Three-gluings of elliptic curves (Revised slides) Everett W. Howe - PowerPoint PPT Presentation

Three-gluings of elliptic curves (Revised slides) Everett W. Howe Center for Communications Research, La Jolla GeoCrypt 2011 Bastia, Corsica, 20 June 2011 Everett W. Howe Three-gluings of elliptic curves 1 of 29

Motivation Two topics of interest Genus-2 curves with maps to elliptic curves Genus-2 curves with Jacobians isogenous to a product of elliptic curves These are really the same topic . . . Everett W. Howe Three-gluings of elliptic curves 2 of 29

A construction Given: Two elliptic curves E 1 , E 2 over a field k An isomorphism ψ : E 1 [ n ] → E 2 [ n ] for some n > 0, such that ψ is an anti-isometry with respect to the Weil pairing We will produce: A genus-2 curve C (possibly degenerate) Degree- n maps C → E 1 and C → E 2 Everett W. Howe Three-gluings of elliptic curves 3 of 29

A construction Given: Two elliptic curves E 1 , E 2 over a field k An isomorphism ψ : E 1 [ n ] → E 2 [ n ] for some n > 0, such that ψ is an anti-isometry with respect to the Weil pairing Weil E 1 [ n ] × E 1 [ n ] � µ n We will produce: A genus-2 curve C (possibly degenerate) Degree- n maps C → E 1 and C → E 2 Everett W. Howe Three-gluings of elliptic curves 3 of 29

A construction Given: Two elliptic curves E 1 , E 2 over a field k An isomorphism ψ : E 1 [ n ] → E 2 [ n ] for some n > 0, such that ψ is an anti-isometry with respect to the Weil pairing Weil E 1 [ n ] × E 1 [ n ] � µ n Weil E 2 [ n ] × E 2 [ n ] � µ n We will produce: A genus-2 curve C (possibly degenerate) Degree- n maps C → E 1 and C → E 2 Everett W. Howe Three-gluings of elliptic curves 3 of 29

� � A construction Given: Two elliptic curves E 1 , E 2 over a field k An isomorphism ψ : E 1 [ n ] → E 2 [ n ] for some n > 0, such that ψ is an anti-isometry with respect to the Weil pairing Weil E 1 [ n ] × E 1 [ n ] µ n ψ × ψ Weil E 2 [ n ] × E 2 [ n ] � µ n We will produce: A genus-2 curve C (possibly degenerate) Degree- n maps C → E 1 and C → E 2 Everett W. Howe Three-gluings of elliptic curves 3 of 29

� � � A construction Given: Two elliptic curves E 1 , E 2 over a field k An isomorphism ψ : E 1 [ n ] → E 2 [ n ] for some n > 0, such that ψ is an anti-isometry with respect to the Weil pairing Weil E 1 [ n ] × E 1 [ n ] µ n ψ × ψ inv. Weil E 2 [ n ] × E 2 [ n ] � µ n We will produce: A genus-2 curve C (possibly degenerate) Degree- n maps C → E 1 and C → E 2 Everett W. Howe Three-gluings of elliptic curves 3 of 29

Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map Everett W. Howe Three-gluings of elliptic curves 4 of 29

Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map mult. by n � � E 1 × � E 1 × E 2 E 2 Everett W. Howe Three-gluings of elliptic curves 4 of 29

� � Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map mult. by n E 1 × � � E 1 × E 2 E 2 α A Everett W. Howe Three-gluings of elliptic curves 4 of 29

� � � Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map mult. by n E 1 × � � E 1 × E 2 E 2 α b α � A A Everett W. Howe Three-gluings of elliptic curves 4 of 29

� � � Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map mult. by n E 1 × � � E 1 × E 2 E 2 α b α λ � � A A Everett W. Howe Three-gluings of elliptic curves 4 of 29

� � � Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map mult. by n E 1 × � � E 1 × E 2 E 2 α b α λ � � Jac C Jac C Everett W. Howe Three-gluings of elliptic curves 4 of 29

� � � � � � Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map mult. by n � E 1 E 1 mult. by n E 1 × � � E 1 × E 2 E 2 α b α λ � � Jac C Jac C Everett W. Howe Three-gluings of elliptic curves 4 of 29

� � � Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map mult. by n � E 1 E 1 λ � � Jac C Jac C Everett W. Howe Three-gluings of elliptic curves 4 of 29

� � � Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map mult. by n � E 1 E 1 E 1 λ � � � Jac C C Jac C Everett W. Howe Three-gluings of elliptic curves 4 of 29

� � � � � Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map mult. by n � E 1 E 1 E 1 ϕ λ � � C Jac C Jac C �� �� Everett W. Howe Three-gluings of elliptic curves 4 of 29

� � � � � Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map mult. by n � E 1 E 1 E 1 mult. by deg ϕ c ϕ ∗ ϕ ∗ ϕ λ � � C Jac C Jac C �� �� Everett W. Howe Three-gluings of elliptic curves 4 of 29

� � � � � Completing a diagram We have: Graph ( ψ ) ⊂ ( E 1 × E 2 )[ n ] , a maximal isotropic subgroup A = ( E 1 × E 2 ) / Graph ( ψ ) α : E 1 × E 2 → A , the natural map mult. by n � E 1 E 1 E 1 mult. by deg ϕ c ϕ ∗ ϕ ∗ ϕ λ � � C Jac C Jac C �� �� This gives degree- n map ϕ 1 : C → E 1 . Get ϕ 2 similarly. Everett W. Howe Three-gluings of elliptic curves 4 of 29

An old story Theorem Every degree-n map C → E 1 that does not factor through an isogeny arises in this manner. The associated E 2 and ψ : E 1 [ n ] → E 2 [ n ] are unique up to isomorphism. Theorem Every genus- 2 curve with non-simple Jacobian arises in this manner, perhaps in several ways. These results are old. What I just presented is close to what appears in Kani, J. Reine Angew. Math. (1997), which is based on Frey/Kani, in Arithmetic Algebraic Geometry (1991). Everett W. Howe Three-gluings of elliptic curves 5 of 29

An older story Frey and Kani note: They can’t find this construction explicitly in literature, but it ‘seems to be known in principle’. They cite: Serre, Sem. Théorie Nombres Bordeaux (1982/82) Ibukiyama/Katsura/Oort, Compositio Math. (1986) But if we allow for a change in perspective, it’s older than that. Everett W. Howe Three-gluings of elliptic curves 6 of 29

An even older story Kowalevski’s dissertation, written 1874 Published in Acta Math. (1884). Mentions unpublished result of Weierstrass (her advisor): Wenn aus einer Function ϑ ( v 1 , . . . , v ρ | τ 11 , . . . , τ ρρ ) durch irgend eine Transformation k ten Grades eine andere hervorgeht, die ein Produkt aus einer ϑ -Funktion von ( ρ − 1 ) Veränderlichen und einer elliptischen ist, so kann der ersprüngliche Funktion stets durch eine lineare Transformation (bei der k = 1 ist) in eine andere ϑ ( v ′ 1 , . . . , v ′ ρ | τ 11 , . . . , τ ρρ ) verwandelt werden, in der τ 12 = µ k , τ 13 = 0 , . . . , τ 1 ρ = 0 ist, wo µ einer der Zahlen 1 , 2 , . . . , k − 1 bedeutet. Everett W. Howe Three-gluings of elliptic curves 7 of 29

An even older story, continued Similar result, discovered independently by Picard Published in Bull. Math. Soc. France (1883). S’il existe une intégrale de premièr espèce correspondant à la relation algébrique y 2 = x ( 1 − x )( 1 − k 2 x )( 1 − l 2 x )( 1 − m 2 x ) qui ait seulement deux périodes, on pourra trouver un système d’intégrales normales, dont le tableau des périodes sera 1 0 1 G D 1 1 0 G ′ D où D désigne un entier réel et positif. Everett W. Howe Three-gluings of elliptic curves 8 of 29

A question of perspective The result of Frey and Kani shows that degree- n covers of elliptic curves, and “ n -gluings” of two elliptic curves, are essentially the same thing. In the 19th century, there was more interest in the former. But I think 19th-century mathematicians would have recognized Frey and Kani’s result. Everett W. Howe Three-gluings of elliptic curves 9 of 29

Explicit examples of genus-2 covers Legendre’s special ultra-elliptic integrals (1828) Traité des fonctions elliptiques , 3 ième supplement, §12 Shows that several integrals involving the expression � x ( 1 − x 2 )( 1 − k 2 x 2 ) can be evaluated in terms of elliptic integrals. Everett W. Howe Three-gluings of elliptic curves 10 of 29