A Lie algebra method for rational parametrization of Severi–Brauer surfaces

Abstract

It is well known that a Severi-Brauer surface has a rational point if and only if it is isomorphic to the projective plane. Given a Severi-Brauer surface, we study the problem to decide whether such an isomorphism to the projective plane, or such a rational point, does exist; and to construct such an isomorphism or such a point in the affirmative case. We give an algorithm using Lie algebra techniques. The algorithm has been implemented in Magma.

Point counting on curves using a gonality preserving lift

“…For higher genus we have written an experimental version of ConvertScroll in Magma, which can be found in the file convertscroll.m. It blindly relies on Schicho's function ParametrizeScroll, which implements the Lie algebra method from [18]. Unfortunately the latter is only guaranteed to work in characteristic zero, and indeed one runs into trouble when naively applying ParametrizeScroll over finite fields of very small characteristic; empirically however, we found that p > g suffices for a slight modification of ParametrizeScroll to work consistently.…”

Computing the Cassels–Tate pairing on the 3-Selmer group of an elliptic curve

“…We then search for a zero-divisor in A by looking at short vectors in this lattice. Once a zero-divisor is found, it is easy to find an isomorphism A ∼ = Mat 3 (K), as described for example in [19,Section 5]. In [11, Paper III, Section 6] it is shown that if K = Q then the shortest vector in the lattice is a zero-divisor.…”

“…The problem of trivialising an n × n matrix algebra (that is, given structure constants for an L-algebra known to be isomorphic to Mat n (L), find such an isomorphism explicitly) is equivalent in the case n = 2 to solving a conic. For n > 2, this problem has been studied in [19,Section 5], [11,Paper III,Section 6], [24], with the result that practical algorithms are available if both n and the discriminant of the number field L are sufficiently small. However, since for us L is the field of definition of a 3-torsion point (which typically has degree 8), we have so far only been able to compute a few small examples.…”

Explicit $n$-descent on elliptic curves III. Algorithms

“…Since our methods are of independent interest, we have made this section self-contained. For instance our methods could be used to improve the algorithm in [29] for parametrising Brauer-Severi surfaces.…”