New PDF release: Lectures on Theta II Birkhaeuser

Algebraic Geometry

By Mumford D Tata

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2. 1. Ej(z) is an IF,-linear polynomial of degree rj. 2. Ej(a) = 0 for all a E A(j). 3. E ~ ( T J )= I. Proof. Part 1 follows from the power series definition of Ej(z). To see Part 2, put z = a E A(j). Then ec(alogc(x)) = Ca(ec(logc(x))) = Ca(x) is a polynomial in x of degree rdeg(") < rj. Thus Ej(a) = 0. Now set a = TI. Then the above formula and our knowledge of CTj (x) imply that Ej(Tj) = 1 giving Part 3. 1. Let L be a field. We say that L is an A-field if and only if there is a morphism L:A -,L.

1. The prime ideal p which is the kernel of z is called the chamcteristic of 3. We say 3 has generic chamcteristic if and only if p = (0); otherwise we say that p is finite and 3 has finite characteristic. 1 obviously agrees with our previous definition in the case A = Pr[T]. As in Section 1, over 3 we have the ring 3 { 7 ) , with T the rthpower has a simple zero at each point of c-' Lz with derivative c. Thus v and the result is now easily established. 0 Remarks. 6. 1 . Let L be a lattice as above with associated Drinfeld module E A via can be summarized by the commutative diagram: C,/L 5 C,/L 4.

The reader will note that, as a quite general rule, the results are exactly what one would expect from the analogy with elliptic curves. F be an A-field and let 3 be a fixed algebraic closure. Let q5 and @ be two Drinfeld modules over 3of fixed rank d > 0. Recall that a morphism from 4 to tjt over 3 is an clement P(7) E 3 { r ) with for all a E A. " If "Endr(q5);" it is a subring of 317) under composition. The modules Horn(#, $), End(4) are always understood to be considered over 5. 1. Let P E 3{r) be a morphism from an isomorphism if and only if deg P(T) = 0.

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