Moduli spaces of genus 2 curves with split Jacobians through K3 - PowerPoint PPT Presentation

Moduli spaces of genus 2 curves with split Jacobians through K3 surfaces Abhinav Kumar Stony Brook October 22, 2015 Abhinav Kumar Genus 2 curves with split Jacobians 1 / 33

Introduction Let C be a genus 2 curve over a field k , and J = Jac ( C ) its Jacobian. Recall that we say J is simple if it does not contain a proper abelian subvariety. Otherwise we say that J is reducible or decomposable or split. Abhinav Kumar Genus 2 curves with split Jacobians 2 / 33

Introduction Let C be a genus 2 curve over a field k , and J = Jac ( C ) its Jacobian. Recall that we say J is simple if it does not contain a proper abelian subvariety. Otherwise we say that J is reducible or decomposable or split. The only possibility for a genus 2 curve is that J is isogenous to a product E × E ′ of elliptic curves. Equivalently, there is a degree n map C → E to some elliptic curve, for some natural number n . Abhinav Kumar Genus 2 curves with split Jacobians 2 / 33

Introduction Let C be a genus 2 curve over a field k , and J = Jac ( C ) its Jacobian. Recall that we say J is simple if it does not contain a proper abelian subvariety. Otherwise we say that J is reducible or decomposable or split. The only possibility for a genus 2 curve is that J is isogenous to a product E × E ′ of elliptic curves. Equivalently, there is a degree n map C → E to some elliptic curve, for some natural number n . Goal: give explicit examples of genus 2 curves with split Jacobians, and explicit descriptions of moduli spaces of such ( C , E ). Abhinav Kumar Genus 2 curves with split Jacobians 2 / 33

Examples We give a few examples of split (over Q ) Jacobians. Example Classical example: X 0 (37) is a genus 2 curve with equation y 2 = x 6 + 8 x 5 − 20 x 4 + 28 x 3 − 24 x 2 + 12 x − 4 Its Jacobian J 0 (37) is (2 , 2)-isogenous to a product of elliptic curves with j -invariants 37 / 54 2 and − 37 2 / 54 3 . Abhinav Kumar Genus 2 curves with split Jacobians 3 / 33

Examples II Example Let C be the curve y 2 = ( x 3 + 420 x − 5600)( x 3 + 42 x 2 + 1120) Then J ( C ) is (3 , 3)-isogenous to a product of elliptic curves with j -invariants − 2 7 · 7 2 and − 2 5 · 7 · 17 3 . Abhinav Kumar Genus 2 curves with split Jacobians 4 / 33

Examples II Example Let C be the curve y 2 = ( x 3 + 420 x − 5600)( x 3 + 42 x 2 + 1120) Then J ( C ) is (3 , 3)-isogenous to a product of elliptic curves with j -invariants − 2 7 · 7 2 and − 2 5 · 7 · 17 3 . Abhinav Kumar Genus 2 curves with split Jacobians 4 / 33

Examples II Example Let C be the curve y 2 = ( x 3 + 420 x − 5600)( x 3 + 42 x 2 + 1120) Then J ( C ) is (3 , 3)-isogenous to a product of elliptic curves with j -invariants − 2 7 · 7 2 and − 2 5 · 7 · 17 3 . A morphism of degree 3 to the elliptic curve y 2 1 = x 3 1 + 4900 x 2 1 + 7031500 x 1 + 2401000000 is given by y 1 = 49000( x 3 − 21 x 2 − 140) y x 1 = − 882000( x − 14) x 3 + 420 x − 5600 , . ( x 3 + 420 x − 5600) 2 Abhinav Kumar Genus 2 curves with split Jacobians 4 / 33

Examples III Example The genus two curve defined by y 2 = ( x 3 − 10 x 2 − 816 x − 204)( x 3 + 4 x 2 − 893 x − 13861) has Jacobian (7 , 7)-isogenous to a product of elliptic curves with j -invariants 2 19 3 2 / 5 5 and 2 8 3 2 7 2 139 3 / 5 2 . Abhinav Kumar Genus 2 curves with split Jacobians 5 / 33

Examples III Example The genus two curve defined by y 2 = ( x 3 − 10 x 2 − 816 x − 204)( x 3 + 4 x 2 − 893 x − 13861) has Jacobian (7 , 7)-isogenous to a product of elliptic curves with j -invariants 2 19 3 2 / 5 5 and 2 8 3 2 7 2 139 3 / 5 2 . Example The genus two curve defined by y 2 = ( x 3 +496 x 2 +52302 x − 2673552)( x 3 +584 x − 271740 x +24634652) has Jacobian (11 , 11)-isogenous to a product of elliptic curves with j -invariants 2 12 29 3 3 6 5 2 43 and 2 1 231 3 1709 3 15391 3 . 3 5 5 3 43 2 109 11 Abhinav Kumar Genus 2 curves with split Jacobians 5 / 33

A moduli space example Theorem A birational model for the surface parametrizing ( C , E ) related by a degree 7 map is given by z 2 = − 16 s 4 r 4 + 2 s (20 s 2 + 17 s − 1) r 3 − (44 s 3 + 57 s 2 + 18 s − 1) r 2 + 2 s (15 s + 17) r + s 2 . It is a singular (i.e. of Picard number 20 ) elliptic K3 surface. Abhinav Kumar Genus 2 curves with split Jacobians 6 / 33

A tautological curve example Example Over the moduli space � L 3 , which has equation z 2 = 11664 r 2 − 8(54 s 3 + 27 s 2 − 72 s + 23) r + ( s − 1) 4 (2 s − 1) 2 , a tautological family of genus 2 curves is given by � � x 3 + 3(3 s − 1) x 2 − 2( m − 27)(9 s − 5) 2 y 2 = ( m + 27) � � x 3 − 3( m − 27)(9 s − 5) 2 x + ( m − 27)(9 s − 5) 3 × 4( m + 27) 4( m + 27) � � z − ( s − 1) 2 (2 s − 1) where m = / (4 r ). Abhinav Kumar Genus 2 curves with split Jacobians 7 / 33

Background Clasically, such pairs ( C , E ) arose in the reduction of hyperelliptic integrals to elliptic ones. Studied by Legendre, Jacobi, Bolza, Humbert, etc. Example � � dx dy √ = � x 6 + 2 x 4 + 5 x 2 + 1 y ( y 3 + 2 y 2 + 5 y + 1) 2 � − dz = √ z 3 + 5 z 2 + 2 z + 1 2 More recently, their moduli spaces were studied from a computational perspective by Shaska and his collaborators, and also from an arithmetic standpoint by Frey, Kani, etc. Abhinav Kumar Genus 2 curves with split Jacobians 8 / 33

Facts 1 Let C → E of degree n . Then J ( C ) → E × E ′ by an ( n , n )-isogeny. And conversely. Abhinav Kumar Genus 2 curves with split Jacobians 9 / 33

Facts 1 Let C → E of degree n . Then J ( C ) → E × E ′ by an ( n , n )-isogeny. And conversely. 2 Then J ( C ) → E × E ′ an ( n , n )-isogeny over a field k . It gives rise to E [ n ] ∼ = E ′ [ n ] an anti-isometry (under Weil pairing), Galois invariant. Abhinav Kumar Genus 2 curves with split Jacobians 9 / 33

Facts 1 Let C → E of degree n . Then J ( C ) → E × E ′ by an ( n , n )-isogeny. And conversely. 2 Then J ( C ) → E × E ′ an ( n , n )-isogeny over a field k . It gives rise to E [ n ] ∼ = E ′ [ n ] an anti-isometry (under Weil pairing), Galois invariant. 3 There is a map of degree n induced on the hyperelliptic quotients φ C E ψ P 1 P 1 Abhinav Kumar Genus 2 curves with split Jacobians 9 / 33

Usual method The standard approach to studying the moduli space is by looking at the covers P 1 → P 1 of degree n which lift to a degree n map from a genus 2 curve to an elliptic curve. Abhinav Kumar Genus 2 curves with split Jacobians 10 / 33

Usual method The standard approach to studying the moduli space is by looking at the covers P 1 → P 1 of degree n which lift to a degree n map from a genus 2 curve to an elliptic curve. (This becomes some condition on the branch points of ψ . Without any conditions we’d have a Hurwitz space). Abhinav Kumar Genus 2 curves with split Jacobians 10 / 33

Usual method The standard approach to studying the moduli space is by looking at the covers P 1 → P 1 of degree n which lift to a degree n map from a genus 2 curve to an elliptic curve. (This becomes some condition on the branch points of ψ . Without any conditions we’d have a Hurwitz space). Then do some invariant theory to mod out by choices. The previous state of the art: knew explicit models of the moduli spaces for n ≤ 5. Abhinav Kumar Genus 2 curves with split Jacobians 10 / 33

New approach Space of J ( C ) reducible by ( n , n )-isogeny is a hypersurface in A 2 , the Humbert surface H n 2 of discriminant n 2 . Abhinav Kumar Genus 2 curves with split Jacobians 11 / 33

New approach Space of J ( C ) reducible by ( n , n )-isogeny is a hypersurface in A 2 , the Humbert surface H n 2 of discriminant n 2 . Reason : decomposition corresponds to real multiplication by quadratic ring of discriminant n 2 . The choice of E corresponds to a double cover, the Hilbert modular surface Y − ( n 2 ) of discriminant n 2 . Abhinav Kumar Genus 2 curves with split Jacobians 11 / 33

New approach Space of J ( C ) reducible by ( n , n )-isogeny is a hypersurface in A 2 , the Humbert surface H n 2 of discriminant n 2 . Reason : decomposition corresponds to real multiplication by quadratic ring of discriminant n 2 . The choice of E corresponds to a double cover, the Hilbert modular surface Y − ( n 2 ) of discriminant n 2 . Previously, in joint work with Elkies, we developed a method to parametrize Hilbert modular surfaces of fundamental discriminant D as moduli spaces of K3 surfaces. I’ll go over this method next, and then describe the additional ideas in the new work. Abhinav Kumar Genus 2 curves with split Jacobians 11 / 33

� Elliptic K3 surfaces with Shioda-Inose structure We say that a K3 surface X has a Shioda-Inose structure if it has a symplectic involution ι such that the quotient map is a Kummer surface Km( A ) and the 2 : 1 quotient maps X A ❅ � ⑦ ⑦ ⑦ ⑦ ❅ ❅ ❅ Y induce a Hodge isometry of transcendental lattices T X ∼ = T A . Abhinav Kumar Genus 2 curves with split Jacobians 12 / 33