By Haruzo Hida

This e-book presents a entire account of the idea of moduli areas of elliptic curves (over integer earrings) and its software to modular types. the development of Galois representations, which play a primary position in Wiles' evidence of the Shimura - Taniyama conjecture, is given. additionally, the publication provides an summary of the facts of numerous modularity result of two-dimensional Galois representations (including that of Wiles), in addition to a number of the author's new ends up in that course. during this new moment variation, a close description of Barsotti - Tate teams (including formal Lie teams) is further to bankruptcy 1. As an software, a down-to-earth description of formal deformation concept of elliptic curves is included on the finish of bankruptcy 2 (in order to make the facts of regularity of the moduli of elliptic curve extra conceptual), and in bankruptcy four, even though restricted to dull situations, newly included are Ribet's theorem of complete photograph of modular p-adic Galois illustration and its generalization to 'big' lambda-adic Galois representations below light assumptions (a new results of the author). although the various notable advancements defined above is out of the scope of this introductory publication, the writer offers a flavor of modern-day study within the sector of quantity conception on the very finish of the booklet (giving a great account of modularity concept of abelian Q-varieties and Q-curves).

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37] P. Deligne and D. Mumford. The irreducibility of the space of curves of given ´ genus. Inst. Hautes Etudes Sci. Publ. , 36:75–109, 1969. V. Dolgachev. Lectures on Invariant Theory, volume 296 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2003. V. Dolgachev and Y. Hu. Variation of geometric invariant theory quotients. ´ Inst. Hautes Etudes Sci. Publ. , 87:5–56, 1998. V. Dolgachev and S. Kond¯o. Moduli of K3 surfaces and complex ball quotients. In Arithmetic and Geometry Around Hypergeometric Functions, volume 260 of Progr.

In our case KX = OX (d − 4), d − 4 > 0, 46 Chapter 2. Compact Moduli of Surfaces and Vector Bundles and KX deforms so, OX (1) deforms as required. 4 Expected dimension We can compute the expected dimension of M using the Hirzebruch–Riemann– Roch formula. Let X be a smooth projective surface. Write c1 = c1 (TX ) = −KX and (−1)i dimR H i (X, R). c2 = c2 (TX ) = e(X) = i For F a vector bundle on X the Hirzebruch–Riemann–Roch formula states that χ(F ) = (ch(F ) · td(X))2 , where ch(F ) = rk(F ) + c1 (F ) + 1 c1 (F )2 − 2c2 (F ) 2 is the Chern character, and 1 1 td(X) = 1 + c1 + (c21 + c2 ) 2 12 is the Todd class.

6]. Let n ∈ N and a0 , . . , an ∈ N. We write P = P(a0 , . . , an ) for the weighted projective space P(a0 , . . , an ) = (An+1 \ {0})/Gm, Gm λ : (X0 , . . , Xn ) −→ (λa0 X0 , . . , λan Xn ). We always assume that gcd(a0 , . . , ai , . . , an ) = 1 for all i. Then P(a0 , . . , an ) is a normal projective variety covered by aﬃne charts (Xi = 0) = An / a1i (a0 , . . , ai , . . , an ), a /ai where the aﬃne orbifold coordinates are given by xji = Xj /Xi j have P(a0 , . . , an ) = Proj k[X0 , .