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Solving conics over function fields

Mark van Hoeij;  John Cremona — 2006

Journal de Théorie des Nombres de Bordeaux

Let F be a field whose characteristic is not  2 and K = F ( t ) . We give a simple algorithm to find, given a , b , c ∈ K * , a nontrivial solution in  K (if it exists) to the equation a X 2 + b Y 2 + c Z 2 = 0 . The algorithm requires, in certain cases, the solution of a similar equation with coefficients in F ; hence we obtain a recursive algorithm for solving diagonal conics over ℚ ( t 1 , ⋯ , t n ) (using existing algorithms for such equations over  ℚ ) and over 𝔽 q ( t 1 , ⋯ , t n ) .

On a theorem of Mestre and Schoof

John E. Cremona;  Andrew V. Sutherland — 2010

Journal de Théorie des Nombres de Bordeaux

A well known theorem of Mestre and Schoof implies that the order of an elliptic curve E over a prime field 𝔽 q can be uniquely determined by computing the orders of a few points on E and its quadratic twist, provided that q > 229 . We extend this result to all finite fields with q > 49 , and all prime fields with q > 29 .

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