicit bijection of gashed puzzles that is analogous to the jeu de taquin algorithm but more complicated.