Elkies trinomial curves

Elkies trinomial curve C₁₆₈

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), (1/11)x⁷-14x+3² (the Erbach-Fisher-McKay polynomial) and two new polynomials with Galois group PSL(2,7),

37^2x^7-28x+3^2

and

(499^2/113)x^7-212x+3^4

On the other hand, the equation of curve 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, 19⁴53x⁸+19x+2 which have Galois group AL(8), and x⁸+324x+567, which comes from two different rational points and has Galois group PSL(2, 7) again, this time as the Galois group of a polynomial of degree eight.

References

Bruin, Nils; Elkies, Noam (2002). "Trinomials ax⁷+bx+c and ax⁸+bx+c with Galois Groups of Order 168 and 8⋅168". Algorithmic Number Theory: 5th International Symposium, ANTS-V. Lecture Notes in Computer Science, vol. 2369, Springer-Verlag. pp. 172–188. MR 2041082.

Erbach, D. W.; Fisher, J.; McKay, J. (1979). "Polynomials with PSL(2,7) as Galois group". Journal of Number Theory 11 (1): 69–75. doi:10.1016/0022-314X(79)90020-9. MR 0527761.

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