Elliptic factors in Jacobians of hyperelliptic curves with certain - PowerPoint PPT Presentation

Elliptic factors in Jacobians of hyperelliptic curves with certain automorphism groups Jennifer Paulhus Grinnell College paulhusj@grinnell.edu www.math.grinnell.edu/ ∼ paulhusj

My original interest in Jacobian variety decomposition was motivated by the following question. Question Given a genus g, what is the largest integer t such that there is some curve X of genus g with J X ∼ E t × A for some elliptic curve E and an abelian variety A? The dim ( J X ) = g , so the largest t can possibly be is g .

Non-hyperelliptic Curves Auto. Jacobian Genus Group Decomposition J X ∼ E 4 4 ( 72 , 40 ) J X ∼ E 5 5 ( 160 , 234 ) J X ∼ E 6 6 ( 72 , 15 ) J X ∼ E 7 PSL ( 2 , 7 ) 7 J X ∼ E 8 8 ( 336 , 208 ) J X ∼ E 3 1 × E 6 9 ( 192 , 955 ) 2 J X ∼ E 10 10 ( 360 , 118 ) J X ∼ E 14 14 PSL ( 2 , 13 )

Decomposition Techniques

X a curve of genus g J X its Jacobian Variety G the automorphism group of X The techniques work for curves defined over any field. But a field must be specified to compute the automorphism group of the curve. We assume all curves are defined over an algebraically closed field of characteristic zero.

From a theorem of Wedderburn we know that Q [ G ] ∼ � = M n i (∆ i ) i where ∆ i are division rings. π i , j ∈ Q [ G ] with the zero matrix in every component except the i th component where it has a 1 in the j , j position and zeros elsewhere. Apply the natural map of Q -algebras e : Q [ G ] → End ( J X ) ⊗ Z Q a result of Kani-Rosen: � J X ∼ e ( π i , j ) J X . i , j

What are these e ( π i , j ) J X ? Recall : We want to find elliptic curve factors. For a special Q -character χ dim e ( π i , j ) J X = 1 2 � χ, χ i � where the χ i are the irreducible Q -characters.

Take the quotient map from X to Y = X / G , branched at s points with monodromy g 1 , . . . , g s ∈ G . g 1 · g 2 · · · g s = 1 G and � g 1 , g 2 , . . . , g s � = G Definition A Hurwitz character of a group G is a character of the form: s � � � χ = 2 χ triv + 2 ( g Y − 1 ) χ � 1 G � + χ � 1 G � − χ � g i � i = 1 χ � g i � is the character of G induced from the trivial character of � g i � and χ triv is the trivial character of G . ( Remember later : two elements in the same conjugacy class generate the same induced character.)

map from X to Y = X / G , branched at s points with monodromy g 1 , . . . , g s ∈ G s � � � χ = 2 χ triv + 2 ( g Y − 1 ) χ � 1 G � + χ � 1 G � − χ � g i � i = 1 To compute a Hurwitz character we need to know: signature – ( m 1 , m 2 , . . . , m s ) where m i is order of g i . monodromy – g 1 , . . . , g s To compute the dimension of the factor, we also need the irreducible Q -characters

Q [ G ] ∼ � � M n i (∆ i ) and J X ∼ e ( π i , j ) J X = i i , j Recall : We want to find lots of isogenous elliptic curves. Theorem (P ., ’07) With notation as above, e ( π i , j ) J X is isogenous to e ( π i , k ) J X . If there is some i with 1 2 � χ, χ i � = 1, then there are n i isogenous elliptic curves in the factorization of J X .

Low Genus Results

Brandt and Stichtenoth (’86) and Shaska (’03) completely classify all automorphism groups of hyperelliptic curves of any genus over an algebraically closed field of characteristic zero. ω the hyperelliptic involution, then the reduced automorphism group ( G / � ω � ) must be a dihedral group, a cyclic group, A 4 , S 4 , or A 5 . We consider hyperelliptic curves with reduced automorphism group one of A 4 , S 4 , or A 5 .

signature monodromy irreducible Q -characters Signatures are in Shaska’s paper. Recall : two elements in the same conjugacy class will generate the same induced character. For small cases we can search through the group to find elements of the group satisfying the monodromy conditions Character tables for these groups are well known.

Theorem (P .) The hyperelliptic curve of genus 4 with affine model √ X : y 2 = x ( x 4 − 1 )( x 4 + 2 − 3 x 2 + 1 ) has a Jacobian variety that decomposes as E 2 1 × E 2 2 for two elliptic curves E i . Theorem (P .) The genus 5 hyperelliptic curve with affine model X : y 2 = x ( x 10 + 11 x 5 − 1 ) has J X ∼ E 5 for the elliptic curve E : y 2 = x ( x 2 + 11 x − 1 ) .

Genus Auto. Group Dim. Jac. Decomp. E 2 1 × E 2 4 SL 2 ( 3 ) 0 2 E 3 × A 2 5 A 4 × C 2 1 E 2 1 × E 3 W 2 =( 48 , 30 ) 0 2 E 5 A 5 × C 2 0 E 2 1 × E 4 6 GL 2 ( 3 ) 0 2 E 1 × E 3 2 × E 3 7 A 4 × C 2 1 3 A 2 2 , 1 × A 2 8 SL 2 ( 3 ) 1 2 , 2 E 4 × A 2 W 3 =( 48 , 28 ) 0 2 E 3 × A 3 9 A 4 × C 2 1 2 E 1 × E 2 2 × A 3 W 2 0 2 E 4 1 × E 5 A 5 × C 2 0 2 A 2 2 × A 2 10 SL 2 ( 3 ) 1 3

Help From a Computer Program

Thomas Breuer wrote a program which classifies all automorphism groups of Riemann Surfaces for a given genus g . It requires a search through a database of all groups up to a certain order. In the late 1990s he ran it in GAP3 for genus up to 48. GAP had complete classification of groups up to order 1000 (except 512 and 768). Branching data is computed in the execution of his algorithms but was not recorded.

I rewrote the program in MAGMA and added functionality to output monodromy data. input the known signature and known automorphism group of a curve and output elements of the monodromy Breuer devised a (recursive) algorithm to handle higher genus but never implemented it. I have also started to implement the higher genus algorithm.

Automorp. Jacobian Genus Group Dimen. Decomposition A 2 × A 3 11 A 4 × C 2 2 3 E 3 × A 2 , 1 × A 3 S 4 × C 2 1 2 , 2 A 2 2 × A 2 12 SL 2 ( 3 ) 1 4 A 2 2 , 1 × A 4 W 3 0 2 , 2 E × A 3 , 1 × A 3 13 A 4 × C 2 2 3 , 2 A 2 3 × A 2 14 SL 2 ( 3 ) 2 4 A 4 2 × A 2 GL 2 ( 3 ) 1 3 E 4 1 × E 6 2 × A 2 SL 2 ( 5 ) 0 2 A 3 2 × A 3 15 A 4 × C 2 2 3 E × E 2 2 × A 3 S 4 × C 2 1 4 E 4 1 × E 5 2 × A 3 A 5 × C 2 0 2

Automorp. Jacobian Genus Group Dimen. Decomposition A 2 3 × A 2 16 SL 2 ( 3 ) 2 5 E × A 4 , 1 × A 3 17 A 4 × C 2 3 4 , 2 E × A 2 2 × A 3 W 2 1 4 A 2 3 × A 2 18 SL 2 ( 3 ) 2 6 A 2 3 , 1 × A 4 GL 2 ( 3 ) 1 3 , 2 E × A 3 2 × A 3 A 4 × C 2 19 3 4 A 2 4 × A 2 SL 2 ( 3 ) 20 3 6 A 2 2 , 1 × A 2 2 , 2 × A 4 W 3 1 3 E 4 × A 2 2 , 1 × A 6 SL 2 ( 5 ) 0 2 , 2

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