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

May 18, 2023 280 likes •871 views

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 .

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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

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Structure of E(K)

Mordell, Weil Let E be an elliptic curve over a number field K. Then E(K) ≃ Zr +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

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Structure of E(K)

Mordell, Weil Let E be an elliptic curve over a number field K. Then E(K) ≃ Zr +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

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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

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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 ran = ords=1L(E/K,s). Conjecturally, r = ran.

Yara Elias On the Selmer group

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Kummer sequence

Exact sequence of GK modules Let K = imaginary quadratic field. Consider the short exact sequence of modules

Ep E

p

E 0.

Yara Elias On the Selmer group

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Kummer sequence

Exact sequence of GK modules Let K = imaginary quadratic field. Consider the short exact sequence of modules

Ep E

p

E 0.

Descent exact sequence Taking Galois cohomology in GK, we obtain

E(K)/pE(K)

δ

H1(K,Ep) H1(K,E)p 0.

Yara Elias On the Selmer group

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Selmer group and Shafarevich-Tate group

Local cohomology For a place v of K, K ֒ → Kv induces Gal(Kv/Kv) − → Gal(K/K).

E(K)/pE(K)

δ

  • H1(K,Ep)
  • ρ
  • H1(K,E)p
  • r
  • ∏v E(Kv)/pE(Kv) δ

∏v H1(Kv,Ep) ∏v H1(Kv,E)p

Ø

Yara Elias On the Selmer group

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Selmer group and Shafarevich-Tate group

Local cohomology For a place v of K, K ֒ → Kv induces Gal(Kv/Kv) − → Gal(K/K).

E(K)/pE(K)

δ

  • H1(K,Ep)
  • ρ
  • H1(K,E)p
  • r
  • ∏v E(Kv)/pE(Kv) δ

∏v H1(Kv,Ep) ∏v H1(Kv,E)p

Definition Selp(E/K) = ker(ρ) Ø(E/K)p = ker(r)

Yara Elias On the Selmer group

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Importance of the Selmer group

Information on the algebraic rank r

E(K)/pE(K)

δ

Selp(E/K) Ø(E/K)p

relates r to the size of Selp(E/K). Ø

Yara Elias On the Selmer group

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Importance of the Selmer group

Information on the algebraic rank r

E(K)/pE(K)

δ

Selp(E/K) Ø(E/K)p

relates r to the size of Selp(E/K). Shafarevich-Tate conjecture The Shafarevich group Ø(E/K) is conjecturally finite = ⇒ Selp(E/K) = δ(E(K)/pE(K)) for all but finitely many p.

Yara Elias On the Selmer group

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From analytic to algebraic rank

Gross, Zagier L′(E/K,1) = ∗ height(yK), where yK ∈ E(K) Heegner point of conductor 1. Hence, ran = 1 = ⇒ r ≥ 1.

Yara Elias On the Selmer group

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From analytic to algebraic rank

Gross, Zagier L′(E/K,1) = ∗ height(yK), where yK ∈ E(K) Heegner point of conductor 1. Hence, ran = 1 = ⇒ r ≥ 1. Kolyvagin If yK is of infinite order in E(K) then Selp(E/K) has rank 1 and so does E(K). Hence, ran = 1 = ⇒ r = 1 & ran = 0 = ⇒ r = 0.

Yara Elias On the Selmer group

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From analytic to algebraic rank

Gross, Zagier L′(E/K,1) = ∗ height(yK), where yK ∈ E(K) Heegner point of conductor 1. Hence, ran = 1 = ⇒ r ≥ 1. Kolyvagin If yK is of infinite order in E(K) then Selp(E/K) has rank 1 and so does E(K). Hence, ran = 1 = ⇒ r = 1 & ran = 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

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From algebraic to analytic rank

Skinner, Urban Let rp = rk(HomZp(Selp∞(E/K),Q/Z)), rp = 0 = ⇒ ran = 0. Ø

Yara Elias On the Selmer group

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From algebraic to analytic rank

Skinner, Urban Let rp = rk(HomZp(Selp∞(E/K),Q/Z)), rp = 0 = ⇒ ran = 0. Skinner For certain elliptic curves, r = 1 & Ø < ∞ = ⇒ ran = 1.

Yara Elias On the Selmer group

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From algebraic to analytic rank

Skinner, Urban Let rp = rk(HomZp(Selp∞(E/K),Q/Z)), rp = 0 = ⇒ ran = 0. Skinner For certain elliptic curves, r = 1 & Ø < ∞ = ⇒ ran = 1. Wei Zhang For large classes of elliptic curves, rp = 1 = ⇒ ran = 1.

Yara Elias On the Selmer group

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Probabilistic result

Bhargava, Shankar Av Sel5(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

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Probabilistic result

Bhargava, Shankar Av Sel5(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

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From elliptic curve to modular form

Generalization E f, Tp(E) A f = newform of even weight A = p-adic Galois representation associated to f, higher-weight analog of the Tate module Tp(E)

Yara Elias On the Selmer group

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From elliptic curve to modular form

Generalization E f, Tp(E) A f = newform of even weight A = p-adic Galois representation associated to f, higher-weight analog of the Tate module Tp(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

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Set up

Algebraic Hecke character ψ : A×

K −

→ C× Hecke character of K of infinity type (r,0) = ⇒ there is an E.C A defined over the Hilbert class field K1 of K with CM by OK.

Yara Elias On the Selmer group

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Set up

Algebraic Hecke character ψ : A×

K −

→ C× Hecke character of K of infinity type (r,0) = ⇒ there is an E.C A defined over the Hilbert class field K1 of K with CM by OK. Ring of coefficients and prime p Let OF be the ring of integers of F = Q(a1,a2,··· ,b1,b2,···), where the ai’s are the coefficients of f and the bi’s are the coefficients of θψ. Let p be a prime with (p,NDφ(N)NAr!) = 1, where NA is the conductor of A.

Yara Elias On the Selmer group

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Motive associated to f and ψ.

Galois representations associated to f and A f Vf, the f-isotypic part of the p-adic ´ etale realization of the motive associated to f by Deligne. A VA, the p-adic ´ etale realization of the motive associated to A. Vf and VA give rise (by extending scalars appropriately) to free OF ⊗Zp-modules of rank 2.

Yara Elias On the Selmer group

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Motive associated to f and ψ.

Galois representations associated to f and A f Vf, the f-isotypic part of the p-adic ´ etale realization of the motive associated to f by Deligne. A VA, the p-adic ´ etale realization of the motive associated to A. Vf and VA give rise (by extending scalars appropriately) to free OF ⊗Zp-modules of rank 2. Galois representation associated to f and A V = Vf ⊗OF ⊗Zp VA(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

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Generalized Heegner cycles (Bertolini, Darmon, Prasana)

Level N structure Heegner hypothesis = ⇒ there is an ideal N of OK satisfying OK/N ≃ Z/NZ = ⇒ level N structure on A, that is a point of exact order N defined over the ray class field L1 of K of conductor N .

Yara Elias On the Selmer group

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Generalized Heegner cycles (Bertolini, Darmon, Prasana)

Level N structure Heegner hypothesis = ⇒ there is an ideal N of OK satisfying OK/N ≃ Z/NZ = ⇒ level N structure on A, that is a point of exact order N defined over the ray class field L1 of K of conductor N . GHC of conductor i Consider (ϕi,Ai) where Ai is an E.C defined over K1 with level N structure and ϕi : A − → Ai is an isogeny over K. codimension r +1 cycle on V Υϕi = Graph(ϕi)r ⊂ (A×Ai)r ≃ (Ai)r ×Ar GHC ∆ϕi = erΥϕi of conductor i defined over Li = L1Ki, where Ki = ring class field of K of conductor i.

Yara Elias On the Selmer group

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Selmer group

Definition The Selmer group S ⊆ H1(L1,V℘

1/p)

consists of the cohomology classes whose localizations at a prime v of L1 lie in H1(Lur

1,v/L1,v,V℘

1/p) for v not dividing NNAp

H1

f (L1,v,V℘

1/p) for v dividing p

where L1,v is the completion of L1 at v, and H1

f (L1,v,V℘

1/p)

is the finite part of H1(L1,v,V℘

1/p). Yara Elias On the Selmer group

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Analog of the transition map

Kuga-Sato like variety Wr = (E ×XN ···×XN E )c,s = Kuga-Sato variety of dimension r +1. X = Wr ×XN Ar.

Yara Elias On the Selmer group

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Analog of the transition map

Kuga-Sato like variety Wr = (E ×XN ···×XN E )c,s = Kuga-Sato variety of dimension r +1. X = Wr ×XN Ar. Chow group CHr(X/L1)0 = r-th Chow group of X over L1 = group of homologically trivial cycles on X of codimension r modulo rational equivalence.

Yara Elias On the Selmer group

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Analog of the transition map

Kuga-Sato like variety Wr = (E ×XN ···×XN E )c,s = Kuga-Sato variety of dimension r +1. X = Wr ×XN Ar. Chow group CHr(X/L1)0 = r-th Chow group of X over L1 = group of homologically trivial cycles on X of codimension r modulo rational equivalence. p-adic Abel-Jacobi map φ : CHr(X/L1)0 − → H1(L1,V)

Yara Elias On the Selmer group

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Analog of the BSD conjecture

Beilinson-Bloch’s conjecture rank(Im(φ)) = ords=r+1L(f ⊗θψ,s).

Yara Elias On the Selmer group

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Analog of the BSD conjecture

Beilinson-Bloch’s conjecture rank(Im(φ)) = ords=r+1L(f ⊗θψ,s). Conjectures on Φ Ker(Φ) = 0 Im(Φ) = S.

Yara Elias On the Selmer group

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Analog of the BSD conjecture

Beilinson-Bloch’s conjecture rank(Im(φ)) = ords=r+1L(f ⊗θψ,s). Conjectures on Φ Ker(Φ) = 0 Im(Φ) = S. Nekovar (ψ of infinity type (0,0) ) Assuming Φ(Heegner cycle) is not torsion, rank(Im(Φ)) = 1. Results of Brylinski and Gross-Zagier p-adic analog of Beilinson-Bloch (Perrin-Riou).

Yara Elias On the Selmer group

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Main theorem

Assumptions (p,NDφ(N)NAr!) = 1 G = Gal

  • L1(V℘

1/p)

  • L1
  • ≃ Aut(V℘

1/p)

V℘

1/p is a simple Aut(V℘ 1/p)-module

the eigenvalues of the generator Fr(v) of Gal(Lur

1,v/L1,v)

acting on V℘

1 are not equal to 1 modulo p for v dividing

NNA

Yara Elias On the Selmer group

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Main theorem

Assumptions (p,NDφ(N)NAr!) = 1 G = Gal

  • L1(V℘

1/p)

  • L1
  • ≃ Aut(V℘

1/p)

V℘

1/p is a simple Aut(V℘ 1/p)-module

the eigenvalues of the generator Fr(v) of Gal(Lur

1,v/L1,v)

acting on V℘

1 are not equal to 1 modulo p for v dividing

NNA Statement If Φ(∆ϕ1) = 0, then the Selmer group S has dimension 1 over OF,℘

1/p, the localization of OF at ℘

1 mod p.

Yara Elias On the Selmer group

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Kolyvagin prime

Kolyvagin prime A rational prime ℓ is a Kolyvagin prime if −D ℓ

  • = −1, aℓ ≡ bℓ ≡ ℓ+1 ≡ 0

mod p.

Yara Elias On the Selmer group

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Kolyvagin prime

Kolyvagin prime A rational prime ℓ is a Kolyvagin prime if −D ℓ

  • = −1, aℓ ≡ bℓ ≡ ℓ+1 ≡ 0

mod p. Conductors of GHC Let n = ∏ℓ be a squarefree integer where the ℓ′s are Kolyvagin

  • primes. Then

Gn = Gal(Ln/L1) ≃ Gal(Kn/K1) ≃ ∏

ℓ

Gal(Kℓ/K1). Let σℓ be a generator of the cyclic group Gal(Kℓ/K1) of order ℓ+1.

Yara Elias On the Selmer group

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Euler system properties

Set up Consider isogenous pairs (An,ϕn), (Am,ϕm) where n = ℓm for an odd prime ℓ.

Yara Elias On the Selmer group

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Euler system properties

Set up Consider isogenous pairs (An,ϕn), (Am,ϕm) where n = ℓm for an odd prime ℓ. Global compatibilies TℓΦ(∆ϕm) = corLn,LmΦ(∆ϕn) = aℓbℓΦ(∆ϕm).

Yara Elias On the Selmer group

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Euler system properties

Set up Consider isogenous pairs (An,ϕn), (Am,ϕm) where n = ℓm for an odd prime ℓ. Global compatibilies TℓΦ(∆ϕm) = corLn,LmΦ(∆ϕn) = aℓbℓΦ(∆ϕm). Local compatibilities resλnΦ(∆ϕn) = Frobℓ(Ln/Lm) resλmΦ(∆ϕm).

Yara Elias On the Selmer group

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Euler system properties

Set up Consider isogenous pairs (An,ϕn), (Am,ϕm) where n = ℓm for an odd prime ℓ. Global compatibilies TℓΦ(∆ϕm) = corLn,LmΦ(∆ϕn) = aℓbℓΦ(∆ϕm). Local compatibilities resλnΦ(∆ϕn) = Frobℓ(Ln/Lm) resλmΦ(∆ϕm). We denote by yn the image of Φ(∆ϕn) ∈ H1(Ln,V) in H1(Ln,Vp).

Yara Elias On the Selmer group

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Lifting the cohomology classes

Proposition The restriction map resL1,Ln : H1(L1,Vp) − → H1(Ln,Vp)Gn is an isomorphism.

Yara Elias On the Selmer group

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Lifting the cohomology classes

Proposition The restriction map resL1,Ln : H1(L1,Vp) − → H1(Ln,Vp)Gn is an isomorphism. Operators Let Trℓ =

ℓ

∑

i=0

σi

ℓ,

Dℓ =

ℓ

∑

i=1

iσi

ℓ.

Define Dn = ∏

ℓ|n

Dℓ ∈ Z[Gn].

Yara Elias On the Selmer group

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Kolyvagin cohomology classes

Proposition Dnyn ∈ H1(Ln,Vp)Gn. = ⇒ Dnyn can be lifted to P(n) ∈ H1(L1,Vp).

Yara Elias On the Selmer group

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Kolyvagin cohomology classes

Proposition Dnyn ∈ H1(Ln,Vp)Gn. = ⇒ Dnyn can be lifted to P(n) ∈ H1(L1,Vp). Local properties of P(n) Let v be a prime of L1. If v|NAN, then resv(P(n)) is trivial. If v ∤ NANnp, then resv(P(n)) lies in H1(Lur

1,v/L1,v,Vp).

Yara Elias On the Selmer group

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Extension by Kolyvagin classes

Global pairing The restriction map where L = L1(Vp) r : H1(L1,Vp) − → H1(L,Vp)G = HomG(Gal(Q/L),Vp) is injective and induces the evaluation pairing [ , ] r(S)×Gal(Q/L) − → Vp.

Yara Elias On the Selmer group

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Extension by Kolyvagin classes

Global pairing The restriction map where L = L1(Vp) r : H1(L1,Vp) − → H1(L,Vp)G = HomG(Gal(Q/L),Vp) is injective and induces the evaluation pairing [ , ] r(S)×Gal(Q/L) − → Vp. Notation GalS(Q/L) = annihilator of r(S) LS = extension of L fixed by GalS(Q/L) GS = Gal(LS/L) I = Gal(LS/L(y1))

Yara Elias On the Selmer group

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Choice of a pertinent Kolyvagin class

Proposition There is a Kolyvagin prime q such that Frobq(LS/Q) = τh, h ∈ Gal(LS/L), hτ+1 / ∈ I and resβy1 = 0 for some prime β in L1 above q.

Yara Elias On the Selmer group

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Choice of a pertinent Kolyvagin class

Proposition There is a Kolyvagin prime q such that Frobq(LS/Q) = τh, h ∈ Gal(LS/L), hτ+1 / ∈ I and resβy1 = 0 for some prime β in L1 above q. Scheme LS H0 = L(y1)

I

  • H1 = L(P(q))

L = L1(Vp)

Vp

  • GS
  • Vp
  • Yara Elias

On the Selmer group

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Local Tate duality

Proposition P(n) belongs to the (−1)ω(n)ε-eigenspace where ω(n) is the number of distinct prime factors of n.

Yara Elias On the Selmer group

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Local Tate duality

Proposition P(n) belongs to the (−1)ω(n)ε-eigenspace where ω(n) is the number of distinct prime factors of n. Local pairing Using local Tate duality, we have a perfect local pairing . , . λ ′ : H1(Lur

1,λ ′/L1,λ ′,Vp)×H1(Lur 1,λ ′,Vp) −

→ Z/p. The action of complex conjugation induces non-degenerate pairings of eigenspaces.

Yara Elias On the Selmer group

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From local to global information

Reciprocity law We have

∑

λ ′|ℓ|n

sλ ′,resλ ′P(n)λ ′ = 0.

Yara Elias On the Selmer group

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SLIDE 54

From local to global information

Reciprocity law We have

∑

λ ′|ℓ|n

sλ ′,resλ ′P(n)λ ′ = 0. Proposition 1 We have S−ε is of dimension 0 over OF,℘

1/p. Yara Elias On the Selmer group

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SLIDE 55

From local to global information

Reciprocity law We have

∑

λ ′|ℓ|n

sλ ′,resλ ′P(n)λ ′ = 0. Proposition 1 We have S−ε is of dimension 0 over OF,℘

1/p.

Proposition 2 We have S+ε is of dimension 1 over OF,℘

1/p. Yara Elias On the Selmer group

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SLIDE 56

Sketch of proof 1

Consider P(ℓ) where ℓ is a Kolyvagin prime satisfying Frobℓ(LS/Q) = τh, h ∈ GS, h / ∈ Gal(LS/L(y1)). P(ℓ) belongs to the −ε-eigenspace. Let s ∈ S−ε. Then

∑

λ ′|ℓ

resλ ′s,resλ ′P(ℓ)−ε

λ ′ = 0

= ⇒ resλ ′s = 0 = ⇒ s(G+

S ) = 0 by Cebotarev’s density theorem

= ⇒ s : G−

S −

→ V ±

p

= ⇒ s(G−

S) = s = 0 since V ± p are of rank 2 over OF,℘

1/p and Vp

has no non-trivial G-submodules.

Yara Elias On the Selmer group

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SLIDE 57

Sketch of proof 2

Consider P(ℓq) where ℓ be a Kolyvagin prime such that Frobℓ(LS/Q) = τi, i ∈ Gal(LS/L(y1)) Frobℓ(L(P(q))/Q) = τj, j ∈ Gal(L(P(q))/L), jτ+1 = 1. P(ℓq) belongs to the ε-eigenspace. Let s ∈ S+ε. Then

∑

λ ′|λ

resλ ′s,resλ ′P(ℓq)+ε

λ ′ + ∑ β ′|β

resβ ′s,resβ ′P(ℓq)+ε

β ′ = 0

= ⇒ resλ ′s = 0 = ⇒ s(I+) = 0 by Cebotarev’s density theorem = ⇒ s : I− − → V ±

p

= ⇒ s(I−) = s(I) = 0 = ⇒ s ∈ HomG(Gal(LS/L)/I,Vp) ≃ HomG(Vp,Vp) ≃ OF,℘

1/p. Yara Elias On the Selmer group

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SLIDE 58

The end

Thank You!

Yara Elias On the Selmer group