Decomposing Jacobian Varieties Jen Paulhus www.math.grinnell.edu/~paulhusj
Plan of Attack • Why decompose Jacobian Varieties? • How to use group actions to decompose them. • Some results. • An intermediate interlude. • Some work in progress.
Motivating Questions I . For a fixed genus what is the largest positive g integer such that there is some genus curve t g over , and some elliptic curve with Jacobian ℚ E JX ∼ E t × A variety ? so the largest can be is . t g dim( JX ) = g
Motivating Questions II . In genus 2, Jacobians of curves with nontrivial automorphism group decompose into the product of two elliptic curves. Often those elliptic curves have interesting arithmetic properties. III . Ekedahl and Serre [1993] find curves in various genera up to 1297 with a curve X of that genus with JX ∼ E 1 × E 2 × ⋯ × E g
Motivating Questions IV . Once we have a decomposition, what are the factors? Do they have complex multiplication? Are there fixed factors across families of curves? V . Equations of curves are generally hard to come by, so can we answer these questions without access to an equation?
Group Actions The following technique will work over any field, as long as you know the automorphism group of the corresponding curve over that field. Given a compact Riemann surface of genus , g ≥ 2 X G = Aut ( X ) let (or a possible subgroup) and let be the quotient, of genus X / G = X G h .
Riemann’s Existence Theorem A finite group G acts on a compact Riemann surface X of genus g >1 if and only if there are elements of the group a 1 , b 1 , …, a h , b h , c 1 , …, c r which generate the group, satisfy the following equation, h r ∏ ∏ [ a i , b i ] c j = 1 G i =1 j =1 and so that satisfy the Riemann Hurwitz formula m j = | c j | j =1 ( 1 − 1 m j ) . r g = 1 + | G | ( h − 1) + | G | ∑ 2 Signature: Generating vector: ( h ; m 1 , …, m r ) ( a 1 , b 1 , …, a h , b h , c 1 , …, c r )
An idempotent in produces a factor of End ℚ ( JX ) f the Jacobian: . f ( JX ) But the endomorphism ring is complicated, so we start with and translate using the natural map ℚ [ G ] η : ℚ [ G ] → End ℚ ( JX )
From a theorem of Wedderburn we know that ℚ [ G ] ≅ M n 1 ( Δ 1 ) × ⋯ M n s ( Δ s ) where the are division rings. Δ i is the idempotent of with the zero matrix in π i , j ℚ [ G ] every component except the i th component where it has a 1 in the position only. Then j , j 1 ℚ [ G ] = ∑ π i , j . i , j
1 ℚ [ G ] = ∑ π i , j . i , j This translates by a result of Kani and Rosen [1989] to an isogeny JX ∼ η ( π 1,1 ) JX × η ( π 1,2 ) JX × ⋯ × η ( π 1, n 1 ) JX × ⋯ × η ( π s , n s ) JX 1 × ⋯ B n s JX ∼ B n 1 s We call this the group algebra decomposition.
For a special -representation V called the Hurwitz ℚ representation with character χ V dim B i = 1 2 ⟨ χ i , χ V ⟩ where the are the irreducible -characters of G . χ i ℚ is the representation of on V G H 1 ( X , ℤ ) ⊗ ℚ .
Definition Given a branched cover with monodromy X → X G c 1 , …, c r , r ∑ χ V = 2 χ 0 + 2( h − 1) ρ ⟨ 1 G ⟩ + ( ρ ⟨ 1 G ⟩ − ρ ⟨ c i ⟩ ) i =1 where is the character of induced from the ρ ⟨ c i ⟩ G trivial character of the subgroup , and is the χ 0 ⟨ c i ⟩ . trivial character of G .
where dim B i = 1 JX ∼ B n 1 1 × ⋯ × B n s 2 ⟨ χ i , χ V ⟩ s To compute the dimensions of the factors of JX , we need • Automorphism group of X • Monodromy of the cover X → X G • Irreducible -characters ℚ
Some Results I. Hyperelliptic Curves Their automorphism groups are well known [2013]. II. Hurwitz Curves Curves with the largest possible automorphism groups for a fixed genus, with signature [0; 2,3,7], e.g. PSL (2, q ), A n . [2016] and current work with students. III. Completely Decomposable Jacobians with Anita Rojas [2017]
Given a genus curve , its Jacobian variety is g X JX called completely decomposable if JX ∼ E 1 × E 2 × ⋯ × E g where the are (possibly isogenous) elliptic curves. E i Ekedahl and Serre [1993] demonstrate various curves up to genus 1297 with completely decomposable Jacobian varieties. However, there are numerous “gaps” in their data. Yamauchi [2007] gives lists of integers N so that the Jacobian of the modular curve X 0 (N) is completely decomposable. His work adds genus 113, 161, and 205 to the list.
We applied the technique above to many curves up to genus 101, and a few strategically chosen curves up to genus 500, and found 7 new examples for g = {36, 46, 81, 85, 91, 193, 244} . • All curves up to genus 48 [Breuer, 2000] • Curves with automorphism group larger than 4(g-1) for genus up to 101 [Conder, 2010] • Even higher genus using LowIndexNormalSubgroup( , n) Γ
An Intermediate Interlude The rest of the talk is joint work with Anita Rojas from Universidad de Chile.
Group actions will not tell the whole story. Example Y : y 2 = ( x 2 − 4)( x 3 − 3 x + a ) . Consider the curve • Aut ( Y ) ≅ C 2 so can’t decompose by previous technique. X : y 2 = x ( x 6 + ax 3 + 1) • For we find JX ∼ JY × E 1 • We can also compute and so JX ∼ E 2 1 × E 2 JY ∼ E 1 × E 2 .
An Example There is a genus 101 curve with automorphism = C 2 group (800, 980) whose Jacobian 10 ⋊ C 8 decomposes as: JX ∼ E × A × E 2 × E 8 × ⋯ × E 8 12 where A is an abelian variety of dimension 2 .
an irreducible -representation from the i th V i ℂ irreducible -character, and the Schur index. ℚ m i Carocca and Rodriguez [2006] Given a Galois cover and X → X G dim V 1 dim Vs m 1 ms JX ∼ B × ⋯ × B , s 1 if is a subgroup of then the group algebra H G decomposition of is given as JX H dim VH dim VH 1 s m 1 JX H ∼ B × ⋯ × B ms , s 1 where is the subspace of fixed by . V H H V i i
dim VH dim VH dim V 1 dim Vs 1 s m 1 ms JX ∼ B × ⋯ × B m 1 JX H ∼ B × ⋯ × B ms s s 1 1 We can compute as dim V H ⟨ V i , ρ H ⟩ . i We look for large genus decompositions that need not have all elliptic curves in their decomposition …
dim VH dim VH dim V 1 dim Vs 1 s m 1 ms JX ∼ B × ⋯ × B m 1 JX H ∼ B × ⋯ × B ms s s 1 1 We can compute as dim V H ⟨ V i , ρ H ⟩ . i We look for large genus decompositions that need not have all elliptic curves in their decomposition … … then we compute the dimensions of the exponents hoping they are 0 for any i which is not an elliptic curve in the decomposition of JX.
An Example (again) There is a genus 101 curve with automorphism group (800, 980) whose Jacobian decomposes as: JX ∼ E × A × E 2 × E 8 × ⋯ × E 8 12 where A is an abelian variety of dimension 2 . The group has 3 subgroups which produce quotients of genus 51. Using the previous slide, we get: JX H ∼ E × E 2 × E 4 × ⋯ × E 4 12
P. and Rojas (2017) For every integer g in the following list, there is a curve of genus g with completely decomposable Jacobian variety found using a group acting on a curve. 1–29, 30, 31, 32, 33, 34–36 , 37, 39, 40, 41, 42, 43, 44, 45, 46, 47 , 48, 49, 50, 51–52, 53 , 54, 55, 57, 58, 61, 62, 63, 64, 65, 67, 69, 71, 72–73, 79–81, 82 , 85, 89, 91, 93, 95, 97, 103, 105–107, 109, 118, 121, 125, 129, 142, 145, 154, 161, 163, 193, 199, 211, 213, 217, 244, 257, 325, 433 The numbers in pink are new genera, the others are different examples from those Ekedahl and Serre found.
Algorithmifying The rest of the talk is work in early stages. Use prior work of Auffarth, Behn, Lange, Rodríguez, Rojas
a complex torus where is a complex vector A = V A / L A V A space of dimension and is a lattice in . g L A V A A polarization on is a non-degenerate real A alternating form satisfying E and E ( L A × L A ) ⊆ ℤ . E ( iu , iv ) = E ( u , v ) A polarization is of type if there exists a ( d 1 , d 2 , …, d g ) basis for such that the matrix for with respect to L A E that basis has the form E = ( 0 ) . 0 D − D where is the diagonal matrix formed from the D type: ( d 1 , d 2 , …, d g ) .
Such a basis is called symplectic. γ A = { α 1 , …, α g , β 1 , …, β g } The period matrix for is the matrix g × 2 g A Π A = ( D Z ) where is a complex symmetric matrix such g × g Z that is positive definite. ℑ ( Z )
Goal: Given a principally polarized abelian variety , its period matrix, and an idempotent A , compute the period matrix of the corresponding f factor . f ( A ) • If the factor is an elliptic curve: the period matrix will be of the form (1 τ ) . • If the factor is higher dimensional: Auffarth, Lange and Rojas [2017] give criterion on Neron- Severi group to determine additional factorization.
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