Darmon points for fields of mixed signature
- 1. Darmon points for number fields of mixed signature London Number Theory Seminar, UCL February 4th, 2015 Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3 1Universitat de Barcelona 2University of Warwick 3Sheffield University Marc Masdeu Darmon points 0 / 24
- 2. Plan 1 Two Theorems 2 Two Conjectures 3 Two Effective Conjectures 4 Two (or three) Examples Marc Masdeu Darmon points 1 / 24
- 3. 1 Two Theorems 2 Two Conjectures 3 Two Effective Conjectures 4 Two (or three) Examples Marc Masdeu Darmon points 1 / 24
- 4. Two prototypical theorems (I) Fix an integer N ě 1 (a level/conductor) Let f P S2pΓ0pNqqnew, f “ q ` ř ně2 anpfqqn, with anpfq P Z. f ; ωf “ 2πifpzqdz P Ω1 X0pNq, where X0pNq “ Γ0pNqzH. Consider the lattice Λf “ !ş γ ωf ˇ ˇ ˇ γ P H1pX0pNq, Zq ) . Theorem 1 (Eichler–Shimura–Manin) Dη: C{Λf – EpCq, where E{Q is an elliptic curve, condpEq “ N, and #EpFpq “ p ` 1 ´ appfq for all p N. Marc Masdeu Darmon points 2 / 24
- 5. Two prototypical theorems (II) Let K{Q be an imaginary quadratic field. Suppose that | N ùñ split in K. § Heegner hypothesis, to force signpLpE{K, sqq “ ´1. Given τ P K X H, set Pτ “ η ˆż τ i8 ωf ˙ P EpCq. Theorem 2 (Shimura, Gross–Zagier) 1 Pτ P EpHτ q, where Hτ {K is a ring class field attached to τ. 2 TrpPτ q P EpKq nontorsion ðñ L1pE{K, 1q ‰ 0. Generalizes to CM fields (F totally real and K{F totally complex). Marc Masdeu Darmon points 3 / 24
- 6. 1 Two Theorems 2 Two Conjectures 3 Two Effective Conjectures 4 Two (or three) Examples Marc Masdeu Darmon points 3 / 24
- 7. Quaternionic automorphic forms of level N F a number field of signature pr, sq. v1, . . . , vr : F ãÑ R, w1, . . . , ws : F ãÑ C. Consider a coprime factorization N “ Dn with D squarefree. Let B{F be the quaternion algebra such that RampBq “ tq: q | Du Y tvn`1, . . . , vru, pn ď rq. Fix isomorphisms B bFvi – M2pRq, i “ 1, . . . , n; B bFwj – M2pCq, j “ 1, . . . , s. These yield Bˆ{Fˆ ãÑ PGL2pRqn ˆ PGL2pCqs ý Hn ˆ Hs 3. R>0 C H3 R>0 H R PGL2(R) PGL2(C) Marc Masdeu Darmon points 4 / 24
- 8. Quaternionic automorphic forms of level N (II) Fix RB 0 pnq Ă B Eichler order of level n. ΓB 0 pnq “ RB 0 pnqˆ{Oˆ F acts discretely on Hn ˆ Hs 3. Obtain a manifold of (real) dimension 2n ` 3s: Y D 0 pnq “ ΓB 0 pnqz pHn ˆ Hs 3q . Y D 0 pnq is compact ðñ B is division. The cohomology of Y D 0 pnq can be computed via H˚ pY D 0 pnq, Cq – H˚ pΓB 0 pnq, Cq. Hecke algebra TD “ ZrTq : q Ds acts on H˚pΓB 0 pnq, Zq. Definition f P Hn`spΓB 0 pnq, Cq eigen for TD is rational if appfq P Z, @p P TD. Marc Masdeu Darmon points 5 / 24
- 9. Elliptic curves from cohomology classes Conjecture 1 (Taylor, ICM 1994) Let f P Hn`spΓB 0 pnq, Zq be a new, rational eigenclass. Then DEf {F of conductor N “ Dn such that #Ef pOF {pq “ 1 ` |p| ´ appfq @p N. When F is totally imaginary, instead of Ef we may get an abelian surface Af {F with QM defined over F (a.k.a. fake elliptic curve). In fact, fake elliptic curves can only arise when: § F is totally imaginary, and § N is square-full: p | N ùñ p2 | N. Our construction automatically rules out this setting. However: the conjecture above is not effective: it doesn’t give a candidate Ef . Marc Masdeu Darmon points 6 / 24
- 10. Algebraic points from quadratic extensions Suppose E “ Ef is attached to f. Let K{F be a quadratic extension of F. § Assume that N is square-free, coprime to discpK{Fq. Hasse-Weil L-function of the base change of E to K ( psq ąą 0) LpE{K, sq “ ź p|N ` 1 ´ ap|p|´s ˘´1 ˆ ź p N ` 1 ´ ap|p|´s ` |p|1´2s ˘´1 . Modularity of E ùñ § Analytic continuation of LpE{K, sq to C. § Functional equation relating s Ø 2 ´ s. Conjecture 2 (very coarse version of BSD conjecture) Suppose that ords“1 LpE{K, sq “ 1. Then DPK P EpKq of infinite order. Known results rely on existence of Heegner points. Without Heegner points, don’t even have candidate for PK. Marc Masdeu Darmon points 7 / 24
- 11. 1 Two Theorems 2 Two Conjectures 3 Two Effective Conjectures 4 Two (or three) Examples Marc Masdeu Darmon points 7 / 24
- 12. Notation for Darmon points F a number field, K{F a quadratic extension. n ` s “ #tv | 8F : v splits in Ku “ rkZ Oˆ K{Oˆ F . K{F is CM ðñ n ` s “ 0. § If n ` s “ 1 we call K{F quasi-CM. SpE, Kq “ ! v | N8F : v not split in K ) . Sign of functional equation for LpE{K, ¨q should be p´1q#SpE,Kq. § From now on, we assume that this is odd. § #SpE, Kq “ 1 ùñ split automorphic forms, § #SpE, Kq ą 1 ùñ quaternionic automorphic forms. Fix a place ν P SpE, Kq. 1 If ν “ p is finite ùñ non-archimedean construction. 2 If ν is infinite ùñ archimedean construction. Marc Masdeu Darmon points 8 / 24
- 13. Previous constructions of Darmon points Non-archimedean § H. Darmon (1999): F “ Q, split. § M. Trifkovic (2006): F “ Qp ? ´dq ( ùñ K{F quasi-CM), split. § M. Greenberg (2008): F totally real, quaternionic. Archimedean § H. Darmon (2000): F totally real, split. § J. Gartner (2010): F totally real, quaternionic. Generalizations § Rotger–Longo–Vigni: lift to Jacobians of Shimura curves. § Rotger–Seveso: cycles attached to higher weight modular forms. Marc Masdeu Darmon points 9 / 24
- 14. Integration Pairing Let Hν “ the ν-adic upper half plane. That is: § The Poincar´e upper half-plane H if ν is infinite, § The p-adic (upper) half-plane Hp if ν “ p is finite. As a set, we have (up to taking connected component): HνpKνq “ P1 pKνq P1 pFνq. Hν comes equipped with an analytic structure (complex- or rigid-). When ν is infinite, there is a natural PGL2pRq-equivariant pairing Ω1 H ˆ Div0 H Ñ C “ Kν, which sends pω, pτ2q ´ pτ2qq ÞÑ ż τ2 τ1 ω P C. Analogously, Coleman integration gives a natural pairing Ω1 Hν ˆ Div0 Hν Ñ Kν. Marc Masdeu Darmon points 10 / 24
- 15. Rigid one-forms and measures The assignment µ ÞÑ ω “ ż P1pFpq dz z ´ t dµptq induces an isomorphism Meas0pP1pFpq, Zq – Ω1 Hp,Z. Inverse given by ω ÞÑ “ U ÞÑ µpUq “ resApUq ω ‰ . Theorem (Teitelbaum) ż τ2 τ1 ω “ ż P1pFpq log ˆ t ´ τ1 t ´ τ2 ˙ dµptq “ limÝÑ U ÿ UPU log ˆ tU ´ τ1 tU ´ τ2 ˙ µpUq. If the residues of ω are all integers, have a multiplicative refinement: ˆ ż τ2 τ1 ω “ limÝÑ U ź UPU ˆ tU ´ τ2 tU ´ τ1 ˙µpUq P Kˆ p . Marc Masdeu Darmon points 11 / 24
- 16. The group Γ B{F “ quaternion algebra with RampBq “ SpE, Kq tνu. Write N “ Dn (or N “ pDn if ν “ p). Fix Eichler order RB 0 pnq Ă B, and a splitting ιν : RB 0 pnq ãÑ M2pFνq. Let Γ “ RB 0 pnqr1{νsˆ{OF r1{νsˆ ιν ãÑ PGL2pFνq. Example (archimedean) F real quadratic, SpE, Kq “ tνu. This gives B “ M2pFq. Γ “ Γ0pNq “ ` a b c d ˘ P GL2pOF q: c P N ( {Oˆ F Ă PGL2pOF q. Example (non-archimedean) F “ Q, N “ pM, SpE, Kq “ tpu. This gives B “ M2pQq. Γ “ ` a b c d ˘ P GL2pZr1{psq: M | c ( {t˘1u ãÑ PGL2pQq Ă PGL2pQpq. Marc Masdeu Darmon points 12 / 24
- 17. Overview of the construction Need to assume that F has narrow class number 1. We attach to E a cohomology class ΦE P Hn`s ` Γ, Ω1 Hν ˘ . We attach to each embedding ψ: K ãÑ B a homology class Θψ P Hn`s ` Γ, Div0 Hν ˘ . § Well defined up to the image of Hn`s`1pΓ, Zq δ Ñ Hn`spΓ, Div0 Hνq. Cap-product and integration on the coefficients yield an element: Jψ “ ˆ ż Θψ ΦE P Kˆ ν . Jψ is well-defined up to the lattice L “ ! ˆ ş δpθq ΦE : θ P Hn`s`1pΓ, Zq ) . Marc Masdeu Darmon points 13 / 24
- 18. Conjectures Conjecture 1 (Oda, Yoshida, Greenberg, Guitart-M-Sengun) There is an isogeny η: Kˆ ν {L Ñ EpKνq. Dasgupta–Greenberg, Rotger–Longo–Vigni: some non-arch. cases. Completely open in the archimedean case. The Darmon point attached to E and ψ: K Ñ B is: Pψ “ ηpJψq P EpKνq. Conjecture 2 (Darmon, Greenberg, Trifkovic, Gartner, G-M-S) 1 The local point Pψ is global, defined over EpHψq. 2 For all σ P GalpHψ{Kq, σpPψq “ Precpσq¨ψ (Shimura reciprocity). 3 TrHψ{KpPψq is nontorsion if and only if L1pE{K, 1q ‰ 0. Marc Masdeu Darmon points 14 / 24
- 19. Cohomology class attached to E Recall that SpE, Kq and ν determine: Γ “ ιν ` RB 0 pnqr1{νsˆ ˘ Ă PGL2pFνq. Theorem (Darmon, Greenberg, Trifkovic, Gartner, G.–M.–S.) There exists a unique (up to sign) class ΦE P Hn`s ` Γ, Ω1 Hν ˘ such that: 1 TqΦE “ aqΦE for all q N. 2 UqΦE “ aqΦE for all q | N. 3 ΦE is “integrally valued”. Idea: relate Hn`spΓ, Ω1 Hν q with H˚pΓB 0 pNq, Cq (as Hecke modules). Marc Masdeu Darmon points 15 / 24
- 20. Homology classes attached to orders O Ă K Let ψ: O ãÑ RB 0 pnq be an embedding of an order O of K. § Which is optimal: ψpOq “ RB 0 pnq X ψpKq. Consider the group Oˆ 1 “ tu P Oˆ : NmK{F puq “ 1u. § rankpOˆ 1 q “ rankpOˆ q ´ rankpOˆ F q “ n ` s. Choose a basis u1, . . . , un`s P Oˆ 1 for the non-torsion units. § ; ∆ψ “ ψpu1q ¨ ¨ ¨ ψpun`sq P Hn`spΓ, Zq. Kˆ{Fˆ ψ ãÑ Bˆ{Fˆ ιν ãÑ PGL2pFνq ý Hν. § Let τψ be the (unique) fixed point of Kˆ on Hν. Hn`s`1pΓ, Zq δ // Hn`spΓ, Div0 Hνq // Hn`spΓ, Div Hνq deg // Hn`spΓ, Zq Θψ ? // r∆ψ bτψs // r∆ψs Fact: r∆ψs is torsion. § Can pull back a multiple of r∆ψ bτψs to Θψ P Hn`spΓ, Div0 Hνq. § Well defined up to δpHn`s`1pΓ, Zqq. Marc Masdeu Darmon points 16 / 24
- 21. 1 Two Theorems 2 Two Conjectures 3 Two Effective Conjectures 4 Two (or three) Examples Marc Masdeu Darmon points 16 / 24
- 22. Example curve F “ Qprq with r4 ´ r3 ` 3r ´ 1 “ 0. F has signature p2, 1q and discriminant ´1732. N “ pr ´ 2q “ p “ p13. B{F ramified only at all infinite real places of F. There is a rational eigenclass f P S2pΓB 0 pNqq. From f we compute ωf P H1pΓ, Ω1 Hp q and Λf “ qZ f . qf “ 8 ¨ 13 ` 11 ¨ 132 ` 5 ¨ 133 ` 3 ¨ 134 ` ¨ ¨ ¨ ` Op13100q. jE “ 1 13 ´ ´ 4656377430074r3 ` 10862248656760r2 ´ 14109269950515r ` 4120837170980 ¯ . E{F : y2 ` ` r3 ` r ` 3 ˘ xy “ x3 ` ` ` ´2r3 ` r2 ´ r ´ 5 ˘ x2 ` ` ´56218r3 ´ 92126r2 ´ 12149r ` 17192 ˘ x ´ 23593411r3 ` 5300811r2 ` 36382184r ´ 12122562. Marc Masdeu Darmon points 17 / 24
- 23. Non-archimedean cubic Darmon point F “ Qprq, with r3 ´ r2 ´ r ` 2 “ 0. F has signature p1, 1q and discriminant ´59. Consider the elliptic curve E{F given by the equation: E{F : y2 ` p´r ´ 1q xy ` p´r ´ 1q y “ x3 ´ rx2 ` p´r ´ 1q x. E has conductor NE “ ` r2 ` 2 ˘ “ p17q2, where p17 “ ` ´r2 ` 2r ` 1 ˘ , q2 “ prq . Consider K “ Fpαq, where α “ ? ´3r2 ` 9r ´ 6. The quaternion algebra B{F has discriminant D “ q2: B “ Fxi, j, ky, i2 “ ´1, j2 “ r, ij “ ´ji “ k. Marc Masdeu Darmon points 18 / 24
- 24. Non-archimedean cubic Darmon point (II) The maximal order of K is generated by wK, a root of the polynomial x2 ` pr ` 1qx ` 7r2 ´ r ` 10 16 . One can embed OK in the Eichler order of level p17 by: ψ: wK ÞÑ p´r2 ` rqi ` p´r ` 2qj ` rk. We obtain γψ “ 6r2´7 2 ` 2r`3 2 i ` 2r2`3r 2 j ` 5r2´7 2 k, and τψ “ p12g`8q`p7g`13q17`p12g`10q172 `p2g`9q173 `p4g`2q174 `¨ ¨ ¨ This yields Θψ P H1pΓ, Div0 Hpq. After integrating we get: Jψ “ 16`9¨17`15¨172 `16¨173 `12¨174 `2¨175 `¨ ¨ ¨`5¨1720 `Op1721 q, which corresponds to: Pψ “ ´108 ˆ ˆ r ´ 1, α ` r2 ` r 2 ˙ P EpKq. Marc Masdeu Darmon points 19 / 24
- 25. Archimedean cubic Darmon point Let F “ Qprq with r3 ´ r2 ` 1 “ 0. F signature p1, 1q and discriminant ´23. Consider the elliptic curve E{F given by the equation: E{F : y2 ` pr ´ 1q xy ` ` r2 ´ r ˘ y “ x3 ` ` ´r2 ´ 1 ˘ x2 ` r2 x. E has prime conductor NE “ ` r2 ` 4 ˘ of norm 89. K “ Fpαq, with α2 ` pr ` 1q α ` 2r2 ´ 3r ` 3 “ 0. § K has class number 1, thus we expect the point to be defined over K. SpE, Kq “ tσu, where σ: F ãÑ R is the real embedding of F. § Therefore the quaternion algebra B is just M2pFq. The arithmetic group to consider is Γ “ Γ0pNEq Ă PGL2pOF q. Γ acts naturally on the symmetric space H ˆ H3: H ˆ H3 “ tpz, x, yq: z P H, x P C, y P Rą0u. Marc Masdeu Darmon points 20 / 24
- 26. Archimedean cubic Darmon point (II) E ; ωE, an automorphic form with Fourier-Bessel expansion: ωEpz, x, yq “ ÿ αPδ´1OF α0ą0 apδαqpEqe´2πipα0 ¯z`α1x`α2 ¯xq yH pα1yq ¨ ˆ ´dx^d¯z dy^d¯z d¯x^d¯z ˙ Hptq “ ˆ ´ i 2 eiθ K1p4πρq, K0p4πρq, i 2 e´iθ K1p4πρq ˙ t “ ρeiθ . § K0 and K1 are hyperbolic Bessel functions of the second kind: K0pxq “ ż 8 0 e´x coshptq dt, K1pxq “ ż 8 0 e´x coshptq coshptqdt. ωE is a 2-form on Γz pH ˆ H3q. The cocycle ΦE is defined as (γ P Γ): ΦEpγq “ ż γ¨O O ωEpz, x, yq P Ω1 H with O “ p0, 1q P H3. Marc Masdeu Darmon points 21 / 24
- 27. Archimedean cubic Darmon point (III) Consider the embedding ψ: K ãÑ M2pFq given by: α ÞÑ ˆ ´2r2 ` 3r r ´ 3 r2 ` 4 2r2 ´ 4r ´ 1 ˙ Let γψ “ ψpuq, where u is a fundamental norm-one unit of OK. γψ fixes τψ “ ´0.7181328459824 ` 0.55312763561813i P H. § Construct Θ1 ψ “ rγψ bτψs P H1pΓ, Div Hq. Θ1 ψ is equivalent to a cycle ř γi bpsi ´ riq taking values in Div0 H. Jψ “ ÿ i ż si ri ΦEpγiq “ ÿ i ż γi¨O O ż si ri ωEpz, x, yq. We obtain, summing over all ideals pαq of norm up to 400, 000: Jψ “ 0.0005281284234 ` 0.0013607546066i ; Pψ P EpCq. Numerically (up to 32 decimal digits) we obtain: Pψ ? “ ´10 ˆ ` r ´ 1, α ´ r2 ` 2r ˘ P EpKq. Marc Masdeu Darmon points 22 / 24
- 28. Conclusion For all quadratic K{F, proposed an analytic construction of candidate PK P EpKνq whenever ords“1 LpE{K, sq “ 1. Heegner points are a very special case of the construction. The TruthTM covers much more than the CM paradigm. Along the way, give the first systematic method to find equations for elliptic curves using automorphic forms over non-totally-real fields. We have extensive numerical evidence. § However, nothing beyond quasi-CM! Marc Masdeu Darmon points 23 / 24
- 29. What’s next Equations for abelian surfaces of GL2-type (in progress). Computing in H2 and H2, maybe using sharblies? Higher class numbers (technical and computational difficulties). Reductive groups other than GL2? (Please help!) Marc Masdeu Darmon points 24 / 24
- 30. Thank you ! Bibliography, code and slides at: http://www.warwick.ac.uk/mmasdeu/ Marc Masdeu Darmon points 24 / 24
- 31. Cohomology Γ “ RB 0 pnqr1{psˆ {OF r1{psˆ ιp ãÑ PGL2pFpq. MapspE0pT q, Zq – IndΓ ΓD 0 ppmq Z, MapspVpT q, Zq – ´ IndΓ ΓD 0 pmq Z ¯2 . Consider the Γ-equivariant exact sequence 0 // HCpZq // MapspE0pT q, Zq ∆ // MapspVpT q, Zq // 0 ϕ // rv ÞÑ ř opeq“v ϕpeqs So get: 0 Ñ HCpZq Ñ IndΓ ΓD 0 ppmq Z ∆ Ñ ´ IndΓ ΓD 0 pmq Z ¯2 Ñ 0 Marc Masdeu Darmon points 1 / 6
- 32. Cohomology (II) 0 Ñ HCpZq Ñ IndΓ ΓD 0 ppmq Z ∆ Ñ ´ IndΓ ΓD 0 pmq Z ¯2 Ñ 0 Taking Γ-cohomology,. . . Hn`s pΓ, HCpZqq Ñ Hn`s pΓ, IndΓ ΓD 0 ppmq , Zq ∆ Ñ Hn`s pΓ, IndΓ ΓD 0 pmq , Zq2 Ñ . . . and using Shapiro’s lemma: Hn`s pΓ, HCpZqq Ñ Hn`s pΓD 0 ppmq, Zq ∆ Ñ Hn`s pΓD 0 pmq, Zq2 Ñ ¨ ¨ ¨ f P Hn`spΓD 0 ppmq, Zq being p-new ô f P Kerp∆q. Pulling back get Φf P Hn`s pΓ, HCpZqq. Marc Masdeu Darmon points 2 / 6
- 33. Recovering E from Λf Λf “ xqf y gives us qf ? “ qE. Assume ordppqf q ą 0 (otherwise, replace qf ÞÑ 1{qf ). Get jpqf q “ q´1 f ` 744 ` 196884qf ` ¨ ¨ ¨ P Cˆ p . From N guess the discriminant ∆E. § Only finitely-many possibilities, ∆E P SpF, 12q. jpqf q “ c3 4{∆E ; recover c4. Recognize c4 algebraically. 1728∆E “ c3 4 ´ c2 6 ; recover c6. Compute the conductor of Ef : Y 2 “ X3 ´ c4 48X ´ c6 864. § If conductor is correct, check aq’s. Marc Masdeu Darmon points 3 / 6
- 34. Overconvergent Method Starting data: cohomology class Φ P H1pΓ, Ω1 Hp q. Goal: to compute integrals şτ2 τ1 Φγ, for γ P Γ. Recall that ż τ2 τ1 Φγ “ ż P1pFpq logp ˆ t ´ τ1 t ´ τ2 ˙ dµγptq. Expand the integrand into power series and change variables. § We are reduced to calculating the moments: ż Zp ti dµγptq for all γ P Γ. Note: Γ Ě ΓD 0 pmq Ě ΓD 0 ppmq. Technical lemma: All these integrals can be recovered from #ż Zp ti dµγptq: γ P ΓD 0 ppmq + . Marc Masdeu Darmon points 4 / 6
- 35. Overconvergent Method (II) D “ tlocally analytic Zp-valued distributions on Zpu. § ϕ P D maps a locally-analytic function h on Zp to ϕphq P Zp. § D is naturally a ΓD 0 ppmq-module. The map ϕ ÞÑ ϕp1Zp q induces a projection: H1pΓD 0 ppmq, Dq ρ // H1pΓD 0 ppmq, Zpq. P f Theorem (Pollack-Stevens, Pollack-Pollack) There exists a unique Up-eigenclass ˜Φ lifting Φ. Moreover, ˜Φ is explicitly computable by iterating the Up-operator. Marc Masdeu Darmon points 5 / 6
- 36. Overconvergent Method (III) But we wanted to compute the moments of a system of measures. . . Proposition Consider the map Ψ: ΓD 0 ppmq Ñ D: γ ÞÑ ” hptq ÞÑ ż Zp hptqdµγptq ı . 1 Ψ belongs to H1 ´ ΓD 0 ppmq, D ¯ . 2 Ψ is a lift of f. 3 Ψ is a Up-eigenclass. Corollary The explicitly computed ˜Φ “ Ψ knows the above integrals. Marc Masdeu Darmon points 6 / 6