Arithmetic of algebraic varieties

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The main result is obtained in the research on the l-adic representation associated a Hilbert modular form. It is a 2-dimensional l-adic representation of the absolute Galois group G_F of a totally real field F.For a finite place upsilon * l, it is shown by Carayol that the representation of the Weil-Deligne group defined by its restriction to the decomposition group at upsilon corresponds to the upsilon-component of the automorphic representation of GL_2 (A_F) determined by f. Using recent results in p-adic Hodge theory, I formulated a similar statement and proved it. The second year of the project was spent to write a paper and paper is nearly completed.As a byproduct of the proof, I proved the monodromy-weight conjecture for Galois representation associated to modular forms. It seems to have been overlooked even for the case F=Q.The paper is already to appear in a Proceedings.I also obtained some other results. For an algebraic variety X over a field K, the second Stiefel-Whitney class of its l-adic etale cohomology is defined in the second Galois cohomology H^2 (K, Z/2Z). I formulated a conjecture between the class with the Hasse-Witt class of de Rham cohomology and proved it in several cases.