Darmon points for fields of mixed signature seminar

Darmon points for fields of mixed signature
Number Theory Seminar, University of Warwick
Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3
1Institut für Experimentelle Mathematik
2,3University of Warwick
January 27, 2014
Marc Masdeu Darmon points for fields of mixed signature January 27, 2014 0 / 1

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.
Assume for simplicity 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 Modularity conjecture (see Samir’s talk in 4 weeks) =⇒
Analytic continuation of L(E/K, s) to C.
Functional equation relating s ↔ 2 − s.

The BSD conjecture
Brian Birch Sir Peter Swinnerton-Dyer
Coarse version of BSD conjecture
ords=1 L(E/K, s) = rkZ E(K).
So L(E/K, 1) = 0
BSD
=⇒ ∃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:
F = Q (and K/Q imaginary quadratic), and
Heegner hypothesis: | N =⇒ split in K.
This ensures 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 = fE(z)dz =
n≥1
ane2πinz
dz ∈ H0
(Γ0(N)
Γ0(N) = { a b
c d ∈ SL2(Z): N | c}
, Ω1
H).
Given τ ∈ K ∩ H, set Jτ =
τ
∞
ΦE ∈ C.
Well-deﬁned up to the lattice ΛE = γ ΦE | γ ∈ H1 Γ0(N)H, Z .
There exists an isogeny η: C/ΛE → E(C).
Set Pτ = η(Jτ ) ∈ E(C).
Fact: Pτ ∈ E(Hτ ), where Hτ /K is a class ﬁeld attached to τ.
Theorem (Gross-Zagier)
PK = TrHτ /K(Pτ ) nontorsion ⇐⇒ L (E/K, 1) = 0.

Heegner Points: revealing the trick
So why does 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)) gets mapped to:
φ((τ) − (∞)) = Pτ ∈ E(Hτ ).

Computing in practice: an example of Mark Watkins
Let 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 the
L-series. Took less than a day (in 2006). The x-coordinate of the point
has numerator:
3677705371866775066140056423418271700879322694922855847262187700616535463492710 1580536513437032674306114130646450005288670465199839976647884079191530786174150
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0785238169688322765007203996448159721599599329974493411710628985038936400655249 7835877740257534533113775202882210048356163645919345794812074571029660897173224
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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(X0(N)) attached to such K.
In general there is no morphism φ: Jac(X0(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.

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.
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.

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.
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.

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, NE trivial.
Guitart-M. (2012): F quadratic and arbitrary, NE arbitrary.
J. Gartner (2010): F totally real, s ≥ 1.
?

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).
The group Γ acts (non-discretely in general) on Hν.
We attach to E a unique cohomology class
ΦE ∈ Hn
Γ, Ω1
Hν
.
We attach to each embedding ψ: K → B a homology class
Θψ ∈ Hn Γ, Div0
Hν .
Well defined up to the image of Hn+1(Γ, Z)
δ
→ Hn(Γ, Div0
Hν).
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).
Proven in some non-arch. cases (Greenberg, Rotger–Longo–Vigni).
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) = 0.
We predict also the exact number ﬁeld over which Pψ is deﬁned.
Include a Shimura reciprocity law like that of Heegner points.

Goals of this talk

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 ⊂ SL2(Fν):
Let m = l|N, split in K l.
Let RD
0 (m) be an Eichler order of level m inside B.
Fix an embedding ιν : RD
0 (m) → M2(ZF,ν).
Γ = ιν RD
0 (m)[1/ν]×
1 ⊂ SL2(Fν).
e.g. S(E, K) = {p} and ν = p give Γ ⊆ SL2 OF [1
p ] .
e.g. S(E, K) = {∞} and ν = ∞ give Γ ⊆ SL2 (OF ).
Remark: We also write ΓD
0 (m) = RD
0 (m)×
1 .

Path integrals: archimedean setting
H = (P1(C) P1(R))+ has a complex-analytic structure.
SL2(R) acts on H through fractional linear transformations:
a b
c d · z =
az + b
cz + d
, z ∈ H.
We consider holomorphic 1-forms ω ∈ Ω1
H.
Given two points P and Q in H, deﬁne:
Q
P
ω = usual path integral.
Compatibility with the action of SL2(R) on H:
γQ
γP
ω =
Q
P
γ∗
ω.

Path integrals: non-archimedean setting
Hp = P1(Kp) P1(Fp) has a rigid-analytic structure.
SL2(Fp) acts on Hp through fractional linear transformations:
a b
c d · z =
az + b
cz + d
, z ∈ Hp.
We consider rigid-analytic 1-forms ω ∈ Ω1
Hp
.
Given two points P and Q in Hp, deﬁne:
Q
P
ω = Coleman integral.
Compatibility with the action of SL2(Fp) on Hp:
γQ
γP
ω =
Q
P
γ∗
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” s1, . . . , sn ∈ {±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 TlΦE = alΦE for all l N.
2 UqΦE = aqΦE for all q | N.
3 Wσi ΦE = siΦE for all embeddings σi : F → R which split in K.
4 ΦE is “integrally valued”.

Homology
Let ψ: O → RD
0 (m) be an embedding of an order O of K.
Which is optimal: ψ(O) = RD
0 (m) ∩ ψ(K).
Consider the group O×
1 = {u ∈ O× : NmK/F (u) = 1}.
rank(O×
1 ) = rank(O×
) − rank(O×
F ) = n.
Choose a basis u1, . . . , un ∈ O×
1 for the non-torsion units.
; ∆ψ = ψ(u1) · · · ψ(un) ∈ Hn(Γ, Z).
K× acts on Hν through K× ψ
→ B× ιν
→ GL2(Fν).
Let τψ be the (unique) ﬁxed point of K×
on Hν.
Have the exact sequence
Hn+1(Γ, Z)
δ // Hn(Γ, Div0
Hν) // Hn(Γ, Div Hν)
deg
// Hn(Γ, Z)
Θψ
? // [∆ψ ⊗τψ] // [∆ψ]
Fact: [∆ψ] is torsion.
Can pull back a multiple of [∆ψ ⊗τψ] to Θψ ∈ Hn(Γ, Div0
Hν).
Well deﬁned up to δ(Hn+1(Γ, Z)).

Goals of this talk

Commutator Decomposition
Goal
H2(Γ, Z)
δ // H1(Γ, Div0
Hp) // H1(Γ, Div Hp)
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 (as 1-cycles).

Commutator 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 on Fab: g = yxyxyx−2y−3.
To simplify γ ⊗τ0 in H1(Γ, Div Hp), use:
1 gh ⊗D ≡ g ⊗D + h ⊗g−1
D.
2 g−1
⊗D ≡ −g ⊗gD.

Overconvergent Method (I) (F = Q, p = ﬁxed prime)
We have attached to E a cohomology class Φ ∈ H1(Γ, Ω1
Hp
).
Goal: to compute integrals
τ2
τ1
Φγ, for γ ∈ Γ.
Recall that τ2
τ1
Φγ =
P1(Qp)
log
t − τ1
t − τ2
dµγ(t).
Expand the integrand into power series and change variables.
We are reduced to calculating the moments:
Zp
ti
dµγ(t) for all γ ∈ Γ.
Note: Γ ⊇ ΓD
0 (m) ⊇ ΓD
0 (pm).
Technical lemma: All these integrals can be recovered from
Zp
ti
dµγ(t): γ ∈ ΓD
0 (pm) .

Overconvergent Method (II)
D = {locally analytic Zp-valued distributions on Zp}.
ϕ ∈ D maps a locally-analytic function h on Zp to ϕ(h) ∈ Zp.
D is naturally a ΓD
0 (pm)-module.
The map ϕ → ϕ(1Zp ) induces a projection:
ρ: H1
(ΓD
0 (pm), D) → H1
(ΓD
0 (pm), Zp).
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. . .
Proposition
Consider the map Ψ: ΓD
0 (pm) → D:
γ → h(t) →
Zp
h(t)dµγ(t) .
1 Ψ belongs to H1(ΓD
0 (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 are
the examples ??

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 = 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).

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
+ r2
x.
E has prime conductor NE = r2 + 4 of norm 89.
K = F(w), with w2 + (r + 1) w + 2r2 − 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 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 ∈ R0}.

Archimedean cubic Darmon point (II)
E ; ωE, an automorphic form with Fourier-Bessel expansion:
ωE(z, x, y) =
α∈δ−1OF
α00
a(δα)(E)e−2πi(α0 ¯z+α1x+α2 ¯x)
yH (α1y) ·
−dx∧d¯z
dy∧d¯z
d¯x∧d¯z
H(t) = −
i
2
eiθ
K1(4πρ), K0(4πρ),
i
2
e−iθ
K1(4πρ) t = ρeiθ
.
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 ΓH × H3.
The cocycle ΦE is deﬁned as (γ ∈ Γ):
ΦE(γ) =
γ·O
O
ωE(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 →
−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.
Construct Θψ = [γψ ⊗τψ] ∈ H1(Γ, Div H).
Θψ 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).

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 → (−r2
+ r)i + (−r + 2)j + rk.
We obtain γψ = 6r2−7
2 + 2r+3
2 i + 2r2+3r
2 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).

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

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.
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.

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