On computing integral points of a Mordell curve over rational function fields in characteristic >3

Cubic function fields with prescribed ramification

“…However, we have imposed in our definition of a Parshin cover that this image be a three-cycle. We conclude that indeed the monodromy group of ϕ is isomorphic to S 3 , proving (2). This in fact suffices to find all Parshin covers ϕ defined over a given non-algebraically closed base field k that are ramified over a Weierstrass point of X.…”

“…For instance, cubic function fields of fixed (resp. bounded) discriminant are constructed and counted in [2,13] (using class field theory), [22] (using an algorithmic implementation of Kummer theory, applicable to a larger class of field extensions), [14,15] (building on the algorithm of [1]), and [4,17] (building on results of Shanks for cubic number fields, see e.g. [20]).…”

On computing non-Galois cubic global function fields of prescribed discriminant in characteristic $>3$

“…This can be done for each prime π dividing f separately using the Chinese Remainder Theorem. Further speed-ups can be obtained from the observation that rα 3 equals (−r)(−α) 3 thus reducing the number of candidates for r. Now we are in a situation in which we can easily establish a suitable basis for Cl f /J f (compare (3)). We represent the class group Cl in canonical form as a product of cyclic subgroups.…”

“…From the theory we know that Γ is a cyclic cubic Kummer extension of Ω of relative discriminant f 2 . By the Dedekind criterion (see [3]) we conclude that Γ = Ω( The class field theoretical computations yield the following polynomial a root of which generates E: y 3 +(8t 11 + 9t 8 + 5t 5 + 7t 2 ) y 2 +(3t 22 + 6t 16 + 2t 13 + 6t 10 + 3t 7 + 4t 4 ) y +10t 33 + 3t 30 + 2t 27 + 7t 24 + 7t 21 + 9t 18 + 5t 15 + 5t 12 + 8t 9 .…”

“…We note that the extension E/F is Galois -and in that case cyclic -precisely if ∆ is a square in o F . Since this case is straightforward it is treated separately in a different context [3]. Here, we restrict ourselves to non-Galois extensions E/F .…”