By A. Campillo

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In other words, all ideals of Vq are given by Therefore Vq is a discrete valuation ring. We have Vq C nq Iq Vq. If %E Ql[(pn] for a, b E Vq is in the intersection, then a is prime to q for all qlq; thus, a is prime to q. This implies Vq = nqlq V q, and Vq is integrally closed (Exercise 3). Since ( = (pn is defined by the relation 1 + (pn-l + (2 pn - + ... + ((p_l)pn-l = 0, 1 thepn-l(p-1) elements 1,(, ... ,(pn-l(p-l)-1 form a base of Z[(pn] over Z. Then we have n =n Vq = (nqZ(q))l Vq + (nqz(q))( + ...

L) = j(L) for the j-invariant defined by 3 j (w) = j (L) = ~. The following facts are known from the general theory of algebraic curves and are proven in this chapter. • • • • If ¢ : EL ---+ Eu is a holomorphic homomorphism with ¢(O) = 0, then there exists>. 39); Write a morphism ¢ : EL ---+ Eu as ¢(u) = (¢x(u), ¢y(u), 1) using the coordinates of the projective space p2. 35). The function j is an example of a modular function and g2 and g3 are examples of modular forms. Modular forms are a special kind of automorphic forms defined on a more general complex domain.

K U {oo} = P(K) (canonically). If K is not algebraically closed, take an algebraic closure K of K. Defining P(K) by the set of points in P(K) with coordinates in K, we have {Vlvaluation rings of K(x) trivial on K with K(P) = K} 9:"! P(K). 4 Algebraic Curves over a Field 41 The set {Vlvaluation rings of K(x) trivial on K} is called the set of closed points of P. They are associated with a maximal ideal of K[x] or K[X-l]. Of course, if V is associated with (t(x)) and t(x) -=I- x, then the same V is associated with x-deg(t)t(x) E K[x-l] because vp(x) = 0 if t(x) -=I- x.