By Peter F. Stiller

ISBN-10: 3322907082

ISBN-13: 9783322907080

ISBN-10: 3322907104

ISBN-13: 9783322907103

In learning an algebraic floor E, which we suppose is non-singular and projective over the sector of advanced numbers t, it's ordinary to check the curves in this floor. so one can do that one introduces a number of equivalence kin at the workforce of divisors (cycles of codimension one). One such relation is algebraic equivalence and we denote via NS(E) the gang of divisors modulo algebraic equivalence generally known as the N~ron-Severi workforce of the outside E. this can be recognized to be a finitely generated abelian staff which are appeared clearly as a subgroup of two H (E,Z). The rank of NS(E) might be denoted p and is called the Picard variety of E. 2 each divisor determines a cohomology classification in H(E,E) that is of I kind (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 regularly that each rational cohomology category 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 couple divisor. right here we're writing H (E,Z) for two its snapshot less than the ordinary 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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Therefore the above map can be realized as a homomorphism: Egen(K(X» --+ K(X) I-Ar~y (J' • Z. In this way we associate to each K(X)-rational point on Egen , or what is the same to each section of E over X, a rational function Z in K(X). For further details see Stiller (29). From another point of view, choose a Zariski-open subset which the operator p x & K(X) local parameter on all of that the bad fibers of of X on Say L+ dx Q which for the moment we will assume is a good XO. We also fix a base point E/X occur at exactly the set of A (Stiller (28), so that over Xa xo & XO.

E. and ~2 o. 5: the injection Proof: Note f NJ'dX ~ This implies that If Z' A: K{X) z .. and Af .. A , f hZ' + d; .. Z K{X) & K{X) except for Z' for some & K{X) is not in the image z .. e. then f ~ AK{X) of torsion. has trivial periods. both satisfy the inhomogeneous equation and so differ by a complex linear combination of ~l and o ~. We now come to the two crucial results which make all our computations possible. Recall that we have shown that the map Egen{K{X» _ f has exactly the torsion as kernel.

2: where 00 2 are a fixed K-basis in a neighborhood of constants which reflect that s c 1w1(x O) + c 2w2(x O) E Xo 001 viewing wi - 002wi • §2. Xo and are cuts the fiber E over Xo at Xo £/:£001 (x O) + IZw2(x O), and W as 0 A bound on the rank r of Egen(K(X». In this section we will construct a bound on the rank Egen(K(X» X. for our given basic elliptic surface f ;! dX ~ A 1 .... 'ft As mentioned -y = z X annihilating the periods of A is a K-equation easily computed once a model for the generic fiber and a parameter are chosen.

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

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