By Anderson, Ian; Honkala, Iiro

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1: Proof: A 3' dX f&, Y = Z(f)· The kernel of this map if exactly the torsion in (Manin [21). o Thus the image is isomorphic to ~r part of Egen(K(X». isomorphic to ~r where r is the rank of the free Note that the image of in and both images have the same t-span in is true for any subgroup of Egen(K(X» K(X) is also K(X). The same of finite index. At this point we must develop a number of technical results which wi+l aid in the description of the image of this map. Egen(K(X» viewed as a section of BO/X.

L: 1) where Xo & Xo is a base point selected away from the cosingular points for convenience. 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).

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.

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