This document summarizes a talk on constructing Darmon points for elliptic curves over number fields of mixed signature. It discusses: 1. Attaching a cohomology class to the elliptic curve and a homology class to each embedding of the field into a quaternion algebra. 2. Taking the cap product of these classes and integrating over coefficients to obtain elements of the field, well-defined up to a lattice. 3. A conjecture that these elements correspond to points on the elliptic curve over a completion of the field. It then outlines some algorithmic challenges in computing these constructions in practice and provides an example over a cubic number field to illustrate the approach.

1. Darmon points for fields of mixed signature
First Joint International Meeting 2014
Bilbo, July 2nd, 2014
Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3
1Institut f¨ur Experimentelle Mathematik
2,3University of Warwick
Marc Masdeu Darmon points for fields of mixed signature 0 / 19

2. The Hasse-Weil L-function
Let F be a number field (assume h+
F = 1).
Let E/F be an elliptic curve of conductor N = NE.
Let K/F be a quadratic extension of F.
Assume that N is square-free, coprime to disc(K/F).
L-function of E/K (for (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 =⇒
Analytic continuation of L(E/K, s) to C.
Functional equation relating s ↔ 2 − s.
Marc Masdeu Darmon points for fields of mixed signature 1 / 19

3. The BSD conjecture
Bryan Birch
Sir Peter Swinnerton-Dyer
BSD conjecture (coarse version)
ords=1 L(E/K, s) = rkZ E(K).
So L(E/K, 1) = 0
BSD
=⇒ ∃PK ∈ E(K) of infinite order.
How to construct such PK?
Marc Masdeu Darmon points for fields of mixed signature 2 / 19

4. Goals of this talk
1 Sketch a general construction of Darmon points.
2 Explain some 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
”3 Illustrate with an example.
Marc Masdeu Darmon points for fields of mixed signature 3 / 19

5. Our construction: Highlights
Available for arbitrary base number fields F (mixed signature).
We assume h+
F = 1 “for simplicity”.
Comes in both archimedean and non-archimedean flavors.
The Heegner point construction also has a (less known)
non-archimedean counterpart!
All of the previously constructions become particular cases:
1 Archimedean:
Darmon (Rational points on modular elliptic curves, 2002)
Gartner (Canad. J. Math., 2012)
2 Non-archimedean:
Darmon (Ann. of Math. (2), 2001)
Trifkovic (Duke Math. J., 2006)
Greenberg (Duke Math. J., 2009)
Marc Masdeu Darmon points for fields of mixed signature 4 / 19

6. Goals of this talk
1 Sketch a general construction of Darmon points.
2 Explain some 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
”3 Illustrate with an example.
Marc Masdeu Darmon points for fields of mixed signature 5 / 19

7. Basic notation
Recall: if v | ∞F is an infinite place of F, then:
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.
S(E, K) = v | N∞F : v not split in K , s = #S(E, K).
S(E, K) has already appeared this week:
S(E, K) = {∞Q} in Vigni’s talk (Heegner hypothesis)
S(E, K) = {v0} in Guitart’s talk.
Sign of functional equation for L(E/K, s) should be (−1)#S(E,K).
From now on, we assume that s is odd.
In particular, S(E, K) is nonempty!
In this talk: assume S(E, K) contains some finite prime.
Marc Masdeu Darmon points for fields of mixed signature 6 / 19

8. Non-archimedean path integrals
For simplicity: Suppose that |p| = p (totally split prime).
Hp = P1(Cp) P1(Qp) has a rigid-analytic structure.
SL2(Qp) acts on Hp through fractional linear transformations:
a b
c d · z =
az + b
cz + d
, z ∈ Hp.
Use rigid-analytic 1-forms ω ∈ Ω1
Hp
.
Coleman integral to integrate
between τ1 and τ2 in Hp.
Compatibility with the action of SL2(Qp)
on Hp:
γτ2
γτ1
ω =
Q
P
γ∗
ω.
Robert Coleman
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9. Coleman Integration on Hp
ω ∈ Ω1
Hp
completely determined by its “restriction to the boundary”:
∂Hp can be identified with P1
(Qp).
The 1-form ω gives rise to a “boundary measure” µω on P1
(Qp).
Coleman integration on Hp can be defined as:
τ2
τ1
ω =
P1(Qp)
logp
t − τ2
t − τ1
dµω(t) = lim
U
U∈U
logp
tU − τ2
tU − τ1
µω(U).
If µω(U) ∈ Z for all U, have a multiplicative refinement.
Bruhat-Tits tree of GL2(Qp).
Hp having the Bruhat-Tits as retract.
Opens for a covering of size p−3.
tU is any point in U ⊂ P1(Qp).
P1(Qp)
U ⊂ P1
(Qp)
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10. The group Γ
Choose a (finite) prime p ∈ S(E, K).
Let B/F = quaternion algebra with Ram(B) = S(E, K) {p}.
B = M2(F) (split case) ⇐⇒ s = 1.
Let D be the discriminant of B (product of finite ramified primes).
Let m be such that NE = pDm.
Let RD
0 (m) be an Eichler order of level m inside B.
Fix an embedding ιp : RD
0 (m) → M2(ZF,p).
Γ = RD
0 (m)[1/p]×
1
ιp
→ SL2(Fp).
e.g. S(E, K) = {p} gives
Γ = a b
c d ∈ SL2(OF [1/p]): c ∈ m[1/p] .
Set ΓD
0 (m) = RD
0 (m)×
1 .
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11. Overview of the construction
We attach to E a cohomology class
ΦE ∈ Hn
(Γ, Ω1
Hp,Z).
We attach to each embedding ψ: K → B a homology class
Θψ ∈ Hn Γ, Div0
Hp .
Well-defined up to the image of Hn+1(Γ, Z)
δ
→ Hn(Γ, Div0
Hp).
Cap-product and integration on the coefficients yield an element:
Jψ = ×
Θψ
ΦE ∈ K×
p .
Jψ is well-defined up to a multiplicative lattice
L = ×
δ(θ)
ΦE : θ ∈ Hn+1(Γ, Z) .
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12. Conjecture 1 (Greenberg, Guitart-M-Sengun)
There is an isogeny β : K×
p /L → E(Kp).
Dasgupta–Greenberg, Rotger–Longo–Vigni: some non-arch. cases.
Jψ = ×
Θψ
ΦE ∈ K×
p /L.
The Darmon point attached to E and ψ: K → B is:
Pψ = β(Jψ) ∈ E(Kp).
Conjecture 2 (Darmon, Greenberg, Trifkovic, 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.
Predicts also the exact number field over which Pψ is defined.
Understand the Galois action on Pψ in terms class field theory.
Marc Masdeu Darmon points for fields of mixed signature 11 / 19

13. 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 Hp through K× ψ
→ B×
ιp
→ GL2(Qp).
Let τψ be the (unique) fixed point of K×
on Hp.
Have the exact sequence
Hn+1(Γ, Z)
δ // Hn(Γ, Div0
Hp) // Hn(Γ, Div Hp)
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
Hp).
Well defined up to δ(Hn+1(Γ, Z)).
Marc Masdeu Darmon points for fields of mixed signature 12 / 19

14. Goals of this talk
1 Sketch a general construction of Darmon points.
2 Explain some 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
”3 Illustrate with an example.
Marc Masdeu Darmon points for fields of mixed signature 13 / 19

15. Computing in practice (n = 1)
Homology: The main problem is to lift the cycle:
H2(Γ, Z)
δ // H1(Γ, Div0
Hp) // H1(Γ, Div Hp)
deg
// H1(Γ, Z)
Θψ
? // [∆ψ ⊗τψ] // [∆ψ]
Use finite presentation + word problem for Γ (Voight, Page).
Cohomology: Use a generalization of modular symbols.
More interesting for higher n (more geometry).
Integration: Could use the definition:
τ2
τ1
ω =
P1(Qp)
logp
t − τ2
t − τ1
dµω(t) = lim
U
U∈U
logp
tU − τ2
tU − τ1
µω(U).
(1)
Yields exponential algorithm: one p-adic digit requires p times more
work.
Marc Masdeu Darmon points for fields of mixed signature 14 / 19

16. Overconvergent Method (I)
We have attached to E a cohomology class Φ ∈ H1(Γ, Ω1
Hp,Z).
Goal: to compute integrals
τ2
τ1
Φγ, for γ ∈ Γ.
Recall that
τ2
τ1
Φγ =
P1(Qp)
logp
t − τ1
t − τ2
dµγ(t).
Write P1(Qp) = {|z| 1} ∪ p−1
a=0(a + pZp).
Expand the integrands into power series and change variables.
We reduce to calculating the moments:
Zp
ti
dµγ(t) for all γ ∈ Γ, for all i ≥ 0.
Recall ΓD
0 (m) = RD
0 (m)×
1 , and note that Γ ⊇ ΓD
0 (m) ⊇ ΓD
0 (pm).
Marc Masdeu Darmon points for fields of mixed signature 15 / 19

17. 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).
Recall that ΦE comes (Shapiro’s lemma) from ϕE ∈ H1(ΓD
0 (pm), Zp):
ϕE(γ) =
Zp
µγ(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.
Marc Masdeu Darmon points for fields of mixed signature 16 / 19

18. Overconvergent Method (III)
But we needed to compute the moments of a system of measures!
Proposition
Consider the map Ψ: ΓD
0 (pm) → D:
γ → Ψ(γ), Ψ(γ)(h(t)) =
Zp
h(t)dµγ(t).
1 Ψ induces a class [Ψ] ∈ H1(ΓD
0 (pm), D).
2 [Ψ] is a lift of µ.
3 [Ψ] is a Up-eigenclass.
Corollary
The algorithmically computed ˜Φ “knows about” the above integrals.
Marc Masdeu Darmon points for fields of mixed signature 17 / 19

19. Where is
the example ??
Marc Masdeu Darmon points for fields of mixed signature 17 / 19

20. 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.
Marc Masdeu Darmon points for fields of mixed signature 18 / 19

21. Non-archimedean cubic Darmon point (II)
The maximal order of K is:
OK = OF + wKOF , wK satisfying x2
+ (r + 1)x +
7r2 − r + 10
16
= 0.
One can embed OK in the Eichler order of level p17 by:
wK → ψ(wK) = (−r2
+ r)i + (−r + 2)j + rk.
We obtain γψ = ψ(u) = 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 = −108 · r − 1,
α + r2 + r
2
∈ E(K).
Marc Masdeu Darmon points for fields of mixed signature 19 / 19

22. Eskerrik Asko !
Bibliography, code and slides at:
http://www.warwick.ac.uk/mmasdeu/
Marc Masdeu Darmon points for fields of mixed signature 19 / 19

23. 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. G¨artner. 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.
Marc Masdeu Darmon points for fields of mixed signature 19 / 19

24. Aside: an interesting by-product
Let Φ ∈ Hn Γ, Ω1
Hp
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.
Hopefully the equation can be descended to F.
A similar technique (in the archimedean case) used by L. Demb´el´e
to compute equations for elliptic curves with everywhere good
reduction.
Stay tuned!
Marc Masdeu Darmon points for fields of mixed signature 1 / 3

25. 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).
Marc Masdeu Darmon points for fields of mixed signature 2 / 3

26. 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, Div Hp), use:
1 gh ⊗D ≡ g ⊗D + h ⊗g−1
D.
2 g−1
⊗D ≡ −g ⊗gD.
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