Singular Modular Forms and Theta Relations by Eberhard Freitag

By Eberhard Freitag

This examine monograph reviews on contemporary paintings at the concept of singular Siegel modular sorts of arbitrary point. Singular modular types are represented as linear combos of theta sequence. The reader is thought toknow merely the fundamental conception of Siegel modular types.

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D + e)! F (xK )G(xL ) (K,L)∈sh(d,e) . p The rest of the proof is an approximation argument. Consider the discretization i/n , i = 1, . . , n of [0, 1] ; given continuous F , G on [0, 1]d and [0, 1]e , we approximate F , G by step functions as above and construct corresponding polynomials f , g . As n → ∞ , these functions are dense in Lp ([0, 1]d ) and Lp ([0, 1]e ) . 9. The constant c2 (d, e) is c2 (d, e) = d+e d 1/2 . 7. Then 2 F (xK )G(xL ) = 2 (K,L)∈sh(d,e) (K , L )∈sh(d , e ) (K , L )∈sh(d , e ) [0,1]d +e F (xK )G(xL ) F (xK )G(xL ) dx and this is equal to d+e d F 2 2 G 2 2 + (K,L)=(K ,L ) [0,1]d +e F (xK )G(xL ) F (xK )G(xL ) dx.

Choose j ∈ {0, . . , n} such that xj |ϕ(X ) = 0 . Then the vector (f0 , . . , fn ) is proportional to (ϕ∗ x0 /ϕ∗ xj , . . , ϕ∗ xn /ϕ∗ xj ) ∈ K(X)n+1 and we may assume that they are equal. ,n Z where the sums range over all prime divisors Z of X . By the valuative criterion of properness (cf. 10), the domain U of ϕ has a complement of codimension at least 2 . 7. By choosing a trivialization of (ϕ|U )∗ OPn (1) at a generic point of Z , we may view ϕ∗ (xi ) as regular functions in Z . ,n and thus ordZ (ϕ∗ xj ) deg Z.

The global height hλ is independent of the choices of F and of the section s . 7, the global height is independent of F . Its independence from the choice of s can be verified as follows. Let t be another non-zero meromorphic section of O(D) with P ∈ / supp(D(t)). 5 show that λD(s) (P, v) − λD(t) (P, v) = λs/t (P, v) for any v ∈ MF . On the other hand, the product formula shows that the global height of P relative to λs/t is 0, proving the claim. 5. As an immediate consequence the global height relative to the natural local height of a non-zero rational function is identically 0.

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