By Peter F. Stiller

ISBN-10: 3322907082

ISBN-13: 9783322907080

ISBN-10: 3322907104

ISBN-13: 9783322907103

In learning an algebraic floor E, which we think is non-singular and projective over the sphere of advanced numbers t, it's common to review the curves in this floor. in an effort to do that one introduces quite a few equivalence kinfolk at the staff of divisors (cycles of codimension one). One such relation is algebraic equivalence and we denote through NS(E) the gang of divisors modulo algebraic equivalence often called the N~ron-Severi crew of the skin E. this is often identified to be a finitely generated abelian crew that are looked obviously as a subgroup of two H (E,Z). The rank of NS(E) may be denoted p and is named the Picard variety of E. 2 each divisor determines a cohomology type in H(E,E) that's of I style (1,1), that's to assert a category in H(E,9!) which might be seen as a 2 subspace of H(E,E) through the Hodge decomposition. The Hodge Conjecture asserts in most cases that each rational cohomology category of kind (p,p) is algebraic. In our case this is often the Lefschetz Theorem on (I,l)-classes: each cohomology category 2 2 is the category linked to a couple divisor. right here we're writing H (E,Z) for two its snapshot lower than the traditional mapping into H (E,t). hence NS(E) modulo 2 torsion is Hl(E,n!) n H(E,Z) and th 1 b i f h -~ p measures e a ge ra c half zero t e cohomology.

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Because Notice that 5 always contains at least one point J is non-constant and any pole of J singularity. will be a true Moreover, it is easy to show that if contain at least 3 points. isomorphic to ~ X =pII: Thus the universal cover of then 5 must Xo will be The period mapping will be ramified at exactly the subset of cosingular points where the exponent difference is 2 or more (as opposed to 1 which is the only other possibility) and the order of M 31 ramification will be that difference (see remarks at the end of §2 above).

Let (resp. ~(B) ~(BO» be the sheaf of germs of sections of BO)' Note that any section of HO(X, ~(B» HO(X, ~(BO» E/X has image in is in 1-1 correspondence with Likewise Egen(K(X» of finite For all of our discussion in this section there will be no loss of generality in working with Egen(K(X» Obviously Egen(K(X». corresponds to a torsion free subgroup index (Shioda [27]). of B. B (resp. Egen(K(X» or any other subgroup of finite index, and we will frequently do so. We denote by f the line bundle on X obtained by taking the tangent space at ~ to each fiber.

A(J,A) At this point, we calculate the local behaviour of in X- xn. 3 where P .. #J_ Q• Now the "parameter" Ji) (d])2 dx J2(] _ 1)2 x ]dJ t d dx log A dx does not provide a good local parameter everywhere and could therefore introduce complications. if d2] - ]-2 dx is a local parameter at any point in locally it retains the above form with P .. Q = etc. e. 26 We see that the global "parameter" singularities. x plays no role in determining the Thus the above form can be used in local calculations.

### Automorphic Forms and the Picard Number of an Elliptic Surface by Peter F. Stiller

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