Darmon points for fields of mixed signature

Darmon points for fields of mixed signature Number Theory Seminar, Cambridge University Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3 1 Institut für Experimentelle Mathematik 2,3 University of Warwick May 13, 2014 Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 0 / 35

The Hasse-Weil L-function Let F be a number field. Let E /F be an elliptic curve of conductor N = N E. Let K/F be a quadratic extension of F. Assume that N is square-free, coprime to disc(k/f ). a p (E) = 1 + p #E(F p ). Hasse-Weil L-function of the base change of E to K (R(s) >> 0) L(E/K, s) = ( 1 ap p s) 1 ( 1 a p p s + p 1 2s) 1. p N p N Assume that E is modular = Analytic continuation of L(E/K, s) to C. Functional equation relating s 2 s. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 1 / 35

The BSD conjecture Bryan Birch Coarse version of BSD conjecture Sir Peter Swinnerton-Dyer ord s=1 L(E/K, s) = rk Z E(K). So L(E/K, 1) = 0 BSD = P K E(K) of infinite order. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 2 / 35

The main tool for BSD: Heegner points Kurt Heegner Exist for F totally real and K/F totally complex (CM extension). I recall the definition of Heegner points in the simplest setting: F = Q (and K/Q imaginary quadratic), and Heegner hypothesis: l N = l split in K. These ensure that ord s=1 L(E/K, s) is odd (so 1). Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 3 / 35

Heegner Points (K/Q imaginary quadratic) Γ 0 (N) = { ( ) a b c d SL2 (Z): N c} Attach to E a holomorphic 1-form on H = {z C : I(z) > 0}. Φ E = 2πif E (z)dz = 2πi a n e 2πinz dz H 0 (Γ 0 (N), Ω 1 H). n 1 τ Given τ K H, set J τ = Φ E C. i { ( )} Well-defined up to the lattice Λ E = γ Φ E γ H 1 Γ 0 (N)\H, Z. There exists an isogeny (Weierstrass uniformization) η : C/Λ E E(C). Set P τ = η(j τ ) E(C). Fact: P τ E(H τ ), where H τ /K is a ring class field attached to τ. Theorem (Gross-Zagier) P K = Tr Hτ /K(P τ ) nontorsion L (E/K, 1) 0. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 4 / 35

Heegner Points: revealing the trick Why did this work? 1 The Riemann surface Γ 0 (N)\H has an algebraic model X 0 (N) /Q. 2 There is a morphism φ defined over Q: φ: Jac(X 0 (N)) E. 3 The CM point (τ) ( ) Jac(X 0 (N))(H τ ) gets mapped to: φ((τ) ( )) = P τ E(H τ ). Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 5 / 35

Computing in practice: an example of Mark Watkins Let E be the elliptic curve of conductor N E = 66157667: E : y 2 + y = x 3 5115523309x 140826120488927. Watkins worked with 460 digits of precision and 600M terms of the L-series. Took less than a day (in 2006). The x-coordinate of the point has numerator: 36777053718667750661400564234182717008793226949228558472621877006165354634927101580536513437032674306114130646450005288670465199839976647884079191530786174150 72739338026281573250924797082687602171017553858718167805487654785022844156276828471927526818990949626599378706300367603592935770218062374839710749312284163465 07852381696883227650072039964481597215995993299744934117106289850389364006552497835877740257534533113775202882210048356163645919345794812074571029660897173224 37033770105616573500859064029709029870912150626669726646199320182539736999955086814229431275632217741073053282806475960497536924235099356803072693704991160726 41097827468479512837941192989412144907943309029865829912295694015235199387427463761071907702040105138183490127866378892547110594555551738109049119276198990318 55149292325338589831979737026402711049742594116000380601480839982975557506035851728035645241044229165029649347049289119188596869401159325131363345962579503132 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16727762828023246483921500347404889696803754466002975574006558127013908324990321257223041794224979546710070039394431032500967717918210997094334680733501444683 96122825088243240736795841228512083604591663154848919522994493400258965092989359393577217235439331087432419973874470183959253201676376403284079570698454395013 81234605867495003402016724626400855369636521155009147176245904149069225438646928549072337653348704931901764847439772432025275648964681387210234070849306330191 79038041239611544624083258348136637213230084906083526213683231531105290336750385743792050893130528314337942393060136915457253067727886206663888425022179164712 3563828956462530983567929499493346622977494903591722345188975062941907415400740881 Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 6 / 35

Darmon s insight Henri Darmon Drop hypothesis of K/F being CM. Simplest case: F = Q, K real quadratic. However: There are no points on Jac(X 0 (N)) attached to such K. In general there is no morphism φ: Jac(X 0 (N)) E. When F is not totally real, even the curve X0 (N) is missing! Nevertheless, Darmon constructed local points in such cases...... and hoped that they were global. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 7 / 35

Goals of this talk 1 Review some history. 2 Sketch a general construction of Darmon points. 3 Give some details of the construction. 4 Explain the algorithmic challenges we face in their computation. The fun of the subject seems to me to be in the examples. B. Gross, in a letter to B. Birch, 1982 5 Illustrate with fun examples. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 8 / 35

Basic notation Consider an infinite place v F of F. If v is real, then: 1 It may extend to two real places of K (splits), or 2 It may extend to one complex place of K (ramifies). If v is complex, then it extends to two complex places of K (splits). n = #{v F : v splits in K}. K/F is CM n = 0. If n = 1 we call K/F quasi-cm. S(E, K) = { v N F : v not split in K }, s = #S(E, K). Sign of functional equation for L(E/K, s) should be ( 1) #S(E,K). From now on, we assume that s is odd. Fix a place ν S(E, K). 1 If ν = p is finite = non-archimedean case. 2 If ν is infinite = archimedean case. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 9 / 35

Non-archimedean History These constructions are also known as Stark-Heegner points. H. Darmon (1999): F = Q, quasi-cm, s = 1. Darmon-Green (2001): special cases, used Riemann products. Darmon-Pollack (2002): same cases, overconvergent methods. Guitart-M. (2012): all cases, overconvergent methods. M. Trifkovic (2006): F imag. quadratic ( = quasi-cm)), s = 1. Trifkovic (2006): F euclidean, E of prime conductor. Guitart-M. (2013): F arbitrary, E arbitrary. M. Greenberg (2008): F totally real, arbitrary ramification, s 1. Guitart-M. (2013): F = Q, quasi-cm case, s 1. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 11 / 35

Archimedean History Initially called Almost Totally Real (ATR) points. But this name only makes sense in the original setting of Darmon. H. Darmon (2000): F totally real, s = 1. Darmon-Logan (2003): F quadratic norm-euclidean, NE trivial. Guitart-M. (2011): F quadratic and arbitrary, N E trivial. Guitart-M. (2012): F quadratic and arbitrary, N E arbitrary. J. Gartner (2010): F totally real, s 1.? Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 12 / 35

Our construction Xavier Guitart M. Haluk Sengun Available for arbitrary base number fields F (mixed signature). Comes in both archimedean and non-archimedean flavors. All of the previous constructions become particular cases. We can provide genuinely new numerical evidence. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 14 / 35

Overview of the construction We define a quaternion algebra B /F and a group Γ SL 2 (F ν ). The group Γ acts (non-discretely in general) on H ν. We attach to E a cohomology class Φ E H n ( Γ, Ω 1 H ν ). We attach to each embedding ψ : K B a homology class Θ ψ H n ( Γ, Div 0 H ν ). Well defined up to the image of Hn+1 (Γ, Z) δ H n (Γ, Div 0 H ν ). Cap-product and integration on the coefficients yield an element: J ψ = Θ ψ, Φ E K ν. J ψ is well-defined up to a multiplicative lattice { } L = δ(θ), Φ E : θ H n+1 (Γ, Z). Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 15 / 35

Conjectures J ψ = Θ ψ, Φ E K ν /L. Conjecture 1 (Oda, Yoshida, Greenberg, Guitart-M-Sengun) There is an isogeny β : K ν /L E(K ν ). Dasgupta Greenberg, Rotger Longo Vigni: some non-arch. cases. Completely open in the archimedean case. The Darmon point attached to E and ψ : K B is: P ψ = β(j ψ ) E(K ν ). Conjecture 2 (Darmon, Greenberg, Trifkovic, Gartner, G-M-S) 1 The local point P ψ is global, and belongs to E(K ab ). 2 P ψ is nontorsion if and only if L (E/K, 1) 0. We predict also the exact number field over which P ψ is defined. Include a Shimura reciprocity law like that of Heegner points. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 16 / 35

Aside: an interesting by-product Let Φ H n ( Γ, Ω 1 H ν ) be an eigenclass with integer eigenvalues. In favorable situations Φ comes from an elliptic curve E over F. No systematic construction of such curves for non totally real F. We can compute the lattice unram. quadratic ext. of F ν. { } L = δ(θ), Φ : θ H n+1 (Γ, Z) F. ν 2 Suppose that Conjecture 1 is true. From L one can find a Weierstrass equation E ν (F ν 2) = F ν 2 /L. Hopefully the equation can be descended to F. A similar technique (in the archimedean case) used by L. Dembélé to compute equations for elliptic curves with everywhere good reduction. Stay tuned! Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 17 / 35

The group Γ Let B /F = quaternion algebra with Ram(B) = S(E, K) {ν}. B = M2 (F ) (split case) s = 1. Otherwise, we are in the quaternionic case. E and K determine a certain {ν}-arithmetic subgroup Γ SL 2 (F ν ): Let m = l N, split in K l. Let R D 0 (m) be an Eichler order of level m inside B. Fix an embedding ιν : R D 0 (m) M 2 (Z F,ν ). ) Γ = ι ν (R0 D (m)[1/ν] 1 SL 2 (F ν ). e.g. S(E, K) = {p} and ν = p give Γ SL2 ( O F [ 1 p ] ). e.g. S(E, K) = { } and ν = give Γ SL2 (O F ). Remark: We also write Γ D 0 (m) = RD 0 (m) 1. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 19 / 35

Path integrals: archimedean setting H = (P 1 (C) P 1 (R)) + has a complex-analytic structure. SL 2 (R) acts on H through fractional linear transformations: ( a b ) az + b c d z = cz + d, z H. We consider holomorphic 1-forms ω Ω 1 H. Given two points τ 1 and τ 2 in H, define: τ2 τ 1 ω = usual path integral. Compatibility with the action of SL 2 (R) on H: γτ2 γτ 1 ω = τ2 τ 1 γ ω. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 20 / 35

Path integrals: non-archimedean setting H p = P 1 (K p ) P 1 (F p ) has a rigid-analytic structure. SL 2 (F p ) acts on H p through fractional linear transformations: ( a b ) az + b c d z = cz + d, z H p. We consider rigid-analytic 1-forms ω Ω 1 H p. Given two points τ 1 and τ 2 in H p, define: τ2 τ 1 ω = Coleman integral. Compatibility with the action of SL 2 (F p ) on H p : γτ2 γτ 1 ω = Q P γ ω. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 21 / 35

Coleman Integration Coleman integration on H p can be defined as: τ2 ( ) t τ2 ω = log p dµ ω (t) = lim τ 1 P 1 (F p) t τ 1 U U U log p ( tu τ 2 t U τ 1 ) res A(U) (ω). Bruhat-Tits tree of GL 2 (F p ), p = 2. H p having the Bruhat-Tits as retract. Annuli A(U) for a covering of size p 3. t U is any point in U P 1 (F p ). U P 1 (Fp) P 1 (F p) If res A(U) (ω) Z for all U, then have a multiplicative refinement: τ2 ( ) tu τ resa(u) (ω) 2 ω = lim K U p. τ 1 t U τ 1 U U Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 22 / 35

Cohomology Recall that S(E, K) and ν determine: ) Γ = ι ν (R0 D (m)[1/ν] 1 SL 2 (F ν ). Choose signs at infinity ε 1,..., ε n {±1}. Theorem (Darmon, Greenberg, Trifkovic, Gartner, G. M. S.) There exists a unique (up to sign) class Φ E H n ( Γ, Ω 1 ) H ν such that: 1 T q Φ E = a q Φ E for all q N. 2 U q Φ E = a q Φ E for all q N. 3 W σi Φ E = ε i Φ E for all embeddings σ i : F R which split in K. 4 Φ E is integrally valued. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 23 / 35

Homology Let ψ : O R0 D (m) be an embedding of an order O of K. Which is optimal: ψ(o) = R D 0 (m) ψ(k). Consider the group O 1 = {u O : Nm K/F (u) = 1}. rank(o 1 ) = rank(o ) rank(o F ) = n. Choose a basis u 1,..., u n O 1 for the non-torsion units. ψ = ψ(u 1 ) ψ(u n ) H n (Γ, Z). ψ K acts on H ν through K B ιν GL 2 (F ν ). Let τψ be the (unique) fixed point of K on H ν. Have the exact sequence δ H n+1 (Γ, Z) H n (Γ, Div 0 H ν ) H n (Γ, Div H ν ) deg H n (Γ, Z) Θ ψ? [ ψ τ ψ ] [ ψ ] Fact: [ ψ ] is torsion. Can pull back a multiple of [ ψ τ ψ ] to Θ ψ H n (Γ, Div 0 H ν ). Well defined up to δ(hn+1 (Γ, Z)). Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 24 / 35

Cycle Decomposition Goal H 2 (Γ, Z) δ H 1 (Γ, Div 0 H p ) H 1 (Γ, Div H p ) deg H 1 (Γ, Z) Θ ψ? [γ ψ τ ψ ] [γ ψ ] Theorem (word problem) Given a presentation F Γ giving Γ = g 1,..., g s r 1,..., r t, There is an algorithm to write γ Γ as a word in the g i s. Effective version for quaternionic groups: John Voight, Aurel Page. γ [Γ, Γ] = γ has word representation W, with W [F, F ]. We use gh D g D + h g 1 D (modulo 1-boundaries). Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 26 / 35

Cycle Decomposition: example G = R 1, R maximal order on B = B 6. F = X, Y G = x, y x 2 = y 3 = 1. Goal: write g τ as g i D i, with D i of degree 0. Take for instance g = yxyxy. Note that wt(x) = 2 and wt(y) = 3. First, trivialize in F ab : g = yxyxyx 2 y 3. To simplify γ τ 0 in H 1 (Γ, Div H p ), use: 1 gh D g D + h g 1 D. 2 g 1 D g gd. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 27 / 35

Overconvergent Method (I) (F = Q, p = fixed prime) We have attached to E a cohomology class Φ H 1 (Γ, Ω 1 H p ). Goal: to compute integrals τ 2 τ 1 Φ γ, for γ Γ. Recall that τ2 ( ) t τ1 Φ γ = log p dµ γ (t). τ 1 P 1 (Q p) t τ 2 Expand the integrand into power series and change variables. We are reduced to calculating the moments: t i dµ γ (t) for all γ Γ. Z p Note: Γ Γ D 0 (m) ΓD 0 (pm). Technical lemma: All these integrals can be recovered from { } Z p t i dµ γ (t): γ Γ D 0 (pm). Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 28 / 35

Overconvergent Method (II) D = {locally analytic Z p -valued distributions on Z p }. ϕ D maps a locally-analytic function h on Z p to ϕ(h) Z p. D is naturally a Γ D 0 (pm)-module. The map ϕ ϕ(1 Zp ) induces a projection: ρ: H 1 (Γ D 0 (pm), D) H 1 (Γ D 0 (pm), Z p ). Shapiro s lemma allows to associate ϕ E H 1 (Γ D 0 (pm), Z p) to Φ E : ϕ E (γ) = µ γ (t). Z p Theorem (Pollack-Stevens, Pollack-Pollack) There exists a unique U p -eigenclass Φ lifting ϕ E. Moreover, Φ is explicitly computable by iterating the U p -operator. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 29 / 35

Overconvergent Method (III) But we wanted to compute the moments of a system of measures... Proposition Consider the map Ψ: Γ D 0 (pm) D: [ ] γ h(t) h(t)dµ γ (t). Z p 1 Ψ belongs to H 1 (Γ D 0 (pm), D). 2 Ψ is a lift of µ. 3 Ψ is a U p -eigenclass. Corollary The explicitly computed Φ knows the above integrals. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 30 / 35

Examples Where are the examples?? Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 30 / 35

Non-archimedean cubic Darmon point (I) F = Q(r), with r 3 r 2 r + 2 = 0. F has signature (1, 1) and discriminant 59. Consider the elliptic curve E /F given by the equation: E /F : y 2 + ( r 1) xy + ( r 1) y = x 3 rx 2 + ( r 1) x. E has conductor N E = ( r 2 + 2 ) = p 17 q 2, where p 17 = ( r 2 + 2r + 1 ), q 2 = (r). Consider K = F (α), where α = 3r 2 + 9r 6. The quaternion algebra B/F has discriminant D = q 2 : B = F i, j, k, i 2 = 1, j 2 = r, ij = ji = k. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 31 / 35

Non-archimedean cubic Darmon point (II) The maximal order of K is generated by w K, a root of the polynomial x 2 + (r + 1)x + 7r2 r + 10. 16 One can embed O K in the Eichler order of level p 17 by: w K ( r 2 + r)i + ( r + 2)j + rk. We obtain γ ψ = 6r2 7 2 + 2r+3 2 i + 2r2 +3r 2 j + 5r2 7 2 k, and τ ψ = (12g+8)+(7g+13)17+(12g+10)17 2 +(2g+9)17 3 +(4g+2)17 4 + After integrating we obtain: J ψ = 16+9 17+15 17 2 +16 17 3 +12 17 4 +2 17 5 + +5 17 20 +O(17 21 ), which corresponds to: P ψ = 3 ( 2 72 r 1, α + ) r2 + r E(K). 2 Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 32 / 35

Archimedean cubic Darmon point (I) Let F = Q(r) with r 3 r 2 + 1 = 0. F has discriminant 23, and is of signature (1, 1). Consider the elliptic curve E /F given by the equation: E /F : y 2 + (r 1) xy + ( r 2 r ) y = x 3 + ( r 2 1 ) x 2 + r 2 x. E has prime conductor N E = ( r 2 + 4 ) of norm 89. K = F (w), with w 2 + (r + 1) w + 2r 2 3r + 3 = 0. K has class number 1, thus we expect the point to be defined over K. The computer tells us that rkz E(K) = 1 S(E, K) = {σ}, where σ : F R is the real embedding of F. Therefore the quaternion algebra B is just M 2 (F ). The arithmetic group to consider is Hyperbolic 3-space Γ = Γ 0 (N E ) SL 2 (O F ). Γ acts naturally on the symmetric space H H 3 : H H 3 = {(z, x, y): z H, x C, y R >0 }. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 33 / 35

Archimedean cubic Darmon point (II) E ω E, an automorphic form with Fourier-Bessel expansion: ω E (z, x, y) = a (δα) (E)e 2πi(α 0 z+α 1 x+α 2 x) yh (α 1 y) H(t) = α δ 1 O F α 0 >0 ( i 2 eiθ K 1 (4πρ), K 0 (4πρ), i 2 e iθ K 1 (4πρ) ) ( dx d z dy d z d x d z t = ρe iθ. ) K0 and K 1 are hyperbolic Bessel functions of the second kind: K 0 (x) = e x cosh(t) dt, K 1 (x) = 0 0 e x cosh(t) cosh(t)dt. ω E is a 2-form on Γ\ (H H 3 ). The cocycle Φ E is defined as (γ Γ): Φ E (γ) = γ O O ω E (z, x, y) Ω 1 H with O = (0, 1) H 3. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 34 / 35

Archimedean cubic Darmon point (III) Consider the embedding ψ : K M 2 (F ) given by: ( 2r w 2 ) + 3r r 3 r 2 + 4 2r 2 4r 1 Let γ ψ = ψ(u), where u is a fundamental norm-one unit of O K. γ ψ fixes τ ψ = 0.7181328459824 + 0.55312763561813i H. Construct Θ ψ = [γ ψ τ ψ ] H 1 (Γ, Div H). Θ ψ is equivalent to a cycle γ i (s i r i ) taking values in Div 0 H. J ψ = i si r i Φ E (γ i ) = i γi O si O r i ω E (z, x, y). We obtain, summing over all ideals (α) of norm up to 400, 000: J ψ = 0.0005281284234 + 0.0013607546066i P ψ E(C). Numerically (up to 32 decimal digits) we obtain: P ψ? = 10 ( r 1, w r 2 + 2r ) E(K). Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 35 / 35

Thank you! Bibliography, code and slides at: http://www.warwick.ac.uk/mmasdeu/ Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 35 / 35

Bibliography H. Darmon and A. Logan. Periods of Hilbert modular forms and rational points on elliptic curves. Int. Math. Res. Not. (2003), no. 40, 2153 2180. H. Darmon and P. Green. Elliptic curves and class fields of real quadratic fields: Algorithms and evidence. Exp. Math., 11, No. 1, 37-55, 2002. H. Darmon and R. Pollack. Efficient calculation of Stark-Heegner points via overconvergent modular symbols. Israel J. Math., 153:319 354, 2006. J. Gärtner. Darmon points and quaternionic Shimura varieties. Canad. J. Math. 64 (2012), no. 6. X. Guitart and M. Masdeu. Elementary matrix Decomposition and the computation of Darmon points with higher conductor. Math. Comp. (arxiv.org, 1209.4614), 2013. X. Guitart and M. Masdeu. Computation of ATR Darmon points on non-geometrically modular elliptic curves. Exp. Math., 2012. X. Guitart and M. Masdeu. Computation of quaternionic p-adic Darmon points. (arxiv.org, 1307.2556), 2013. X. Guitart, M. Masdeu and M.H. Sengun. Darmon points on elliptic curves over number fields of arbitrary signature. (arxiv.org, 1404.6650), 2014. M. Greenberg. Stark-Heegner points and the cohomology of quaternionic Shimura varieties. Duke Math. J., 147(3):541 575, 2009. D. Pollack and R. Pollack. A construction of rigid analytic cohomology classes for congruence subgroups of SL 3 (Z). Canad. J. Math., 61(3):674 690, 2009. M. Trifkovic. Stark-Heegner points on elliptic curves defined over imaginary quadratic fields. Duke Math. J., 135, No. 3, 415-453, 2006. Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 35 / 35

Available Code SAGE code for non-archimedean Darmon points when n = 1. https://github.com/mmasdeu/darmonpoints Compute with quaternionic modular symbols. Need presentation for units of orders in B (J. Voight, A. Page). Implemented overconvergent method for arbitrary B. We obtain a method to find algebraic points. SAGE code for archimedean Darmon points (in restricted cases). https://github.com/mmasdeu/atrpoints Only for the split (B = M 2 (F )) cases, and: 1 F real quadratic, and K/F ATR (Hilbert modular forms) 2 F cubic (1, 1), and K/F totally complex (cubic automorphic forms). Marc Masdeu Darmon points for fields of mixed signature May 13, 2014 1 / 1