Efficient calculation of theta functions attached to p-adic Schottky groups
- 1. EFFICIENT CALCULATION OF THETA FUNCTIONS ATTACHED TO p-ADIC SCHOTTKY GROUPS SEMINARI DE GEOMETRIA ALGEBRAICA Nov 15, 2024 Marc Masdeu Universitat Autònoma de Barcelona (UAB) – Centre de Recerca Matemàtica (CRM)
- 2. p-adic Schottky groups Marc Masdeu Efficient calculation of Θ 0 / 20
- 3. Setup Let K be a non-archimedean field contained in complete and algebraically-closed CK. PGL2pKq acts on P1pKq via ` a b c d ˘ z “ az`b cz`d . L :“ tz P P1pKq | z “ limnÑ8 γnq with all γn distinctu. A subgroup Γ ď PGL2pKq is discrete if 1 L ‰ P1 pKq. 2 The orbit of all z P P1 pKq has compact closure (automatic if K local). Γ is Schottky if it is free, finitely-generated and discrete. Theorem (Ihara) : if Γ is finitely generated, then Γ is Schottky iff @γ P Γ, γ is hyperbolic. γ P PGL2pKq is hyperbolic if γ „ ` λ 0 0 µ ˘ with |λ| ‰ |µ|. Σ “ tfixed points of Γu. ; Ω “ P1 pKq ∖ Σ. Thus Γ acts on Ω discontinuously. Write Γ “ xγ1, . . . , γgy for a set of generators, and γ´i :“ γ´1 i . Marc Masdeu Efficient calculation of Θ 1 / 20
- 4. Example: quaternionic Schottky groups Let B the rational Hamiltonian quaternion algebra H “ Q ‘ Qi ‘ Qj ‘ Qij, ij “ ´ji, i2 “ j2 “ ´1. Let O “ x1, i, j, 1`i`j`ij 2 yZ Ď H be the Hurwitz quaternions. Let p ” 1 pmod 4q, so H ãÑ M2pQpq. Let Γ be the image of tα P Or1{psˆ | α ” 1 pmod 2qu in PGL2pQpq. Then Γ is a Schottky group of rank p`1 2 (see e.g. Milione’s Ph.D Thesis). In general, groups arising from totally definite quaternion algebras are discontinuous, so they contain finite index normal subgroups which are Schottky. Marc Masdeu Efficient calculation of Θ 2 / 20
- 5. Example: the Tate elliptic curve Suppose Γ “ xγy “ x ´ λ1 0 0 λ2 ¯ y, with |λ1| ă |λ2|. Σ “ t0, 8u ; Ω “ P1pKq ∖ t0, 8u “ Kˆ. γn ¨ z “ qnz, with q “ λ1{λ2. Theorem (Tate): Ω{Γ “ Kˆ{qZ – EqpKq, where Eq : y2 ` xy “ x3 ` a4x ` a6, with a4 “ ´5 ÿ n3qn 1 ´ qn , a6 “ ÿ n 7n5 ` 5n3 12 qn 1 ´ qn P Zrrqss. The isomorphism is explicit: given z P Ω, can compute pxpzq, ypzqq P EqpKq. Marc Masdeu Efficient calculation of Θ 3 / 20
- 6. Mumford uniformization Generalizes the previous construction to any Γ. Let X{ Spec OK be a totatly split curve (. . . ) of genus g ą 1. Theorem (Mumford): D Schottky group Γ, unique up to conjugacy, such that Xan “ Ω{Γ. JacpXqan – pKˆqg{Λ, where Λ can be explicitly described from Γ (Gerritzen, Oesterlé). Manin–Drinfeld and Gerritzen–v.d. Put describe Λ via Theta functions. ‹ Teitelbaum (hyperelliptic genus 2 curves) and Morrison–Ren compute some examples. Marc Masdeu Efficient calculation of Θ 4 / 20
- 7. Make your own Schottky group For a P K and ρ P |Cˆ K|, define open and closed discs as: Bpa, ρq “ tx P CK | |x ´ a| ă ρu, Bpa, ρq` “ tx P CK | |x ´ a| ď ρu. Lemma: Let B1 “ Bpa1, ρ1q and B´1 “ Bpa´1, ρ´1q be two open discs such that B` 1 X B` ´1 “ H and ρ1ρ´1 P |Kˆ|. Then Dγ P PGL2pKq such that γpP1 ∖ B1q “ B` ´1, γpP1 ∖ B` 1 q “ B´1. A recipe for a Scottky group 1 Choose 2g balls B˘1, . . . B˘g in P1pCKq, such that § ρpBiqρpB´iq P |Kˆ |, § The closed balls B` ˘1, . . . , B` ˘g are mutually disjoint. 2 Choose γ1, . . . γg according to the lemma, and set Γ “ xγ1, . . . , γgy. Proposition: Γ is a Schottky group, freely generated by tγ1, . . . γgu. Marc Masdeu Efficient calculation of Θ 5 / 20
- 8. Theta functions Marc Masdeu Efficient calculation of Θ 5 / 20
- 9. Theta functions (Γ and Ω as before) Θpz; a, bq “ ź γPΓ z ´ γa z ´ γb “ lim nÑ8 ź γPΓďn z ´ γa z ´ γb . ‹ Given D “ ř nP P P Div0 pΩq, consider ϕDpzq “ ś pz ´ PqnP . Set Θpz; Dq “ ś γPΓ ϕγDpzq. Properties of Θpz; Dq 1 Additivity: Θpz, D1 ` D2q “ Θpz, D1qΘpz, D2q. 2 Γ-invariance: Θpz, αDq “ Θpz, Dq, for all α P Γ. 3 div Θpz, Dq “ ΓD. § In particular uα :“ Θpz, a, αaq is analytic. § Easy to check: uα does not depend on a. 4 uαβpzq “ uαpzquβpzq for all α, β P Γ. 5 Θpz, Dq “ uαpDqΘpαz, Dq for all α P Γ. Marc Masdeu Efficient calculation of Θ 6 / 20
- 10. Applications An (incomplete) list of things you can compute with Θpz; a, bq: Given X hyperelliptic and totally split, the group Γ such that X “ Ω{Γ (Kadziela’s lifting algorithm). § And the uniformization map φ: Ω{Γ Ñ X, as z ÞÑ pxpzq, ypzqq “ pΘpz, Rq, Θpz, Sqq P XpKq. Given Γ, the Jacobian JacpΩ{Γq: (Teitelbaum, Morrison–Ren). § Set χpα, βq “ uαpz0q uαpβz0q . § JacpXΓq “ pKˆ qg {Λ, with Λ “ χpΓ, Γq. The canonical map Ω{Γ ãÑ Pg´1, given as z ÞÑ pdlog uγ1 : dlog uγ2 : ¨ ¨ ¨ : dlog uγg q. Schneider’s p-adic heights (Werner, Müller–Kaya–van der Put–M., in progress). Apply the above to quadratic Chabauty at “bad” primes (?) Marc Masdeu Efficient calculation of Θ 7 / 20
- 11. Efficient calculation Marc Masdeu Efficient calculation of Θ 7 / 20
- 12. Today’s goal Goal: approximate the quantity Θďnpz, Dq :“ ź γPΓďn ϕγDpzq, Γďn “ tγ P Γ | ℓpγq ď nu. Naive idea: run through all elements γ P Γďn and evaluate ϕγDpzq for each of them. Θďnpz, Dq “ n ź k“0 ź γPΓk ϕγDpzq, Γk “ tγ P Γ | ℓpγq “ ku. STOP #Γďn “ gp2g´1qn´1 g´1 ùñ naive algorithm is exponential. Marc Masdeu Efficient calculation of Θ 8 / 20
- 13. Basic idea (joint with X. Xarles) Set Γi k “ tγ P Γk | ℓpγq “ k and γ starts with γiu “ tγiγj1 ¨ ¨ ¨ γjk´1 u. Basic observation: for all n ě 1, we have Γi n`1 “ ž j‰´i γiΓj n. Since ϕγDpzq ¨ “ ϕDpγ´1zq, then: We have Θi n`1pz, Dq ¨ “ ź j‰´i Θj npγ´1 i z, Dq, @n ě 1. § We may adjust the constant by dividing out by the leading term. Putting it together, Θďnpz, Dq “ n ź k“0 Θkpz, Dq “ Θď1pzq n ź k“2 ź i Θi kpz, Dq. However: deg Θďn grows exponentially with n, so to compute it as an element in Kpzqˆ we still need exponential time and space. Marc Masdeu Efficient calculation of Θ 9 / 20
- 14. Fundamental domains Definition: A fundamental domain for Γ is a set F “ P1pKq ∖ Ť 1ď˘iďg Bi with Bi open balls, such that: 1 The closed balls B` ˘1, . . . B` ˘g are pairwise disjoint. 2 There are generators γ1, . . . γg of Γ such that γipP1 ∖ B` ´iq “ Bi for all 1 ď ˘i ď g. Morrison–Ren give an algorithm to compute: 1 A fundamental domain F for any Schottky group Γ (together with generators). 2 Given z P Ω, find α P Γ and z0 P F` such that αz “ z0. 3 Given γ P Γ, express it as a word in the generators. Marc Masdeu Efficient calculation of Θ 10 / 20
- 15. γ1 γ−1 B1 B2 B−2 B−1 F+ γ2 γ−2 Marc Masdeu Efficient calculation of Θ 11 / 20
- 16. Simplifications Goal: Compute Θpz, Dq, for arbitrary z P Ω and D P Div0 pΩq. Lemma: The map Div0 pF`qΓ Ñ Div0 pΩqΓ is surjective. § Idea: if a “ γia0 with a0 P F` , pick z0 P F` such that γiz0 P F` , and paq´pb0q “ pγia0q´pγiz0q`pγiz0q´pb0q „Γ pa0q´pz0q`pγiz0q´pb0q. ‹ Hence may assume that D is supported on F` . ‹ Using Θpz, Dq “ uαpDqΘpαz, Dq, we may assume that z P F`. To compute χi,j “ χpγi, γjq “ uγi pz0q uγi pγjz0q we may take z0 P BB´j so that γjz0 P BBj. From now on, we will assume that: z P F` and D P Div0 pF` q. Marc Masdeu Efficient calculation of Θ 12 / 20
- 17. A polynomial time algorithm Consider the 2g-tuple of functions ⃗ Fk “ ´ Θj kpzq ¯ j P ś j Kpzqˆ. Θj kpzq has zeros and poles inside Bj, so analytic on Bc j . Therefore ⃗ Fk can be thought of as a tuple in R “ ś j OpBc j qˆ. Key fact: OpBc i q is an affinoid algebra, its elements are represented by power series. For each i ‰ ´j, since γipBjq Ď Bi, we have a commutative diagram OpBc j qˆ γi // OpBc i qˆ KpBc j qˆ γi // ? OO KpBc i qˆ ? OO fpzq // fpγ´1 i zq Induces ∇: R Ñ R, ∇pfiqi “ p ś j‰´i γifjqi. Therefore, can compute ∇⃗ Fk “ ⃗ Fk`1 only using 2g power series! Marc Masdeu Efficient calculation of Θ 13 / 20
- 18. Marc Masdeu Efficient calculation of Θ 14 / 20
- 19. Time to compute Θpz; a, bq (iterative, g “ 2, p “ 3) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 precision (digits) 0 10 20 30 40 50 60 70 time (s) Marc Masdeu Efficient calculation of Θ 15 / 20
- 20. Comparison with naive algorithm (g “ 2, p “ 3) 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 precision (digits) 2 9 2 7 2 5 2 3 2 1 21 23 25 time (s) iterative naive Marc Masdeu Efficient calculation of Θ 16 / 20
- 21. Timings (varying g, p “ 11) 50 65 80 95 110 125 140 precision (digits) 2 5 2 3 2 1 21 23 25 27 time (s) g = 2 iterative naive 50 65 80 95 precision (digits) 2 2 20 22 24 26 g = 3 50 65 80 precision (digits) 20 21 22 23 24 25 26 g = 4 Marc Masdeu Efficient calculation of Θ 17 / 20
- 22. An archimedean version Can adapt these ideas to compute with classical Schottky groups. Consider 2g open balls in P1pCq, written B˘1, . . . , B˘g. § We assume that they have mutually disjoint closures. Let γi be the matrix mapping isomorphically Bc ´i Ñ B` i . The group Γ “ xγ1, . . . , γgy is free of rank g. For each i, write Γpiq “ tγ P Γ | tpγq ‰ γ˘iu. Goal: Calculate an approximation to ωipzq “ ÿ σPΓpiq 1 z ´ σpAq ´ 1 z ´ σpBq , i “ 1, . . . , g, where A, B are the fixed points of γi. Note that ωi “ dlog ś σPΓpiq z´σpAq z´σpBq . The same ideas described before work through. Marc Masdeu Efficient calculation of Θ 18 / 20
- 23. Archimedean version Marc Masdeu Efficient calculation of Θ 19 / 20
- 24. Final thoughts Recap 1 p-adic Theta functions are a fundamental object when working with Schottky groups. 2 A naive algorithm for their computation is impractical. 3 When they come from modular groups, algorithms exist that use the Up Hecke operator to give efficient computations of these Theta functions (Darmon–Vonk, Guitart–M.–Xarles, Negrini). 4 Inspired by these algorithms, we describe iterative construction of Θpz; a, bq as a product of affinoid functions. 5 One can easily compute with affinoid functions (bounded power series). 6 Our algorithm is completely general, it does not use any other structure on Γ other than being Schottky. What’s next 1 Compare with modular approaches (when available). 2 Apply it to formulate p-adic BSD for abelian varieties with bad reduction at p (in progress w/ S.Müller, E.Kaya and M. van der Put). Marc Masdeu Efficient calculation of Θ 20 / 20
- 25. Thank you! Image: Ross Hilbert