# Download PDF by Gerard Van Der Geer: Hilbert modular surfaces

By Gerard Van Der Geer

During the last 15 years vital effects were accomplished within the box of Hilbert Modular types. even though the most emphasis of this ebook is at the geometry of Hilbert modular surfaces, either geometric and mathematics features are handled. An abundance of examples - in reality an entire bankruptcy - completes this useful presentation of the topic. This Ergebnisbericht will quickly turn into an indispensible device for graduate scholars and researchers during this box.

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**Example text**

A straightforward imitation of the argument just presented allows one to check that the formally differentiated series F'(x) converges uniformly, and likewise for all higher order derivatives. It follows that the series F defines a Cm function on [0, oo). The simplest way to see that F is real analytic on (0,oo) is to think of x as a complex variable and verify directly that the complex derivative exists (the estimates that we just discussed make this easy). Alternatively, one may refine the estimates in the above paragraphs to majorize the jthderivative of F by an expression of the form C ~j j !

If there is a 6 > 0 such that p(t) _> 6 for all t E (a, b), then f is real analytic on I. Before proving the theorem, we consider a weaker result the proof of which illustrates the basic technique. 2 With the same notation as in the theorem, if [c, d] c (a, b) with c < d and p(t) > 0 for each t E [c,d], then there is a non-empty open subinterval of [c, d] on which f is real analytic. Proof: Setting for l = 1 , 2 , . . , we note that each Fc is closed. By hypothesis we have so by the Baire Category Theorem some F4 must contain a non-empty open subinterval of [c, 4.

5 Let { a j ) be a given sequence of real or complex numl bers. Then there is a function f that i s Cm on [O, 1) and ~ e a analytic on (0,1 ) and such that f(j)(0) = aj , and f b ) ( l )= 0, all j. Proof: Let h(x) be a non-negative Cm function on W which is s u p ported in [O, 11,real analytic in (0,I ) , and satisfies S h(x)dx = 1. Set 5 H ( x ) = 1- h(t)dt. Then H is C'O on W,real analytic on (0,I), and Choosing F according to the previous lemma so that F ( ~ ) ( o=) aj for j = 0,1,2,. aj for every j and F b ) ( l )= 0 for all j.