On the Selmer group associated to a modular form and an algebraic - PowerPoint PPT Presentation

On the Selmer group associated to a modular form and an algebraic Hecke character. Yara Elias McGill University April 7, 2015 Yara Elias On the Selmer group

Structure of E ( K ) Mordell, Weil Let E be an elliptic curve over a number field K . Then E ( K ) ≃ Z r + E ( K ) tor where r = the algebraic rank of E E ( K ) tors = the finite torsion subgroup of E ( K ) . Yara Elias On the Selmer group

Structure of E ( K ) Mordell, Weil Let E be an elliptic curve over a number field K . Then E ( K ) ≃ Z r + E ( K ) tor where r = the algebraic rank of E E ( K ) tors = the finite torsion subgroup of E ( K ) . Questions arising Is E ( K ) finite? How do we compute r ? Could we produce a set of generators for E ( K ) / E ( K ) tors ? Yara Elias On the Selmer group

Main insight in the field Wiles, Breuil, Conrad, Diamond, Taylor For K = Q , L ( E / K , s ) has analytic continuation to all of C and satisfies L ∗ ( E / K , 2 − s ) = w ( E / K ) L ∗ ( E / K , s ) . Yara Elias On the Selmer group

Main insight in the field Wiles, Breuil, Conrad, Diamond, Taylor For K = Q , L ( E / K , s ) has analytic continuation to all of C and satisfies L ∗ ( E / K , 2 − s ) = w ( E / K ) L ∗ ( E / K , s ) . Birch, Swinnerton-Dyer’s conjecture The analytic rank of E / K is defined as r an = ord s = 1 L ( E / K , s ) . Conjecturally, r = r an . Yara Elias On the Selmer group

Kummer sequence Exact sequence of G K modules Let K = imaginary quadratic field. Consider the short exact sequence of modules p � E p � E � E � 0 . 0 Yara Elias On the Selmer group

Kummer sequence Exact sequence of G K modules Let K = imaginary quadratic field. Consider the short exact sequence of modules p � E p � E � E � 0 . 0 Descent exact sequence Taking Galois cohomology in G K , we obtain δ � E ( K ) / pE ( K ) � H 1 ( K , E p ) � H 1 ( K , E ) p � 0 . 0 Yara Elias On the Selmer group

Ø � � � � � � � Selmer group and Shafarevich-Tate group Local cohomology For a place v of K , K ֒ → K v induces Gal ( K v / K v ) − → Gal ( K / K ) . δ � E ( K ) / pE ( K ) H 1 ( K , E p ) H 1 ( K , E ) p 0 0 ρ r � ∏ v E ( K v ) / pE ( K v ) δ � ∏ v H 1 ( K v , E p ) � ∏ v H 1 ( K v , E ) p � 0 0 Yara Elias On the Selmer group

� � � � � � � Selmer group and Shafarevich-Tate group Local cohomology For a place v of K , K ֒ → K v induces Gal ( K v / K v ) − → Gal ( K / K ) . δ � E ( K ) / pE ( K ) H 1 ( K , E p ) H 1 ( K , E ) p 0 0 ρ r � ∏ v E ( K v ) / pE ( K v ) δ � ∏ v H 1 ( K v , E p ) � ∏ v H 1 ( K v , E ) p � 0 0 Definition Sel p ( E / K ) = ker ( ρ ) Ø ( E / K ) p = ker ( r ) Yara Elias On the Selmer group

Ø Importance of the Selmer group Information on the algebraic rank r δ � E ( K ) / pE ( K ) � Sel p ( E / K ) � Ø ( E / K ) p � 0 0 relates r to the size of Sel p ( E / K ) . Yara Elias On the Selmer group

Importance of the Selmer group Information on the algebraic rank r δ � E ( K ) / pE ( K ) � Sel p ( E / K ) � Ø ( E / K ) p � 0 0 relates r to the size of Sel p ( E / K ) . Shafarevich-Tate conjecture The Shafarevich group Ø ( E / K ) is conjecturally finite = ⇒ Sel p ( E / K ) = δ ( E ( K ) / pE ( K )) for all but finitely many p . Yara Elias On the Selmer group

From analytic to algebraic rank Gross, Zagier L ′ ( E / K , 1 ) = ∗ height ( y K ) , where y K ∈ E ( K ) � Heegner point of conductor 1. Hence, r an = 1 = ⇒ r ≥ 1 . Yara Elias On the Selmer group

From analytic to algebraic rank Gross, Zagier L ′ ( E / K , 1 ) = ∗ height ( y K ) , where y K ∈ E ( K ) � Heegner point of conductor 1. Hence, r an = 1 = ⇒ r ≥ 1 . Kolyvagin If y K is of infinite order in E ( K ) then Sel p ( E / K ) has rank 1 and so does E ( K ) . Hence, r an = 1 = ⇒ r = 1 & r an = 0 = ⇒ r = 0 . Yara Elias On the Selmer group

From analytic to algebraic rank Gross, Zagier L ′ ( E / K , 1 ) = ∗ height ( y K ) , where y K ∈ E ( K ) � Heegner point of conductor 1. Hence, r an = 1 = ⇒ r ≥ 1 . Kolyvagin If y K is of infinite order in E ( K ) then Sel p ( E / K ) has rank 1 and so does E ( K ) . Hence, r an = 1 = ⇒ r = 1 & r an = 0 = ⇒ r = 0 . Remark Both of these theorems require the modularity of elliptic curves proved by Wiles, Breuil, Diamond, Conrad and Taylor. Yara Elias On the Selmer group

Ø From algebraic to analytic rank Skinner, Urban Let r p = rk ( Hom Z p ( Sel p ∞ ( E / K ) , Q / Z )) , r p = 0 = ⇒ r an = 0 . Yara Elias On the Selmer group

From algebraic to analytic rank Skinner, Urban Let r p = rk ( Hom Z p ( Sel p ∞ ( E / K ) , Q / Z )) , r p = 0 = ⇒ r an = 0 . Skinner For certain elliptic curves, r = 1 & Ø < ∞ = ⇒ r an = 1 . Yara Elias On the Selmer group

From algebraic to analytic rank Skinner, Urban Let r p = rk ( Hom Z p ( Sel p ∞ ( E / K ) , Q / Z )) , r p = 0 = ⇒ r an = 0 . Skinner For certain elliptic curves, r = 1 & Ø < ∞ = ⇒ r an = 1 . Wei Zhang For large classes of elliptic curves, r p = 1 = ⇒ r an = 1 . Yara Elias On the Selmer group

Probabilistic result Bhargava, Shankar Av Sel 5 ( E ( Q )) = 6 . = ⇒ average rank of E.C over Q ordered by height ≤ 1 = ⇒ at least 4 / 5 of E.C over Q have rank 0 or 1 and at least 1 / 5 of of E.C over Q have rank 0 Yara Elias On the Selmer group

Probabilistic result Bhargava, Shankar Av Sel 5 ( E ( Q )) = 6 . = ⇒ average rank of E.C over Q ordered by height ≤ 1 = ⇒ at least 4 / 5 of E.C over Q have rank 0 or 1 and at least 1 / 5 of of E.C over Q have rank 0 Bhargava, Skinner, Wei Zhang At least 66 % of E.C over Q satisfy BSD and have finite Shafarevich group. Yara Elias On the Selmer group

From elliptic curve to modular form Generalization E � f , T p ( E ) � A f = newform of even weight A = p -adic Galois representation associated to f , higher-weight analog of the Tate module T p ( E ) Yara Elias On the Selmer group

From elliptic curve to modular form Generalization E � f , T p ( E ) � A f = newform of even weight A = p -adic Galois representation associated to f , higher-weight analog of the Tate module T p ( E ) Notation f normalized newform of level N ≥ 5 and even weight r + 2 ≥ 2. √ K = Q ( − D ) imaginary quadratic field with odd discriminant satisfying the Heegner hypothesis with | O × K | = 2 . Yara Elias On the Selmer group

Set up Algebraic Hecke character → C × Hecke character of K of infinity type ( r , 0 ) ψ : A × K − = ⇒ there is an E.C A defined over the Hilbert class field K 1 of K with CM by O K . Yara Elias On the Selmer group

Set up Algebraic Hecke character → C × Hecke character of K of infinity type ( r , 0 ) ψ : A × K − = ⇒ there is an E.C A defined over the Hilbert class field K 1 of K with CM by O K . Ring of coefficients and prime p Let O F be the ring of integers of F = Q ( a 1 , a 2 , ··· , b 1 , b 2 , ··· ) , where the a i ’s are the coefficients of f and the b i ’s are the coefficients of θ ψ . Let p be a prime with ( p , ND φ ( N ) N A r !) = 1 , where N A is the conductor of A . Yara Elias On the Selmer group

Motive associated to f and ψ . Galois representations associated to f and A f � V f , the f -isotypic part of the p -adic ´ etale realization of the motive associated to f by Deligne. A � V A , the p -adic ´ etale realization of the motive associated to A . V f and V A give rise (by extending scalars appropriately) to free O F ⊗ Z p -modules of rank 2. Yara Elias On the Selmer group

Motive associated to f and ψ . Galois representations associated to f and A f � V f , the f -isotypic part of the p -adic ´ etale realization of the motive associated to f by Deligne. A � V A , the p -adic ´ etale realization of the motive associated to A . V f and V A give rise (by extending scalars appropriately) to free O F ⊗ Z p -modules of rank 2. Galois representation associated to f and A V = V f ⊗ O F ⊗ Z p V A ( r + 1 ) V ℘ 1 its localization at a prime ℘ 1 in F dividing p , is a four dimensional representation of Gal ( Q / Q ) . Yara Elias On the Selmer group

Generalized Heegner cycles (Bertolini, Darmon, Prasana) Level N structure Heegner hypothesis = ⇒ there is an ideal N of O K satisfying O K / N ≃ Z / N Z = ⇒ level N structure on A , that is a point of exact order N defined over the ray class field L 1 of K of conductor N . Yara Elias On the Selmer group

Generalized Heegner cycles (Bertolini, Darmon, Prasana) Level N structure Heegner hypothesis = ⇒ there is an ideal N of O K satisfying O K / N ≃ Z / N Z = ⇒ level N structure on A , that is a point of exact order N defined over the ray class field L 1 of K of conductor N . GHC of conductor i Consider ( ϕ i , A i ) where A i is an E.C defined over K 1 with level N structure and ϕ i : A − → A i is an isogeny over K . � codimension r + 1 cycle on V Υ ϕ i = Graph ( ϕ i ) r ⊂ ( A × A i ) r ≃ ( A i ) r × A r � GHC ∆ ϕ i = e r Υ ϕ i of conductor i defined over L i = L 1 K i , where K i = ring class field of K of conductor i . Yara Elias On the Selmer group