# A general theory of fibre spaces with structure sheaf by A Grothendieck

By A Grothendieck

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45 A. Garcia and H. 4. It was studied in detail in [8]. , q= 3 for some prime power . We define the tower W5 over Fq recursively by the equation Y −Y −1 = 1 − X + X −( −1) . 18)) of the tower W4 . The tower in [8] is given recursively by U +U −1 1−V . 20) define in fact the same tower. 2. 19) as follows: f (Y ) = 1 − f 1 X X −1 with f (T ) := T − T −1 . 21) One verifies easily that the polynomial f (T ) satisfies the equation T · (f (T ) +1 − f (T ) + 1) = (T − 1) 2+ +1 + 1. 13 and therefore we consider the set S := {α ∈ Fq | (α − 1) Since q = 3, 2+ +1 = −1} \ {0}.

Case p = 5 and f (X, Y ) = (4X + 1)Y 2 + (X 2 + X + 2)Y + X + 3. - Case p = 7 and f (X, Y ) = (X 2 + 6)Y 2 + XY + X 2 + 4. All four towers above were shown to be elliptic modular by Elkies (see the appendix in [33]). We finish this section with the remark that all optimal recursive towers presented here support the “modularity conjecture” for such towers; a conjecture which was proposed by N. Elkies in [14]. 5. , towers such that there exist places which are wildly ramified in F. 2). Typical examples of wild towers are the so-called Artin-Schreier towers F = (F0 , F1 , F2 , .

Of Li ) and Pi+1 is a place of Mi+1 (resp. of Li+1 ) above Pi , then the different exponent of Pi+1 |Pi is given by d(Pi+1 |Pi ) = 2 · (e(Pi+1 |Pi ) − 1). Proof. i) All solutions of the equation γ + γ = 0 are in Fq , and the maps y → y+γ yield distinct automorphisms of F over Fq (x). Hence the extension F/Fq (x) is Galois of degree , and the Galois group is abelian. 4) and the same argument as above shows that the extension F/Fq (y) is also abelian of degree . 4). iii) The existence of intermediate fields Mi (resp.