Numerical Evidence for Darmon Points

Numerical Evidence for Darmon Points
M`etodes Efectius en Geometria Algebraica
Xavier Guitart 1 Marc Masdeu 2
1Universitat Polit`ecnica de Catalunya
2Columbia University
June 4, 2013
Marc Masdeu Numerical Evidence for Darmon Points June 4, 2013 1 / 28

The Problem
Problem
Given an algebraic curve C defined over Q, find points on
C, defined on prescribed algebraic extensions K of Q.

Mordell-Weil
Louis Mordell Andr´e Weil

Basic Setup
E = elliptic curve deﬁned over Q, and K = Q(
√
dK).
If dK > 0 then K is called real quadratic, and
If dK < 0 then K is called imaginary quadratic.
Theorem (Mordell–Weil)
E(K) = (torsion) ⊕ Zrkalg(E,K)
The algebraic rank rkalg(E, K) is hard to determine.
Attach to E and K an L-function L(E/K, s) as follows.
Let N = conductor(E).
If p is a rational prime, ap(E) = 1 + p − #E(Fp).
If v is a closed point of Spec OK, |v| = size of the residue ﬁeld κ(v).
L(E/K, s) =
v|N
1 − a|v||v|−s −1
×
v N
1 − a|v||v|−s
+ |v|1−2s −1
Modularity (Wiles et al) =⇒ analytic continuation of L(E/K, s) to C.

Birch and Swinnerton-Dyer
Brian Birch
Peter Swinnerton-Dyer

Birch and Swinnerton-Dyer
Conjecture (BSD, rough version)
ords=1 L(E/K, s) = rkalg(E, K).
The left-hand side is the analytic rank, rkan(E, K).
Consequence
rkan(E, K) ≥ 1 =⇒ ∃PK ∈ E(K) of inﬁnite order.
Fact
Deﬁne S(K, N) = { | N : inert in K}.Then rkan(E, K) is ≥ 1 when:
1 K is imaginary quadratic and #S(K, N) is even, or
2 K is real quadratic and #S(K, N) is odd.

Heegner vs Darmon (points)
Kurt Heegner Henri Darmon

Fact
The analytic rank rkan(E, K) is ≥ 1 when:
In case 1 , one can construct Heegner points on E(K).
Attached to elements τ ∈ K ∩ H.
Gross–Zagier formula =⇒ they have inﬁnite order if rkan(E, K) = 1.
If S(K, N) = ∅, they can be obtained by C-analytic methods.
If S(K, N) = ∅, can use p-adic methods.

Fact
The analytic rank rkan(E, K) is ≥ 1 when:
In case 2 , Heegner points are not available.
Note that in this case, K ∩ H = ∅!
In 2001, Darmon gave a construction (a.k.a “Stark-Heegner”).
p-adic analytic =⇒ a priori they lie in E(Kp).
Proof of their algebraicity completely open so far.
Darmon’s construction restricted to S(K, N) = {p}.
Greenberg extended to S(K, N) of arbitrary odd size.
We call them Darmon points.

Goals
In this talk we will:
1 Explain what Darmon Points are,
2 Explain how to calculate them, and
“The fun of the subject seems to me to be in the examples.
B. Gross, in a letter to B. Birch, 1982
”3 See some fun examples!

Integration on the p-adic upper-half plane
Deﬁnition
The p-adic upper half plane is the rigid-analytic space
Hp = P1
(Cp) P1
(Qp) ( = Cp Qp ).
This is the p-adic analogue of H± = C R.
Ω1
Hp
= space of rigid-analytic one-forms on Hp.
Coleman integral: allows to make sense of
τ2
τ1
ω, for τ1, τ2 ∈ Hp and ω ∈ Ω1
Hp
.
If the residues of ω are all integers, have a multiplicative reﬁnement:
×
τ2
τ1
ω = lim
−→
U U∈U
tU − τ2
tU − τ1
µ(U)
where µ(U) = resA(U) ω.

The p-adic upper half plane
×
τ2
τ1
ω = lim
−→
U U∈U
tU − τ2
tU − τ1
µ(U)
where µ(U) = resA(U) ω.
Bruhat-Tits tree of
GL2(Qp) with p = 2.
Hp having the
Bruhat-Tits as retract.
Annuli A(U) for U a
covering of size p−3.
tU is any point in
U ⊂ P1(Qp).

Darmon points `a la Greenberg
Henri Darmon
Matthew Greenberg

Darmon points `a la Greenberg
Assumption
1 K is real quadratic.
2 S(K, N) = { : | pD} has odd cardinality.
Write N = pDM, and B/Q = quaternion algebra of discriminant D.
Fix ιp : B → M2(Qp).
B×
→ GL2(Qp) acts on Hp by:
a b
c d τ =
aτ + b
cτ + d
.
R an Eichler order of level M (and such that ιp(R) is “nice”
ιp(R) ⊂ a b
c d ∈ M2(Zp) : vp(c) ≥ 1 .
).
Γ = R[1
p ]
×
1
(elements in R[1
p ] of reduced norm 1) → SL2(Qp).
If D = 1, then B = M2(Q) and Γ = a b
c d ∈ SL2(Z[1/p]) : M | c .

Cohomology
Let Ω1
Hp,Z = rigid-analytic differentials having integral residues.
Can attach to E a unique (up to sign) class [ΦE] ∈ H1 Γ, Ω1
Hp,Z .
Uses Hecke action and Jacquet–Langlands correspondence.
Theorem (M. Greenberg)
There exists a unique class [ΦE] ∈ H1(Γ, Ω1
Hp,Z) satisfying:
1 T [ΦE] = a [ΦE] for all primes pDM;
2 U [ΦE] = a [ΦE] for all | DM;
3 Wp[ΦE] = ap[ΦE] (Atkin-Lehner involution);
4 W∞[ΦE] = [ΦE] (Involution at ∞).
Hecke action can be made explicit
Suitable for computation.

Homology
Start with an embedding ψ: K → B.
ψ induces an action of K× on Hp via ιp.
1 Let τψ ∈ Hp be the unique ﬁxed point of K×
2 Set γψ = ψ( 2
), where O×
K = {±1} × .
ψ ; ˜Θψ = [γψ ⊗τψ] ∈ H1(Γ, Div Hp).
Key exact sequence:
H1(Γ, Div0
Hp) // H1(Γ, Div Hp)
deg
// H1(Γ, Z)
Θψ
? // ˜Θψ
// torsion
Challenge: pull
a multiple of
˜Θψ back to Θψ ∈ H1(Γ, Div0
Hp).
Requires algorithms adapted to the nature of Γ.

Conjecture
Recall our data
1 Θψ = [ γ γ ⊗Dγ] ∈ H1(Γ, Div0
Hp)
2 ΦE ∈ Z1(Γ, Ω1
Hp,Z)
Jψ =
γ
×
Dγ
(ΦE)γ ∈ K×
p .
Jψ is well deﬁned modulo powers of the Tate parameter qE
Conjecture (Darmon (D = 1), Greenberg (D 1))
1 Pψ = ΨTate(Jψ) belongs to E(Kab)
2 If rkan(E, K) = 1, then Pψ has inﬁnite order.

Numerical Evidence
Conjecture (Darmon (D = 1), Greenberg (D 1))
1 Pψ = ΨTate(Jψ) belongs to E(Kab)
2 If rkan(E, K) = 1, then Pψ has inﬁnite order.
Numerical Evidence
M = 1 M 1
D = 1 Darmon–Green (2002) Guitart–M. (2012)
(B = M2(Q)) Darmon–Pollack (2006)
D 1 Guitart–M. (2013)

Other directions
E deﬁned over other number ﬁeld F = Q.
1 F totally real: Greenberg (2009).
No numerical evidence.
2 F quadratic imaginary: Trifkovic (2006).
Numerical evidence when F is Euclidean.
3 F arbitrary: Guitart–Sengun–M. (in progress).
Lift to Jac(X0(N)): Rotger-Longo-Vigni (2012).
Higher weight modular forms: Rotger-Seveso (2012).

Calculating the Cycle
Key Exact Sequence
H1(Γ, Div0
Hp) // H1(Γ, Div Hp)
deg
// H1(Γ, Z)
Θψ
? // ˜Θψ
// torsion
M = 1 M 1
D = 1 Adjacent Cusps Elementary Matrix Decomposition
(B = M2(Q)) (Manin Trick) Go Go
D 1 Commutator Decomposition Go

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.
We trivialize on Fab: g = yxyxy(x−2)(y−3).
To simplify g ⊗τ0 in H1(Γ, Div Hp), use:
1 γ1γ2 ⊗D = γ1 ⊗D + γ2 ⊗γ−1
1 D.
2 γ−1
⊗D = −γ ⊗γD.

Implementation
We have written SAGE code to do all of the above.
Depend on:
1 Overconvergent method for D = 1 (R.Pollack).
2 Finding a presentation for units of orders in B (J.Voight).
Currently depends on MAGMA.
Overconvergent methods (adapted to D 1)
Efﬁcient (polynomial time) integration algorithm.
Apart from checking the conjecture, can use the method to actually
ﬁnding the points.

Examples
Please show them
the examples !

p = 5, Curve 15A1
E : y2
+ xy + y = x3
+ x2
− 10x − 10
dK h P+
13 1 −
√
13 + 1, 2
√
13 − 4
28 1 −15
√
7 + 43, 150
√
7 − 402
37 1 −5
9
√
37 + 5
9, 25
27
√
37 − 70
27
73 1 −17
32
√
73 + 77
32, 187
128
√
73 − 1199
128
88 1 −17
9 , 14
27
√
22 + 4
9
97 1 − 25
121
√
97 + 123
121, 375
2662
√
97 − 4749
2662
133 1 103
9 , 92
27
√
133 − 56
9
172 1 −1923
1681, 11781
68921
√
43 + 121
1681
193 1 1885
288
√
193 + 25885
288 , 292175
3456
√
193 + 4056815
3456

p = 3, Curve 21A1
E : y2
+ xy = x3
− 4x − 1
dK h P+
8 1 −9
√
2 + 11, 45
√
2 − 64
29 1 − 9
25
√
29 + 32
25, 63
125
√
29 − 449
125
44 1 − 9
49
√
11 − 52
49, 54
343
√
11 + 557
343
53 1 − 37
169
√
53 + 184
169, 555
2197
√
53 − 5633
2197
92 1 533
46 , 17325
2116
√
23 − 533
92
137 1 − 1959
11449
√
137 + 242
11449, 295809
2450086
√
137 − 162481
2450086
149 1 − 261
2809
√
149 + 2468
2809, 8091
148877
√
149 − 101789
148877
197 1 − 79135143
209961032
√
197 + 977125081
209961032, 1439547386313
1075630366936
√
197 − 9297639417941
537815183468
D h hD(x)
65 2 x2 + 61851
6241
√
65 − 491926
6241 x − 403782
6241
√
65 + 3256777
6241

p = 11, Curve 33A1
E : y2
+ xy = x3
+ x2
− 11x
dK h P +
13 1 − 1
2
√
13 + 3
2
, 1
2
√
13 − 7
2
28 1 22
7
, 55
49
√
7 − 11
7
61 1 − 1
2
√
61 + 5
2
,
√
61 − 11
73 1 − 53339
49928
√
73 + 324687
49928
, 31203315
7888624
√
73 − 290996167
7888624
76 1 −2,
√
19 + 1
109 1 − 143
2
√
109 + 1485
2
, 5577
2
√
109 − 58223
2
172 1 − 51842
21025
, 2065147
3048625
√
43 + 25921
21025
184 1 59488
21609
, 109252
3176523
√
46 − 29744
21609
193 1 94663533349261
678412148664608
√
193 + 1048806825770477
678412148664608
,
147778957920931299317
12494688311813553741184
√
193 + 30862934493092416035541
12494688311813553741184
D h hD(x)
40 2 x2
+ 2849
1681
√
10 − 6347
1681
x − 5082
1681
√
10 + 16819
1681
85 2 x2
+ 119
361
√
85 − 1022
361
x − 168
361
√
85 + 1549
361
145 4 x4
+ 169016003453
83168215321
√
145 − 1621540207320
83168215321
x3
+ − 1534717557538
83168215321
√
145 + 18972823294799
83168215321
x2
+ 5533405190489
83168215321
√
145 − 66553066916820
83168215321
x
+ − 6414913389456
83168215321
√
145 + 77248348177561
83168215321

p = 13, Curve 78A1
Exclusive for MEGA
First examples of Quaternionic Darmon Points.
78 = 2 · 3 · 13, we take p = 13 and D = 6.
E : y2
+ xy = x3
+ x2
− 19x + 685
dK h P+
5 1 10, 18
√
5 − 5
149 1 −10654790
1138489 , 1229396070
1214767763
√
149 + 5327395
1138489
197 1 964090
121 , −67449270
1331
√
197 − 482045
121
293 1 66626
7325 , −23188374
10731125
√
293 − 33313
7325
317 1 36974414
388325 , −226154303712
4308465875
√
317 − 18487207
388325
437 1 971110
339889 , −244302600
198155287
√
437 − 485555
339889
461 1 −146849210
23609881 , 132062657760
114720411779
√
461 + 73424605
23609881
509 1 86626030090
4613262241 , −1193915313019350
313337384670961
√
509 − 43313015045
4613262241
557 1 394312144429886
2528440578125 , 7858947314645355852384
94887001960802734375
√
557 − 197156072214943
2528440578125

Thank you !
Find the slides at: http://www.math.columbia.edu/∼masdeu/

Bibliography
Henri Darmon and Peter Green.
Elliptic curves and class fields of real quadratic fields: Algorithms and evidence.
Exp. Math., 11, No. 1, 37-55, 2002.
Henri Darmon and Robert Pollack.
Efficient calculation of Stark-Heegner points via overconvergent modular symbols.
Israel J. Math., 153:319–354, 2006.
Xavier Guitart and Marc Masdeu.
Elementary matrix Decomposition and the computation of Darmon points with higher conductor.
arXiv.org, 1209.4614, 2012.
Matthew Greenberg.
Stark-Heegner points and the cohomology of quaternionic Shimura varieties.
Duke Math. J., 147(3):541–575, 2009.
David Pollack and Robert Pollack.
A construction of rigid analytic cohomology classes for congruence subgroups of SL3(Z).
Canad. J. Math., 61(3):674–690, 2009.
trifkovic Mak Trifkovic,
Stark-Heegner points on elliptic curves defined over imaginary quadratic fields.
Duke Math. J., 135, No. 3, 415-453, 2006.

Differential forms and measures
Theorem (Teitelbaum?)
The assignment
µ → ω =
P1(Qp)
dz
z − t
dµ(t)
induces an isomorphism Meas0(P1(Qp), Cp) ∼= Ω1
Hp
.
Theorem (Teitelbaum)
τ2
τ1
ω =
P1(Qp)
log
t − τ1
t − τ2
dµ(t).
Proof sketch.
τ2
τ1
ω =
τ2
τ1 P1(Qp)
dz
z − t
dµ(t) =
P1(Qp)
log
t − τ2
t − τ1
dµ(t).

Continued Fractions
Theorem (Manin)
The cusps ∞ and γ∞ can always be connected by a chain of adjacent
cusps ∞ = c0 ∼ c1 ∼ · · · ∼ cr = γ∞.
Effective version: continued fractions.
(γ∞ − ∞) ⊗τ =
i
(gi∞ − gi0) ⊗τ
=
i
(∞ − 0) ⊗τi
=
i
(∞ − 1) ⊗τi + (1 − 0) ⊗τi τi = g−1
i τ
=
i
(∞ − 0) ⊗t−1
τi + (0 − ∞) ⊗tsτi
=
i
(∞ − 0) ⊗(t−1
τi − tsτi)
Back Marc Masdeu Numerical Evidence for Darmon Points June 4, 2013 3 / 4

Elementary Matrices
Theorem (Vaserstein (1972), Guitart–M. (2012) (effective version))
Any matrix γ ∈ Γ1(M) can be decomposed as
γ = l1u1 · · · lrur, ui = 1 xi
0 1 , li = 1 0
yi 1 .
Obtain the following:
(ug∞ − ∞) ⊗τ = (g∞ − ∞) ⊗u−1
τ.
and (lg∞ − ∞) ⊗τ simpliﬁes to:
= (lg∞ − 0) ⊗τ + (0 − ∞) ⊗τ
= (g∞ − 0) ⊗l−1
τ + (0 − ∞) ⊗τ
= (g∞ − ∞) ⊗l−1
τ + (∞ − 0) ⊗l−1
τ + (0 − ∞) ⊗τ
= (g∞ − ∞) ⊗l−1
τ + (∞ − 0) ⊗(l−1
τ − τ).
Iterating this process we reduce to a degree-0 divisor. Back

Numerical Evidence for Darmon Points

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