By Previato E. (ed.)

ISBN-10: 082182810X

ISBN-13: 9780821828106

Our wisdom of items of algebraic geometry similar to moduli of curves, (real) Schubert sessions, primary teams of enhances of hyperplane preparations, toric types, and edition of Hodge constructions, has been stronger lately via rules and structures of quantum box conception, resembling reflect symmetry, Gromov-Witten invariants, quantum cohomology, and gravitational descendants.

These are a number of the issues of this refereed number of papers, which grew out of the targeted consultation, "Enumerative Geometry in Physics," held on the AMS assembly in Lowell, MA, April 2000. This consultation introduced jointly mathematicians and physicists who mentioned at the newest effects and open questions; the entire abstracts are integrated as an Appendix, and in addition integrated are papers by means of a few who couldn't attend.

The assortment presents an summary of state of the art instruments, hyperlinks that attach classical and glossy difficulties, and the most recent wisdom available.

Readership: Graduate scholars and examine mathematicians drawn to algebraic geometry and comparable disciplines.

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By lack of a precise reference, we have included a proof for the case of stacks. 1. Let X be a Deligne–Mumford stack which is smooth and proper over Zp . For every prime l = p and every i ≥ 0, the canonical map of Gal(Qp /Qp )representations Hi (XFp ,ét , Ql ) → Hi (XQ ,ét , Ql ) (1) p is an isomorphism. In particular, Hi (XQ p ,ét , Ql ) is unramified. Qnr p Proof. Denote by the maximal unramified extension of Qp in Qp and let Znr p be ) and denote by s, respectively, η, its closed, its ring of integers.

Therefore, one should not expect that all the functors f ∗ , ⊗, f! , f∗ , Hom, f ! , postulated by Grothendieck for a good cohomological theory, exist for Crys(. . ). In special cases, one may construct some further functors. For instance, in [Bö04] it is shown that for B finite or B = A and any open immersion j : U → X there exists a meaningful functor j# : Crys(U, A) −→ Crys(X, A) (8) in the sense of Definition 6. It corresponds to j∗ in the étale theory. It is not called j∗ , since in [BP04] this name was reserved for a different functor.

If f is a modular function of weight k for (n), then so is f ◦γ for all γ ∈ GL2 (A), because (n) is normal in GL2 (A). Therefore, one can equally consider the expansion n n∈Z an,γ tn (z) of f ◦ γ . Since GL2 (A) acts transitively on the “cusps’’ of \ , following the case of elliptic modular forms, one defines the following. Definition 19. A modular function f of weight k for (n) is called a ⎧ ⎫ ⎧ ⎫ ⎨ modular form ⎬ ⎨n < 0⎬ cusp form ⇐⇒ ∀γ ∈ GL2 (A) ∀n ∈ Z with n < 1 an,γ = 0. ⎩ ⎭ ⎩ ⎭ double-cusp form n<2 By Mk ( (n), C∞ ), we denote the C∞ -vector space of modular forms for (n) of weight k and by Sk ( (n), C∞ ) and Skdc ( (n), C∞ ) its subspaces of cusp and doublecusp forms.

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Advances in Algebraic Geometry Motivated by Physics by Previato E. (ed.)

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