Elkies trinomial curves

In number theory, the Elkies trinomial curves are certain hyperelliptic curves constructed by Noam Elkies which have the property that rational points on them correspond to trinomial polynomials giving an extension of Q with particular Galois groups.

One curve, C₁₆₈, gives Galois group PSL(2,7) from a polynomial of degree seven, and the other, C₁₃₄₄, gives Galois group AL(8), the semidirect product of a 2-elementary group of order eight acted on by PSL(2, 7), giving a transitive permutation subgroup of the symmetric group on eight roots of order 1344.

The equation of the curve C₁₆₈ is

:$y^2\; =\; x(81x^5+396x^4+738x^3+660x^2+269x+48)$

The curve is a plane algebraic curve model for a Galois resolvent for the trinomial polynomial equation x⁷ + bx + c = 0. If there exists a point (x, y) on the (projectivized) curve, there is a corresponding pair (b, c) of rational numbers, such that the trinomial polynomial either factors or has Galois group PSL(2,7), the finite simple group of order 168. The curve has genus two, and so by Faltings theorem there are only a finite number of rational points on it. These rational points were proven by Nils Bruin using the computer program Kash to be the only ones on C168, and they give only four distinct trinomial polynomials with Galois group PSL(2,7): x⁷-7x+3 (the Trinks polynomial), x⁷-154x+99 (the Erbach-Fisher-McKay polynomial) and two new polynomials with Galois group PSL(2,7),

:$x^7-frac\{28\}\{1369\}x+frac\{9\}\{1369\}$

and

:$x^7-frac\{23959\}\{249001\}x+frac\{9153\}\{249001\}$.

The equation of C₁₃₄₄ is

:$y^2\; =\; 2\; x^6\; +\; 4\; x^5\; +\; 36\; x^4\; +\; 16\; x^3\; -\; 45\; x^2\; +\; 190\; x\; +\; 1241$

Once again the genus is two, and by Faltings theorem the list of rational points is finite. It is thought the only rational points on it correspond to polynomials x⁸+16x+28, x⁸+576x+1008, x⁸+x/363527+1/6907013, and x⁸+324x+567, which comes from two different rational points. All but the last have Galois group AL(8), but the last one has Galois group PSL(2, 7) again, this time as the Galois group of a polynomial of degree eight.

References

*cite conference
author = Bruin, Nils; Elkies, Noam
title = Trinomials "ax"⁷+"bx"+"c" and "ax"⁸+"bx"+"c" with Galois Groups of Order 168 and 8⋅168
booktitle = Algorithmic Number Theory: 5th International Symposium, ANTS-V
publisher = Lecture Notes in Computer Science, vol. 2369, Springer-Verlag
date = 2002
pages = 172–188
id = MathSciNet | id = 2041082

*cite journal
author = Erbach, D. W.; Fisher, J.; McKay, J.
title = Polynomials with PSL(2,7) as Galois group
journal = Journal of Number Theory
volume = 11
year = 1979
issue = 1
pages = 69–75
id = MathSciNet | id = 0527761
doi = 10.1016/0022-314X(79)90020-9