Computational Aspects of Modular Forms and Galois Representations by Edixhoven Bas;Couveignes Jean-Marc;de Jong Robin;Merkl Franz;Bosman Johan;

Author:Edixhoven, Bas;Couveignes, Jean-Marc;de Jong, Robin;Merkl, Franz;Bosman, Johan;
Language: eng
Format: epub
Publisher: Princeton University Press

9.1.5 Proposition Let l > 5 be prime a prime number. The rational function bl on X1(5l)[ζ5l] from Proposition 8.2.9 extends to a morphism to

9.2 CONTROLLING Dx − D0

In this subsection, the hypotheses are as follows (unless stated otherwise). We let K be a number field, OK its ring of integers, B := Spec(OK), p : → B a regular, split semistable curve over B whose generic fiber X → Spec K is geometrically irreducible and of genus g ≥ 1. We let D be the closure in of an effective divisor of degree g (also denoted D) on X. We let x be a K-rational torsion point of the Jacobian of X, that is, a torsion element of Pic(X), which has the property that there is a unique effective divisor Dx on X such that x = [Dx − D]. Finally, we let P : B → be a section of p, that is, an element of (B).

We denote by Φx,P the unique finite vertical fractional divisor Φ (that is, with rational coefficients that are not necessarily integral) on such that (Dx − D − Φ, C) = 0 for all irreducible components C of fibers of p, and such that P(B) is disjoint from the support of Φ. It is not difficult to see that a Φ satisfying the first condition exists and that it is unique up to adding multiples of fibers of p (the intersection pairing restricted to the divisors with support in a fiber is negative semidefinite); see Lemme 6.14.1 of [Mor2]. The second condition removes the ambiguity of adding multiples of fibers.

We denote by δs the number of singular points in the geometric fiber at a closed point s of B.

9.2.1 Theorem The OB-module R1p*Oχ(Dx) is a torsion module on B, and we have:

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