By Peter F. Stiller
In learning an algebraic floor E, which we think is non-singular and projective over the sphere of advanced numbers t, it's usual to review the curves in this floor. so as to do that one introduces numerous equivalence kin at the staff of divisors (cycles of codimension one). One such relation is algebraic equivalence and we denote through NS(E) the crowd of divisors modulo algebraic equivalence also known as the N~ron-Severi staff of the skin E. this is often identified to be a finitely generated abelian team which might be appeared clearly as a subgroup of two H (E,Z). The rank of NS(E) may be denoted p and is called the Picard variety of E. 2 each divisor determines a cohomology category in H(E,E) that's of I kind (1,1), that's to claim a category in H(E,9!) that are considered as a 2 subspace of H(E,E) through the Hodge decomposition. The Hodge Conjecture asserts in most cases that each rational cohomology type of kind (p,p) is algebraic. In our case this can be the Lefschetz Theorem on (I,l)-classes: each cohomology category 2 2 is the category linked to a few divisor. right here we're writing H (E,Z) for two its snapshot less than the typical mapping into H (E,t). therefore 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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Additional info for Automorphic Forms and the Picard Number of an Elliptic Surface
X with & a. "', (10 0) 1 ' the t & K(X) E/x and with Egen over (which we will suppose to be a good local x), we arrive at a differential equation as in section 1 Now the point x under consideration may be a cosingular point for this differential equation but as our analysis of the local behavior of these equations shows (see Part II, §2), we can adjust the model (multiplying the periods by a suitable rational function exponents of the differential equation are (O,s) solutions wI' s &Zl, so that the s > 1.
9: Every ~ & Egen(K(X» 0 determines a parabolic cohomology class relative to the monodromy representation of W1(XO,xO) + E/X. SL 2 CE) where Xo - X - S, S the set of singular fibers of J' & Egen(K(X» Moreover the only which determine the trivial class are the torsion elements. Proof: Clear. o We now return to the problem of finding the divisor. assume Egen/K(X) has only multiplicative reduction. We first choose a model for Egen : and a parameter x & K(X). We then obtain the K-equation . dX the periods of -y for this model.
8. o Thus there is a finite dimensional subspace of dimension in K(X) viewed as a complex vector space. r sitting One would like to find a divisor Bt. ) OJ. t. Riemann-Roch then gives an r. In order to find 9\.. and make such an estimate we shall make a geometric assumption. Egen/K(X) Name I y, we assume mUltiplicative reduction, or equivalently that type ~ (see Kodaira (16). most of Parts III and IV. 3. 7: (Existence of Good Local Models) we can choose a model of over K(X) For each point so that the function least one holomorphic single-valued non-vanishing branch at x.
Automorphic Forms and the Picard Number of an Elliptic Surface by Peter F. Stiller