Galois theory of a quaternion group origami
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1 Galois theory of a quaternion group origami Special Session on Automorphisms of Riemann Surfaces and Related Topics Fall Sectional Meeting Loyola University Chicago, Chicago, IL Rachel Joint work with Professor Edray Goins Purdue University October 3, 2015
2 Table of Contents Motivation 1 Motivation 2 3
3 Table of Contents Motivation 1 Motivation 2 3
4 φ : Y X X = Y = G m = multiplicative group = P 1 (C) {0, } φ : Y X φ : x x N Fix a rational point p G m (thought of in the target of the map φ, X).
5 Consider the set V = φ 1 (p) = {x G m φ(x) = p}, i.e. { x Gm x N = p }. f p (x) = x N p First, consider the case that p = 1. Then V is the set of (nonzero) solutions to f p (x) = x N 1. These are the N th roots of unity.
6 Gal(Q(ζ N )/Q) = (Z/NZ). σ i : ζ N ζn i, (i, N) = 1 Q(ζ N ) (Z/NZ) Q
7 When f p (x) = x N p is irreducible, the picture becomes the following: Q( N p, ζ N ) (Z/NZ) Q(ζ N ) (Z/NZ) Q
8 Gal(sf (x N p)/q) is a subgroup of AGL 1 (Z/NZ). There is a Galois representation ρ N,p : G Q AGL 1 (Z/NZ) This is given by σ σ( N d) N d ( a b 0 1 ) such that σ(ζ N ) = ζ p N and = ζ a N, so a (Z/NZ) and b (Z/NZ).
9 Table of Contents Motivation 1 Motivation 2 3
10 Let E be an elliptic curve over Q. Definition An origami is a pair (C, f ) where C is a curve and f : C E is a map branched only above one point. We study automorphisms of origamis and relate these to polynomials over Q.
11 Definition A deck transformation or automorphism of a cover f : C E is a homeomorphism g : C C such that f g = f. Each deck transformation permutes the elements of each fiber. This defines a group action of the the deck transformations on the fibers.
12 Let E be an elliptic curve over Q. Fix a positive integer N. We define multiplication by N on E, denoted [N] to be adding a point to itself N times. We define the N-division points of E: E[N] = { P E(Q) : [N]P = O }.
13 Facts: The N-division points form a group that is isomorphic to (Z/NZ) 2. For example, E[2] (Z/2Z) 2, a Klein 4-group. The Galois group G Q sends division points to division points. We will write Q(E[N]) to mean the field obtained by adjoining all of the coordinates of the N-division points of E to Q.
14 The Galois group of Q(E[N]) over Q is a subgroup of GL 2 (Z/NZ). Q(E[N]) GL 2 (Z/NZ) Q
15 For example, the Galois group of Q(E[2])/Q is a subgroup of the automorphism group of E[2] and After choice of basis, Aut(Z/2Z Z/2Z) S 3. Aut(Z/2Z Z/2Z) GL 2 (Z/2Z).
16 Let E be given by y 2 = x 3 + Ax + B. Fix a point P E(Q) given by P = (z : w : 1). Consider the set V = [N] 1 P = { Q E(Q) [N]Q = P }. For example, when P = O, this set is the set of N-division points. This is no longer a group in general, but we can still adjoin the coordinates of such points to Q and find the Galois group of the extension.
17 Q([N] 1 P) (Z/NZ) 2 Q(E[N]) GL 2 (Z/NZ) Q
18 The Galois group of Q([N] 1 P) over Q is a subgroup of the affine general linear group 1 (Z/NZ) 2 AGL 2 (Z/NZ) GL 2 (Z/NZ) 1 e.g. for N = 2 1 (Z/2Z) 2 S 4 S 3 1 a b e c d f : a, b, c, d, e, f (Z/NZ) and ad bc 0
19 There is a representation ρ N,P : G Q AGL 2 (Z/NZ). Let T 1, T 2 is a basis for E[N]. Suppose σ(t 1 ) = at 1 ct 2 and σ(t 2 ) = bt 1 dt 2. Choose any Q Q such that [N]Q = P. Suppose σ(q) Q = et 1 ft 2. Then the top representation is given by a b e σ c d f 0 0 1
20 Multiplication by 2 Motivation E : y 2 = x 3 + Ax + B The formula for the x-coordinate of [2]P = P P (P = (z : w : 1)) is the following: x 4 2Ax 2 8Bx + A 2 4(x 3 + Ax + B)
21 Proposition Fix a rational point P = (z : w : 1). Consider the extension F P = Q(sf ([2] 1 P)) over Q, where F P /Q is given by the splitting field of the quartic f E,P (x) = (x 4 2Ax 2 8Bx + A 2 ) 4z(x 3 + Ax + B). If this polynomial is irreducible, then Q(sf (F E,P ))/Q is an S 4 -extension. Note that S 4 = AGL 2 (Z/2Z).
22 The example 1 V 4 S 4 S 3 1 is a specific case of a more general theory. Take the semidirect product of a group and its automorphism group where the action of the quotient on the automorphism group of the normal subgroup is the identity. 1 G Hol(G) Aut(G) 1
23 Question: What about non-abelian deck groups G? Example: Q 8 group of quaternions, non-abelian group of order 8. An origami with the example deck group is studied in a paper titled, "An extraordinary origami curve" by Herrlich and Schithüsen.
24 Table of Contents Motivation 1 Motivation 2 3
25 Motivation Cover group G = Q8
26 Motivation Cover group G = Q8
27 f = 8 e = = 16 v = = 4 Formula for Euler characteristic: 2 2g = v e + f = g = 3
28 Riemann-Hurwitz formula Then f : Y Z 2g(Y ) 2 = deg(f ) (2g(Z ) 2) + z Z(e z 1) Using the formula, with g(y ) = 3 g(z ) = 1, we see that there are 4 points in Y above the 2-division points in Z, each with ramification degree 2.
29 In fact, Herrlich and Schithüsen give that Y : y 4 = x 3 + Ax + B. This is an example of a superelliptic curve. The map Y Z is given by. (x, y) (x, y 2 )
30 Let g be the composition Y Z X. To find the g 1 (P) points, we give a formula for multiplication by 2 in terms of the y-coordinates. φ 2 zψ 2 2 degree 4 in x ω 2 wψ 3 2 degree 6 in x (We will think of y as part of the coefficients).
31 The resultant polynomial for P = (z : w : 1) is y 4 8wy 3 + 6(2Az + 3B)y 2 = 0. Plugging in y 2 instead of y gives f E,P = y 8 8wy 6 + 6(2Az + 3B)y 4
32 Theorem (D., Goins) Fix a rational point P = (z : w : 1). Consider the extension F P = Q(sf (f E,P ))/Q given by the splitting field of the polynomial f E,P. If the polynomial is irreducible, then Gal(Q(f E,P )/Q) = Hol(Q 8 ).
33 Q(sf (f E,P )) Hol(Q 8 ) Q
34 1 Q 8 Hol(Q 8 ) Aut(Q 8 ) S 4 1 Therefore, Hol(Q 8 ) is a specific group of order 192. Let = 16(4A B 2 ).
35 Motivation Cover group G = Q8
36 f 1 = y y 3 + (512B Bz Bw z z 3 w w )y w B 2 z B 2 z 3 w B 2 w B B Bz w Bz 6 w Bz 3 w Bz Bw w 6 z w 4 z z z 9 w z 6 w z z 3 w z 3 w
37 Does L 1 really depend on P? There is an isomorphism from L 1 the the splitting field of x 4 4 x 12A. This is a special polynomial because it defines the S 4 extension contained inside of Q(E[4]). What about L 2?
38 Thank you. Questions?
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