Quaternionic rigid meromorphic cocycles
- 1. QUATERNIONIC RIGID MEROMORPHIC COCYCLES COMPUTATIONS WITH ARITHMETIC GROUPS CMS WINTER MEETING December 3, 2020 Marc Masdeu Universitat Autònoma de Barcelona
- 2. Hilbert’s 12th Problem Let K{Q be a number field. Kronecker’s Jugendtraum Describe all abelian extensions of K via “modular functions”. Easiest case: K “ Q. Theorem (Kronecker–Weber (1853, 1886)) Qab “ Ť ně1 Q ´ e 2πi n ¯ “ A transcendental function ( e2πiz ) yields algebraic values at rational arguments! Marc Masdeu Quaternionic rigid meromorphic cocycles 1 / 11
- 3. K{Q imaginary quadratic: CM theory The theory of complex multiplication is not only the most beautiful part of mathematics but also of the whole of science. Marc Masdeu Quaternionic rigid meromorphic cocycles 1 / 11
- 4. CM theory Now, suppose K “ Qp ? Dq with D ă 0. Replace C with H “ tz P C | =pzq ą 0u. SL2pZq acts on H via ` a b c d ˘ z “ az`b cz`d . Consider now SL2pZq-invariant meromorphic functions: SL2pZqzH Ñ C Fact: every such function is a rational function in jpzq “ 1 q ` 744 ` 196884q ` 21493760q2 ` ¨ ¨ ¨ , q “ e2πiz . Theorem (Kronecker, Weber, Takagi, Hasse) If τ satisfies a quadratic polynomial in Zrxs, then jpτq P Qpτqab. Moreover, we have: Kab “ ď τPK ď ně1 K pjpτq, ℘pτ{nqq . Marc Masdeu Quaternionic rigid meromorphic cocycles 2 / 11
- 5. K{Q real quadratic From now on: K “ Qp ? Dq real quadratic (i.e. D ą 0). Problem: The upper-half plane does not contain real points! In 1999, Darmon proposed to look at a p-adic analogue of H. Fix a prime p which is inert in K. So Kp is a quadratic extension of Qp. Write Hp “ Kp r Qp. Set Γ “ SL2pZr1{psq, which acts on Hp via ` a b c d ˘ τ “ aτ`b cτ`d . Induces action on O “ OpHpq “ rigid analytic functions on Hp. And on M “ MpHpq “ FracpOq “ meromorphic functions on Hp. New Problem: E nonconstant Γ-invariant meromorphic functions! Marc Masdeu Quaternionic rigid meromorphic cocycles 3 / 11
- 6. Darmon–Vonk classes Henri Darmon Jan Vonk Marc Masdeu Quaternionic rigid meromorphic cocycles 3 / 11
- 7. Darmon–Vonk classes Fix an embedding K ãÑ R. For γ P Γ and w P K, set: δγpwq “ $ ’ & ’ % 1 σpwq ă γ8 ă w, ´1 σpwq ą γ8 ą w, 0 else. Lemma There is a cohomology class Φˆ τ P H1pΓ, Mˆ{Kˆ p q such that div Φˆ τ pγq “ ÿ w δγpwq ¨ pwq. Obstruction to lifting Φˆ τ to H1pΓ, Mˆq is an element of H2pΓ, Kˆ p q. Modulo 12-torsion, H2 pΓ, Kˆ p q – H1 pΓ0ppq, Kˆ p q. § Related to modular forms of weight 2, level Γ0ppq. Marc Masdeu Quaternionic rigid meromorphic cocycles 4 / 11
- 8. Darmon–Vonk classes (II) Proposition (Darmon–Vonk) Let p be monstruous prime, i.e. p ď 31 or p P t41, 47, 59, 71u. Then ` Φˆ τ {Φˆ pτ ˘12 lifts to a cocycle Jˆ τ P H1 pΓ, Mˆq. Given τ1 P HRM p “ tz P Hp | Qpzq real quadraticu, have: Γτ1 “ StabΓpτ1 q “ xγτ1 y. Assume that τ1 R Γτ. Since γτ1 τ1 “ τ1, the following quantity is well-defined Jppτ, τ1 q “ Jˆ τ pγτ1 qpτ1 q P Kˆ p . Conjecture (Darmon–Vonk) If τ1 R Γτ, the quantity Jppτ, τ1q belongs to QpτqabQpτ1qab. Marc Masdeu Quaternionic rigid meromorphic cocycles 5 / 11
- 9. Quaternionic generalisation Joint work with Xavier Guitart (UB) and Xavier Xarles (UAB). Xavier Guitart Xavier Xarles We generalized the construction of Darmon–Vonk to quaternion algebras over totally real fields, avoiding the S-arithmetic group Γ. Marc Masdeu Quaternionic rigid meromorphic cocycles 5 / 11
- 10. Quaternionic generalisation: homology B indefinite quaternion algebra split at p. Choose splittings ι8 : B ãÑ M2pRq and ιp : B ãÑ M2pQpq. Choose R Ă B maximal order, and set Γ0p1q “ Rˆ 1 (norm-one units). Notation: ∆pτq “ Div Γ0p1qτ, and ∆0pτq “ Div0 Γ0p1qτ. If ψ: O ãÑ R is an optimal embedding (O Ă K an order), γψ “ ψpuq, xuy “ Oˆ 1 {tors. If γψτψ “ τψ, then θ̄ψ “ rγψ bτψs P H1pΓ0p1q, ∆pτψqq. If H1pΓ0p1q, Zq is torsion, then θψ P H1pΓ0p1q, ∆0pτψqq. § θψ considered by M.Greenberg ; quaternionic Darmon points. § If H1pΓ0p1q, Zq non-torsion, may use Hecke operators to lift θ̄ψ. Marc Masdeu Quaternionic rigid meromorphic cocycles 6 / 11
- 11. Quaternionic generalisation: cohomology Fix x P H. ι8pγψq “ ` a b c d ˘ P GL2pRq has two fixed points on P1pRq: τ˘ 8 “ a ´ d ˘ a pd ´ aq2 ` 4bc 2c . ιppγψq fixes two points τ˘ P Hp. Given γ P Γ0p1q, and w˘ “ γwτ˘ 8 P Γ0p1qτ˘ 8, set δγpwq: x γx w+ w− C(x, γx) C(w− , w+ ) δγ(w) = 1 Fact: The set tw` P Γ0p1qτ` 8 | δγpwq ‰ 0u is finite. ; γ ÞÑ ř w δγpwq ¨ pwq P Z1pΓ0p1q, ∆pτqq. ; rϕψs P H1pΓ0p1q, ∆pτqq (independent of the choice of x). If H1pΓ0p1q, Zq is torsion, can lift to H1pΓ0p1q, ∆0pτqq. Marc Masdeu Quaternionic rigid meromorphic cocycles 7 / 11
- 12. Pairings and overconvergent cohomology rϕψ1 s P H1 pΓ0p1q, ∆0pτψ1 qq, θψ2 P H1pΓ0p1q, ∆0pτψ2 qq Obvious pairing: cross-ratio (Weil pairing): pQ ´ P, Q1 ´ P1q ÞÑ pQ1´QqpP1´Pq pQ1´PqpP1´Qq P pQpψ1qQpψ2qqˆ. More subtle: construct Φ P H1pΓ0ppq, Kpxtyq. H1 pΓ0ppq, Kpxtyq` – // H1pΓ0ppq, Kpxtyq 1´Up // // H1pΓ0ppq, Kpxtyqnilp M2pΓ0ppqq Φ P rTr ϕψ1 s P H1pΓ0ppq, Kpptq{Kˆ p q dlogp OO Essentially, Φ “ ř ně0 Un p pTr ϕψ1 q . Have natural pairing Kpxty ˆ ∆0pτψ2 q Ñ Kˆ p , pf, Q ´ Pq ÞÑ expp şQ P fpzqdz. Theorem (Guitart–M.–Xarles) xΦ, cores θψ2 y ˆ xϕψ1 , θψ2 yWeil ¨ “ Jppψ1, ψ2q. Marc Masdeu Quaternionic rigid meromorphic cocycles 8 / 11
- 13. Examples The fun of the subject seems to me to be in the examples. Marc Masdeu Quaternionic rigid meromorphic cocycles 8 / 11
- 14. Large genus example Let p “ 37 (not monstruous). Note that dim H1pΓ0ppq, Qq “ 5. Consider K1 “ Qp ? 57q, K2 “ Qp ? 61q. We get J37pψ1, ψ2q “ 17662185365697700344487705388816 `9911534933329042169401982379602 ˜ 1 ` ? 57 2 ¸ ` Op3720 q We check that J37pψ1, ψ2q is the image of an element α P M “ Qp ? ´3, ? ´19, ? 57q such that ι37pαq » J37pψ1, ψ2q. Ideal generated by α is supported over primes of norms 11 and 17, as predicted by an analogue of the Gross–Zagier factorisation theorem (they could only be 11, 17, 23, 31, 37, 67, 79). Marc Masdeu Quaternionic rigid meromorphic cocycles 9 / 11
- 15. Quaternionic example B “ ´ 6,´1 Q ¯ , discriminant 6. R “ x1, i, 1 ` i ` j, i ` ky. OK1 “ Zp1` ? 53 2 q and ψ1p1` ? 53 2 q “ 1{2 ´ 3{2i ´ 1{2j. § ; γψ1 “ 51{2 ` 21{2i ` 7{2j. OK2 “ Zp ? 23q and ψ2p ? 23q “ 2i ` j. § ; γψ2 “ 1151 ` 480i ` 240j. We get Jppψ1, ψ2q “ 50971141466526826096289662898361868496463698468806135561183036939036 `9674029354607221223815165708202713711819464972332940921086896674730 1 ` ? 53 2 `Op597 q The period Jppψ1, ψ2q satisfies, modulo roots of unity, the polynomial fpxq “ 41177889x4 `7867012x3 `33058502x2 `7867012x`41177889. One has L “ K1K2pfq is the compositum of the narrow class fields of K1 and K2, as predicted by the conjecture. Marc Masdeu Quaternionic rigid meromorphic cocycles 10 / 11
- 16. Exchange of primes (global object?) Idea: Compare Jp with discpBq “ q ¨ r vs Jq with discpBq “ p ¨ r. K1 “ Qp ? 53q (h` K1 “ 1), K2 “ Qp ? 23q (h` K2 “ 2). Calculate J3p53, 23q P Q32 using B of discriminant 10. § 2263329212681251489468 ` 6644010739654744556634 1` ? 53 2 ` Op346 q Calculate J5p53, 23q P Q52 using B of discriminant 6. § 76500603105371600283097752216081 ` 71876326310173029976331143056540 1` ? 53 2 ` Op547 q Compositum M of the narrow class fields is generated by a root of x8 ´ 4x7 ` 84x6 ´ 238x5 ` 1869x4 ´ 3346x3 ` 7260x2 ´ 5626x ` 3497. There is ι3 : M ãÑ Q32 and ι5 : M ãÑ Q52 . We check that there is α P M and units u1, u2 in Qp ? 53, ? 23q Ă M such that: § ι3pαu1q “ J3p53, 23q and ι5pαu2q “ J5p53, 23q. Ideal generated by α is supported over primes of norms 2 and 31. Marc Masdeu Quaternionic rigid meromorphic cocycles 11 / 11
- 17. Merci ! http://www.mat.uab.cat/˜masdeu/ Marc Masdeu Quaternionic rigid meromorphic cocycles 11 / 11
- 18. Example over real quadratic F “ Qp ? 5q, and let w “ 1` ? 5 2 . B{F of discriminant 2OF given by B “ ´ ´w,´2 F ¯ . R “ x1, i, 2w ` 2i ` j, 2w ` 2 ` 2wi ` ky maximal order. p “ p´3w ` 2q, of norm 11. K1 “ Fp ? 1 ´ 2wq, and ψ1 : OK1 ãÑ R given by ? 1 ´ 2w ÞÑ pw ´ 2qi ´ j. § ; γψ1 “ w ´ 2 ` p2w ´ 3qi ` pw ´ 1qj. K2 “ Fp ? 9 ´ 14wq, and ψ2 : OK2 ãÑ R, given by ? 9 ´ 14w ÞÑ p´3w ` 2qi ` pw ´ 2qk. § ; γψ2 “ ´55w ` 88 ` p´50w ` 81qi ` p34w ´ 55qk. Obtain Jppψ1, ψ2q “ 2650833861085011569846208847449970229624664608755690791954838 ` Op11 59 q, Jppψ1, ψ2q satisfies the polynomial 25420x4 ´ 227820x3 ` 2200011x2 ´ 27566220x ` 372174220. This generates a quadratic unramified extension of K1K2. Marc Masdeu Quaternionic rigid meromorphic cocycles 1 / 4
- 19. Tables for D “ 6, p “ 5 Plus table 8 12 53 77 92 93 8 - - 3, 5 2, 3 5 12 - 5 ? 2 - 53 - 5 ? 3, 23, 31 2, 5, 41 77 3, 5 ? ? ? ? 92 2, 3 2 3, 23, 31 ? ? 93 5 - 2, 5, 41 ? ? Minus table 8 12 53 77 92 93 8 H - 3, 5 2, 3 2, 5 12 H 2,5 ? H H 53 - 2, 5 3, 5 2, 3, 23, 31 2, 5, 41 77 3, 5 ? 3, 5 ? ? 92 2, 3 H 2, 3, 23, 31 ? ? 93 2, 5 H 2, 5, 41 ? ? Marc Masdeu Quaternionic rigid meromorphic cocycles 2 / 4
- 20. Tables for D “ 10, p “ 3 Plus table 5 8 53 77 92 5 - - 3 2, 3 8 - - 3, 5 2, 3 53 - - 3, 5, 31 2, 3, 23, 31 77 3 3, 5 3, 5, 31 ? 92 2, 3 2, 3 2, 3, 23, 31 ? Minus table 5 8 53 77 92 5 - - 3 3 8 - - 3, 5 2, 3 53 - - ? 3, 23, 31 77 3 3, 5 ? ? 92 3 2, 3 3, 23, 31 ? Marc Masdeu Quaternionic rigid meromorphic cocycles 3 / 4
- 21. Tables for D “ 22, p “ 3 Plus table 8 29 44 77 8 - - 2 29 - 3 2 44 - 3 2, 11 77 2 2 2, 11 Minus table 8 29 44 77 8 - H ? 29 - ? ? 44 H ? ? 77 ? ? ? Marc Masdeu Quaternionic rigid meromorphic cocycles 4 / 4