darmon points for fields of mixed signature

Darmon points for fields of mixed signatureNumber Theory Seminar, Cambridge University

Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3

1Institut fur Experimentelle Mathematik

2,3University 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 = NE .Let K/F be a quadratic extension of F .

I Assume that N is square-free, coprime to disc(K/F ).

Hasse-Weil L-function of the base change of E to K (<(s) >> 0)

L(E/K, s) =∏p|N

(1− ap|p|−s

)−1 ×∏p-N

(1

ap(E) = 1 + |p| −#E(Fp).

− ap|p|−s + |p|1−2s)−1.

Assume that E is modular =⇒I Analytic continuation of L(E/K, s) to C.I Functional equation relating s↔ 2− s.


The BSD conjecture

Bryan Birch Sir Peter Swinnerton-Dyer

Coarse version of BSD conjecture

ords=1 L(E/K, s) = rkZE(K).

So L(E/K, 1) = 0BSD=⇒ ∃PK ∈ E(K) of infinite order.


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:

I F = Q (and K/Q imaginary quadratic), andI Heegner hypothesis: ` | N =⇒ ` split in K.

F These ensure that ords=1 L(E/K, s) is odd (so ≥ 1).


Heegner Points (K/Q imaginary quadratic)

Attach to E a holomorphic 1-form on H = z ∈ C : =(z) > 0.

ΦE = 2πifE(z)dz = 2πi∑n≥1

ane2πinzdz ∈ H0(Γ0(N)

Γ0(N) = (a bc d

)∈ SL2(Z) : N | c

,Ω1H).

Given τ ∈ K ∩H, set Jτ =

∫ τ

i∞ΦE ∈ C.

Well-defined up to the lattice ΛE =∫

γ ΦE | γ ∈ H1

(Γ0(N)\H,Z

).

I There exists an isogeny (Weierstrass uniformization)

η : C/ΛE → E(C).

I Set Pτ = η(Jτ ) ∈ E(C).Fact: Pτ ∈ E(Hτ ), where Hτ/K is a ring class field attached to τ .

Theorem (Gross-Zagier)

PK = TrHτ/K(Pτ ) nontorsion ⇐⇒ L′(E/K, 1) 6= 0.


Heegner Points: revealing the trick

Why did this work?

1 The Riemann surface Γ0(N)\H has an algebraic model X0(N)/Q.

2 There is a morphism φ defined over Q:

φ : Jac(X0(N))→ E.

3 The CM point (τ)− (∞) ∈ Jac(X0(N))(Hτ ) gets mapped to:

φ((τ)− (∞)) = Pτ ∈ E(Hτ ).


Computing in practice: an example of Mark WatkinsLet E be the elliptic curve of conductor NE = 66157667:

E : y2 + y = x3 − 5115523309x− 140826120488927.

Watkins worked with 460 digits of precision and 600M terms of theL-series. Took less than a day (in 2006). The x-coordinate of the pointhas numerator:3677705371866775066140056423418271700879322694922855847262187700616535463492710158053651343703267430611413064645000528867046519983997664788407919153078617415072739338026281573250924797082687602171017553858718167805487654785022844156276828471927526818990949626599378706300367603592935770218062374839710749312284163465078523816968832276500720399644815972159959932997449341171062898503893640065524978358777402575345331137752028822100483561636459193457948120745710296608971732243703377010561657350085906402970902987091215062666972664619932018253973699995508681422943127563221774107305328280647596049753692423509935680307269370499116072641097827468479512837941192989412144907943309029865829912295694015235199387427463761071907702040105138183490127866378892547110594555551738109049119276198990318551492923253385898319797370264027110497425941160003806014808399829755575060358517280356452410442291650296493470492891191885968694011593251313633459625795031323398472754224400945538247051892256536774595128631179117218385529343091245081344933664374080939243620397499119074169735041423221117570585842007250226321161647201649986417295226774605259994990779421258204288795260637356926859910185168629387960475973239865371541712483169437963732171919939969937146546295368843960579247909386476566632815961781457221160982165009303338243218067269370181901361905565732088070483553355670787931266569286578590367793505932745987173797308807240343018677394437498418094567158841937203289014615526598826284058422097567571678166621399450818646421085335959899757162592592401528340509406544796171476859225008569444498220453860921224090969785448172188478976405134778065983291776042463808123777390491844755507773416209859765703930378802827649670195524084007307548226764414817153853440019798322326524148883358655673772143604560032969616681774819448090662574425967723478296641269729319041016852811289447800746467967609424309596170222574798740894035649650388853798178669200489298145202684936775070590737659026716380873664884967028363262685745931232451074203488781017631238933476570202755912488242478005942708620520821859733932900091898676772594580806760650987034535395255769756395437005076256407298723407894063143944684005844559206833619762001218344307512339014732284974905619980784862510749935288713187974033480873704269009975564425770812549105721851078566051398773310150428421211060806907435781732684894004990568983126219539479670123584145477528970810970917957442036976840460662556632012422927601267598712660045163774329619172720402171470835633999876124205952757920338556769918233682548621595584500438080514815332972700352873822470382792932239463850701180823069589872686033969240544031038574440588486055874154005176700326311212061277324813403910882777964885444157381565530147684062461546660051396904280851450982725007914162147746734845018267225005270911649442625371695958489316807540967747128604905727462240940311870432045261072392010796034682975228951065985674370150833487978753641627976939688198041395488857512826871522370782603587052302844262030644936842506142828799181077337962070672500038239594129356776240932360470386373655773263995890088045077860119731559277310730347065365574614438066227076224110878093718721572104568368924936138367920267618203822171654819989241236047827879232297391719205754470070995016783807950770131133259898013857299939208183016544242513395646068768201219283722462133998592132827925111680439534438397939011399741944793002975660976645391993846519084361887324288183733023830463888594279378938418880142666851776166056447837041357949318307502656863359340665652409440494482130055919971289855607602603992142786359126343515867623548693540215307461899928995825545976321083096385692969648000469830727362384831490147146008960565520296427479914190634547491420595642742982546549258938664049551469033300244757461635437149962496524201711710542317263364935415869714317789440514810596337383994114185743238117709497297268436126729250006313556598341642005544413154510034334524662047071238116636236628372968629480617587599286317636619851856158018862057707210320063041448677873470583163922956715800916558720872094859132869301288586404425891254542685803974845719210123188723116248983176156076281764600974413363235490318282359656362779508273280875479395111123742164365842033792484501226474060940351711307406637235476759398859593638811358930351020183894442127461462503283482426106735240223789949783920200988147219745020626928157366892297590658220939427953187053452755989894263352359355056053114113015603211922694308617337435444029085864973053536009094312149332025225287171092144929593300160658102876231441792884666648885406227023467042137524563725744495639792157824065669378853529458719945417708388719305422203077716714984665181087226221094216767415449456954035098669531672776282802324648392150034740488969680375446600297557400655812701390832499032125722304179422497954671007003939443103250096771791821099709433468073350144468396122825088243240736795841228512083604591663154848919522994493400258965092989359393577217235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Darmon’s insight

Henri DarmonDrop hypothesis of K/F being CM.

I Simplest case: F = Q, K real quadratic.However:

I There are no points on Jac(X0(N)) attached to such K.I In general there is no morphism φ : Jac(X0(N))→ E.I When F is not totally real, even the curve X0(N) is missing!

Nevertheless, Darmon constructed local points in such cases. . .I . . . and hoped that they were global.


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.


Basic notation

Consider an infinite place v | ∞F of F .I If v is real, then:

1 It may extend to two real places of K (splits), or2 It may extend to one complex place of K (ramifies).

I 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.I 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).I 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.


Goals of this talk





Non-archimedean History

These constructions are also known as Stark-Heegner points.

H. Darmon (1999): F = Q, quasi-CM, s = 1.I Darmon-Green (2001): special cases, used Riemann products.I Darmon-Pollack (2002): same cases, overconvergent methods.I Guitart-M. (2012): all cases, overconvergent methods.

M. Trifkovic (2006): F imag. quadratic ( =⇒ quasi-CM)), s = 1.I Trifkovic (2006): F euclidean, E of prime conductor.I Guitart-M. (2013): F arbitrary, E arbitrary.

M. Greenberg (2008): F totally real, arbitrary ramification, s ≥ 1.I Guitart-M. (2013): F = Q, quasi-CM case, s ≥ 1.


Archimedean History

Initially called Almost Totally Real (ATR) points.I But this name only makes sense in the original setting of Darmon.

H. Darmon (2000): F totally real, s = 1.I Darmon-Logan (2003): F quadratic norm-euclidean, NE trivial.I Guitart-M. (2011): F quadratic and arbitrary, NE trivial.I Guitart-M. (2012): F quadratic and arbitrary, NE arbitrary.

J. Gartner (2010): F totally real, s ≥ 1.I ?


Goals of this talk





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.


Overview of the construction

We define a quaternion algebra B/F and a group Γ ⊂ SL2(Fν).I The group Γ acts (non-discretely in general) on Hν .

We attach to E a cohomology class

ΦE ∈ Hn(Γ,Ω1

Hν).

We attach to each embedding ψ : K → B a homology class

Θψ ∈ Hn

(Γ,Div0Hν

).

I Well defined up to the image of Hn+1(Γ,Z)δ→ Hn(Γ,Div0Hν).

Cap-product and integration on the coefficients yield an element:

Jψ = 〈Θψ,ΦE〉 ∈ K×ν .

Jψ is well-defined up to a multiplicative lattice

L =〈δ(θ),ΦE〉 : θ ∈ Hn+1(Γ,Z)

.


Conjectures

Jψ = 〈Θψ,ΦE〉 ∈ K×ν /L.

Conjecture 1 (Oda, Yoshida, Greenberg, Guitart-M-Sengun)There is an isogeny β : K×ν /L→ E(Kν).

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:

Pψ = β(Jψ) ∈ E(Kν).

Conjecture 2 (Darmon, Greenberg, Trifkovic, Gartner, G-M-S)1 The local point Pψ is global, and belongs to E(Kab).2 Pψ is nontorsion if and only if L′(E/K, 1) 6= 0.

We predict also the exact number field over which Pψ is defined.Include a Shimura reciprocity law like that of Heegner points.


Aside: an interesting by-product

Let Φ ∈ Hn(Γ,Ω1

Hν)

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

L =〈δ(θ),Φ〉 : θ ∈ Hn+1(Γ,Z)

unram. quadratic ext. of Fν .

⊂ F×ν2.

Suppose that Conjecture 1 is true.From L one can find a Weierstrass equation Eν(Fν2) ∼= F×

ν2/L.

I Hopefully the equation can be descended to F .

A similar technique (in the archimedean case) used by L. Dembeleto compute equations for elliptic curves with everywhere goodreduction.Stay tuned!


Goals of this talk





The group Γ

Let B/F = quaternion algebra with Ram(B) = S(E,K) r ν.I B = M2(F ) (split case) ⇐⇒ s = 1.I Otherwise, we are in the quaternionic case.

E and K determine a certain ν-arithmetic subgroup Γ ⊂ SL2(Fν):I Let m =

∏l|N, split in K l.

I Let RD0 (m) be an Eichler order of level m inside B.

I Fix an embedding ιν : RD0 (m) →M2(ZF,ν).

Γ = ιν

(RD

0 (m)[1/ν]×1

)⊂ SL2(Fν).

I e.g. S(E,K) = p and ν = p give Γ ⊆ SL2

(OF [ 1p ]

).

I e.g. S(E,K) = ∞ and ν =∞ give Γ ⊆ SL2 (OF ).

Remark: We also write ΓD0 (m) = RD

0 (m)×1 .


Path integrals: archimedean setting

H = (P1(C) r P1(R))+ has a complex-analytic structure.SL2(R) acts on H through fractional linear transformations:

(a bc d

)· z =

az + b

cz + d, z ∈ H.

We consider holomorphic 1-forms ω ∈ Ω1H.

Given two points τ1 and τ2 in H, define:∫ τ2

τ1

ω = usual path integral.

Compatibility with the action of SL2(R) on H:∫ γτ2

γτ1

ω =

∫ τ2

τ1

γ∗ω.


Path integrals: non-archimedean setting

Hp = P1(Kp) r P1(Fp) has a rigid-analytic structure.SL2(Fp) acts on Hp through fractional linear transformations:(

a bc d

)· z =

az + b

cz + d, z ∈ Hp.

We consider rigid-analytic 1-forms ω ∈ Ω1Hp

.Given two points τ1 and τ2 in Hp, define:∫ τ2

τ1

ω = Coleman integral.

Compatibility with the action of SL2(Fp) on Hp:∫ γτ2

γτ1

ω =

∫ Q

Pγ∗ω.


Coleman Integration

Coleman integration on Hp can be defined as:∫ τ2

τ1

ω =

∫P1(Fp)

logp

(t− τ2

t− τ1

)dµω(t) = lim−→

U

∑U∈U

logp

(tU − τ2

tU − τ1

)resA(U)(ω).

Bruhat-Tits tree of GL2(Fp), |p| = 2.Hp having the Bruhat-Tits as retract.Annuli A(U) for a covering of size |p|−3.tU is any point in U ⊂ P1(Fp).

P1(Fp)

U ⊂ P1(Fp)

If resA(U)(ω) ∈ Z for all U , then have a multiplicative refinement:

×∫ τ2

τ1

ω = lim−→U

∏U∈U

(tU − τ2

tU − τ1

)resA(U)(ω)

∈ K×p .


Cohomology

Recall that S(E,K) and ν determine:

Γ = ιν

(RD

0 (m)[1/ν]×1

)⊂ SL2(Fν).

Choose “signs at infinity” ε1, . . . , εn ∈ ±1.

Theorem (Darmon, Greenberg, Trifkovic, Gartner, G.–M.–S.)There exists a unique (up to sign) class

ΦE ∈ Hn(Γ,Ω1

Hν)

such that:

1 TqΦE = aqΦE for all q - N.2 UqΦE = aqΦE for all q | N.3 WσiΦE = εiΦE for all embeddings σi : F → R which split in K.4 ΦE is “integrally valued”.


Homology

Let ψ : O → RD0 (m) be an embedding of an order O of K.

I Which is optimal: ψ(O) = RD0 (m) ∩ ψ(K).

Consider the group O×1 = u ∈ O× : NmK/F (u) = 1.I rank(O×1 ) = rank(O×)− rank(O×F ) = n.

Choose a basis u1, . . . , un ∈ O×1 for the non-torsion units.I ; ∆ψ = ψ(u1) · · ·ψ(un) ∈ Hn(Γ,Z).

K× acts on Hν through K×ψ→ B×

ιν→ GL2(Fν).

I Let τψ be the (unique) fixed point of K× on Hν .Have the exact sequence

Hn+1(Γ,Z)δ // Hn(Γ,Div0Hν) // Hn(Γ,DivHν)

deg // Hn(Γ,Z)

Θψ ? // [∆ψ ⊗ τψ] // [∆ψ]

Fact: [∆ψ] is torsion.I Can pull back a multiple of [∆ψ ⊗ τψ] to Θψ ∈ Hn(Γ,Div0Hν).I Well defined up to δ(Hn+1(Γ,Z)).


Goals of this talk





Cycle Decomposition

Goal

H2(Γ,Z)δ // H1(Γ,Div0Hp) // H1(Γ,DivHp)

deg // H1(Γ,Z)

Θψ ? // [γψ ⊗ τψ] // [γψ]

Theorem (word problem)Given a presentation F Γ giving

Γ = 〈g1, . . . , gs | r1, . . . , rt〉,

There is an algorithm to write γ ∈ Γ as a word in the gi’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−1D (modulo 1-boundaries).


Cycle Decomposition: example

G = R×1 , R maximal order on B = B6.

F = 〈X,Y 〉 G = 〈x, y | x2 = y3 = 1〉.

Goal: write g ⊗ τ as∑gi ⊗Di, with Di of degree 0.

Take for instance g = yxyxy. Note that wt(x) = 2 and wt(y) = 3.First, trivialize in Fab: g = yxyxyx−2y−3.To simplify γ ⊗ τ0 in H1(Γ,DivHp), use:

1 gh⊗D ≡ g ⊗D + h⊗g−1D.2 g−1 ⊗D ≡ −g ⊗gD.

g ⊗ τ0

= yxyxyx−2y−3 ⊗ τ0

= y ⊗ τ0

+ xyxyx−2y−3 ⊗y−1τ0 = y ⊗ τ0

+ xyxyx−2y−3 ⊗ τ1 = y ⊗ τ0 + x⊗ τ1

+ yxyx−2y−3 ⊗x−1τ1 = y ⊗ τ0 + x⊗ τ1

+ yxyx−2y−3 ⊗ τ2 = y ⊗ τ0 + x⊗ τ1

+ y ⊗ τ2 + xyx−2y−3 ⊗y−1τ2 = y ⊗ (τ0 + τ2) + x⊗ τ1

+ xyx−2y−3 ⊗ τ3 = y ⊗ (τ0 + τ2) + x⊗ τ1

+ x⊗ τ3 + yx−2y−3 ⊗x−1τ3 = y ⊗ (τ0 + τ2) + x⊗ (τ1 + τ3)

+ yx−2y−3 ⊗ τ4 = y ⊗ (τ0 + τ2) + x⊗ (τ1 + τ3)

+ y ⊗ τ4 + x−2y−3 ⊗y−1τ4 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3)

+ x−2y−3 ⊗ τ5 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3)

+ x−1 ⊗ τ5 + x−1y−3 ⊗xτ5 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3)

− x⊗xτ5 + x−1y−3 ⊗xτ5 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3)

− x⊗ τ6 + x−1y−3 ⊗ τ6 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6)

+ x−1 ⊗ τ6 + y−3 ⊗xτ6 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6)

− x⊗xτ6 + y−3 ⊗xτ6 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6)

− x⊗ τ7 + y−3 ⊗ τ7 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6 − τ7)

+ y−3 ⊗ τ7 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6 − τ7)

+ y−1 ⊗ τ7 + y−2 ⊗yτ7 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗yτ7 + y−2 ⊗yτ7 = y ⊗ (τ0 + τ2 + τ4) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗ τ8 + y−2 ⊗ τ8 = y ⊗ (τ0 + τ2 + τ4 − τ8) + x⊗ (τ1 + τ3 − τ6 − τ7)

+ y−2 ⊗ τ8 = y ⊗ (τ0 + τ2 + τ4 − τ8) + x⊗ (τ1 + τ3 − τ6 − τ7)

+ y−1 ⊗ τ8 + y−1 ⊗yτ8 = y ⊗ (τ0 + τ2 + τ4 − τ8) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗yτ8 + y−1 ⊗yτ8 = y ⊗ (τ0 + τ2 + τ4 − τ8) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗ τ9 + y−1 ⊗ τ9 = y ⊗ (τ0 + τ2 + τ4 − τ8 − τ9) + x⊗ (τ1 + τ3 − τ6 − τ7)

+ y−1 ⊗ τ9 = y ⊗ (τ0 + τ2 + τ4 − τ8 − τ9) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗yτ9 = y ⊗ (τ0 + τ2 + τ4 − τ8 − τ9) + x⊗ (τ1 + τ3 − τ6 − τ7)

− y ⊗ τ10 = y ⊗ (τ0 + τ2 + τ4 − τ8 − τ9 − τ10) + x⊗ (τ1 + τ3 − τ6 − τ7)


Overconvergent Method (I) (F = Q, p = fixed prime)

We have attached to E a cohomology class Φ ∈ H1(Γ,Ω1Hp).

Goal: to compute integrals∫ τ2τ1

Φγ , for γ ∈ Γ.Recall that ∫ τ2

τ1

Φγ =

∫P1(Qp)

logp

(t− τ1

t− τ2

)dµγ(t).

Expand the integrand into power series and change variables.I We are reduced to calculating the moments:∫

Zp

tidµγ(t) for all γ ∈ Γ.

Note: Γ ⊇ ΓD0 (m) ⊇ ΓD

0 (pm).Technical lemma: All these integrals can be recovered from∫

Zptidµγ(t) : γ ∈ ΓD

0 (pm)

.


Overconvergent Method (II)

D = locally analytic Zp-valued distributions on Zp.I ϕ ∈ D maps a locally-analytic function h on Zp to ϕ(h) ∈ Zp.I D is naturally a ΓD

0 (pm)-module.The map ϕ 7→ ϕ(1Zp) induces a projection:

ρ : H1(ΓD0 (pm),D)→ H1(ΓD

0 (pm),Zp).

Shapiro’s lemma allows to associate ϕE ∈ H1(ΓD0 (pm),Zp) to ΦE :

ϕE(γ) =

∫Zpµγ(t).

Theorem (Pollack-Stevens, Pollack-Pollack)

There exists a unique Up-eigenclass Φ lifting ϕE .

Moreover, Φ is explicitly computable by iterating the Up-operator.


Overconvergent Method (III)

But we wanted to compute the moments of a system of measures. . .

PropositionConsider the map Ψ: ΓD

0 (pm)→ D:

γ 7→[h(t) 7→

∫Zph(t)dµγ(t)

].

1 Ψ belongs to H1(ΓD0 (pm),D).

2 Ψ is a lift of µ.3 Ψ is a Up-eigenclass.

Corollary

The explicitly computed Φ knows the above integrals.


Examples

Where arethe examples ??


Non-archimedean cubic Darmon point (I)

F = Q(r), with r3 − r2 − r + 2 = 0.F has signature (1, 1) and discriminant −59.Consider the elliptic curve E/F given by the equation:

E/F : y2 + (−r − 1)xy + (−r − 1) y = x3 − rx2 + (−r − 1)x.

E has conductor NE =(r2 + 2

)= p17q2, where

p17 =(−r2 + 2r + 1

), q2 = (r) .

Consider K = F (α), where α =√−3r2 + 9r − 6.

The quaternion algebra B/F has discriminant D = q2:

B = F 〈i, j, k〉, i2 = −1, j2 = r, ij = −ji = k.


Non-archimedean cubic Darmon point (II)

The maximal order of K is generated by wK , a root of the polynomial

x2 + (r + 1)x+7r2 − r + 10

16.

One can embed OK in the Eichler order of level p17 by:

wK 7→ (−r2 + r)i+ (−r + 2)j + rk.

We obtain γψ = 6r2−72 + 2r+3

2 i+ 2r2+3r2 j + 5r2−7

2 k, and

τψ = (12g+8)+(7g+13)17+(12g+10)172+(2g+9)173+(4g+2)174+· · ·

After integrating we obtain:

Jψ = 16+9·17+15·172+16·173+12·174+2·175+· · ·+5·1720+O(1721),

which corresponds to:

Pψ = −3

2× 72×

(r − 1,

α+ r2 + r

2

)∈ E(K).


Archimedean cubic Darmon point (I)

Let F = Q(r) with r3 − r2 + 1 = 0.F has discriminant −23, and is of signature (1, 1).Consider the elliptic curve E/F given by the equation:

E/F : y2 + (r − 1)xy +(r2 − r

)y = x3 +

(−r2 − 1

)x2 + r2x.

E has prime conductor NE =(r2 + 4

)of norm 89.

K = F (w), with w2 + (r + 1)w + 2r2 − 3r + 3 = 0.I K has class number 1, thus we expect the point to be defined over K.I The computer tells us that rkZE(K) = 1

S(E,K) = σ, where σ : F → R is the real embedding of F .I Therefore the quaternion algebra B is just M2(F ).

The arithmetic group to consider is

Γ = Γ0(NE) ⊂ SL2(OF ).

Γ acts naturally on the symmetric space H

Hyperbolic 3-space

×H3:

H×H3 = (z, x, y) : z ∈ H, x ∈ C, y ∈ R>0.


Archimedean cubic Darmon point (II)

E ; ωE , an automorphic form with Fourier-Bessel expansion:

ωE(z, x, y) =∑

α∈δ−1OFα0>0

a(δα)(E)e−2πi(α0z+α1x+α2x)yH (α1y) ·(−dx∧dzdy∧dzdx∧dz

)

H(t) =

(− i

2eiθK1(4πρ),K0(4πρ),

i

2e−iθK1(4πρ)

)t = ρeiθ.

I K0 and K1 are hyperbolic Bessel functions of the second kind:

K0(x) =

∫ ∞0

e−x cosh(t)dt, K1(x) =

∫ ∞0

e−x cosh(t) cosh(t)dt.

ωE is a 2-form on Γ\ (H×H3).The cocycle ΦE is defined as (γ ∈ Γ):

ΦE(γ) =

∫ γ·O

OωE(z, x, y) ∈ Ω1

H with O = (0, 1) ∈ H3.


Archimedean cubic Darmon point (III)

Consider the embedding ψ : K →M2(F ) given by:

w 7→(−2r2 + 3r r − 3

r2 + 4 2r2 − 4r − 1

)Let γψ = ψ(u), where u is a fundamental norm-one unit of OK .γψ fixes τψ = −0.7181328459824 + 0.55312763561813i ∈ H.

I Construct Θ′ψ = [γψ ⊗ τψ] ∈ H1(Γ,DivH).Θ′ψ is equivalent to a cycle

∑γi ⊗ (si − ri) taking values in Div0

H.

Jψ =∑i

∫ si

ri

ΦE(γi) =∑i

∫ γi·O

O

∫ si

ri

ωE(z, x, y).

We obtain, summing over all ideals (α) of norm up to 400, 000:

Jψ = 0.0005281284234 + 0.0013607546066i; Pψ ∈ E(C).

Numerically (up to 32 decimal digits) we obtain:

Pψ?= −10×

(r − 1, w − r2 + 2r

)∈ E(K).


Thank you !

Bibliography, code and slides at:http://www.warwick.ac.uk/mmasdeu/


http://www.warwick.ac.uk/mmasdeu/

BibliographyH. 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. Gartner. 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 SL3(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.


Available Code

SAGE code for non-archimedean Darmon points when n = 1.

https://github.com/mmasdeu/darmonpoints

I Compute with “quaternionic modular symbols”.F Need presentation for units of orders in B (J. Voight, A. Page).

I Implemented overconvergent method for arbitrary B.I We obtain a method to find algebraic points.

SAGE code for archimedean Darmon points (in restricted cases).

https://github.com/mmasdeu/atrpoints

I Only for the split (B = M2(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).


https://github.com/mmasdeu/darmonpoints

https://github.com/mmasdeu/atrpoints

darmon points for fields of mixed signature

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