Download e-book for kindle: An elementary theory of Eisenstein series by Kubota T.

By Kubota T.

ISBN-10: 0470509201

ISBN-13: 9780470509203

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Extra info for An elementary theory of Eisenstein series

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Proof. Certainly tdim V ≤ n if V is represented as a germ of a subvariety of Cn . On the other hand if V is a holomorphic subvariety of Cm and n = tdim V then rank JG (0) = m−n where g1 , . . , gk is a set of generators for id V . It follows from the implicit function theorem that we may choose holomorphic coordinates for a neighborhood of zero so that the last m − n coordinate functions belong to the set {g1 , . . , gk }. In other words, we may assume that zn+1 , . . , zm ∈ id V . This means that V is contained in an n dimensional subspace of Cm as required.

Z˜n . 7 Definition. A prime ideal is called strictly regular in the variables zm+1 , . . , zn if it ˜ is generated over m M ˜ by the single element z˜m+1 . 8 Lemma. Any non-zero prime ideal P can be made strictly regular in some set of variables zm+1 , . . , zn by a linear change of coordinates. Proof. By a linear change of coordinates we can make P regular in some set of variables ˜ is an algebraic field extension of m M ˜ by the zm+1 , . . , zn . Then, as noted above, n M ˜ is actually generelements zm+1 , .

Zn ], prove it for n O. 2. Prove that if U ⊂ Cn is a connected open set and D ⊂ U is a closed subvariety, then U − D is also connected. 3. Prove that if P ⊂ n H is a prime ideal which is just regular in the variables zm+1 , . . , zn , then the projection Cn = Cm × Cn−m → Cm still exhibits loc P as a finite branched holomorphic cover of pure order r over Cm for some r. 4. In a finite branched holomorphic cover π : V → W prove that each point where the branching order is one is a regular point of the variety V - that is, a point which has a neighborhood in V biholomorphic to a polydisc in Cm for some m.

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An elementary theory of Eisenstein series by Kubota T.


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