- Elkies trinomial curves
In
number theory , the Elkies trinomial curves are certainhyperelliptic curve s constructed byNoam Elkies which have the property that rational points on them correspond to trinomial polynomials giving an extension of Q with particularGalois group s.One curve, C168, gives Galois group
PSL(2,7) from a polynomial of degree seven, and the other, C1344, gives Galois group AL(8), thesemidirect product of a2-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 byFaltings theorem there are only a finite number of rational points on it. These rational points were proven by Nils Bruin using the computer programKash 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
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