Numerical Evidence for Darmon Points M` etodes Efectius en Geometria Algebraica Xavier Guitart 1 Marc Masdeu 2 1 Universitat Polit` ecnica de Catalunya 2 Columbia University June 4, 2013 Marc Masdeu Numerical Evidence for Darmon Points June 4, 2013 1 / 28
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Numerical Evidence for Darmon PointsMetodes Efectius en Geometria Algebraica
Xavier Guitart 1 Marc Masdeu 2
1Universitat Politecnica de Catalunya
2Columbia University
June 4, 2013
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The Problem
ProblemGiven an algebraic curve C defined over Q, find points onC, defined on prescribed algebraic extensions K of Q.
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Mordell-Weil
Louis Mordell Andre Weil
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Basic Setup
E = elliptic curve defined over Q, and K = Q(√dK).
I If dK > 0 then K is called real quadratic, andI 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.
I Let N = conductor(E).I If p is a rational prime, ap(E) = 1 + p−#E(Fp).I If v is a closed point of SpecOK , |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
)−1Modularity (Wiles et al) =⇒ analytic continuation of L(E/K, s) to C.
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Birch and Swinnerton-Dyer
Brian Birch Peter Swinnerton-Dyer
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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).
Consequencerkan(E,K) ≥ 1 =⇒ ∃PK ∈ E(K) of infinite order.
FactDefine 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, or2 K is real quadratic and #S(K,N) is odd.
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Heegner vs Darmon (points)
Kurt Heegner Henri Darmon
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Heegner vs Darmon (points)
FactThe analytic rank 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.
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) 6= ∅, can use p-adic methods.
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Heegner vs Darmon (points)
FactThe analytic rank 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.
In case 2 , Heegner points are not available.I Note that in this case, K ∩H = ∅!
In 2001, Darmon gave a construction (a.k.a “Stark-Heegner”).I p-adic analytic =⇒ a priori they lie in E(Kp).I Proof of their algebraicity completely open so far.I Darmon’s construction restricted to S(K,N) = p.I Greenberg extended to S(K,N) of arbitrary odd size.I We call them Darmon points.
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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!
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Integration on the p-adic upper-half plane
DefinitionThe 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.Ω1Hp = space of rigid-analytic one-forms on Hp.
Coleman integral: allows to make sense of∫ τ2
τ1
ω, for τ1, τ2 ∈ Hp and ω ∈ Ω1Hp .
If the residues of ω are all integers, have a multiplicative refinement:
×∫ τ2
τ1
ω = lim−→U
∏U∈U
(tU − τ2tU − τ1
)µ(U)
where µ(U) = resA(U) ω.
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The p-adic upper half plane
×∫ τ2
τ1
ω = lim−→U
∏U∈U
(tU − τ2tU − τ1
)µ(U)
where µ(U) = resA(U) ω.
Bruhat-Tits tree ofGL2(Qp) with p = 2.Hp having theBruhat-Tits as retract.Annuli A(U) for U acovering of size p−3.tU is any point inU ⊂ P1(Qp).
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Darmon points a la Greenberg
Henri DarmonMatthew Greenberg
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Darmon points a la Greenberg
Assumption1 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).
I B× → GL2(Qp) acts on Hp by:(a bc d
)τ =
aτ + b
cτ + d.
R an Eichler order of level M (and such that ιp(R) is “nice”
ιp(R) ⊂(
a bc d
)∈M2(Zp) : vp(c) ≥ 1
.
).
Γ =(R[1p ]
)×1
(elements in R[1p ] of reduced norm 1) → SL2(Qp).
If D = 1, then B = M2(Q) and Γ =(
a bc d
)∈ SL2(Z[1/p]) : M | c
.
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Cohomology
Let Ω1Hp,Z = rigid-analytic differentials having integral residues.
Can attach to E a unique (up to sign) class [ΦE ] ∈ H1(
Γ,Ω1Hp,Z
).
Uses Hecke action and Jacquet–Langlands correspondence.
Theorem (M. Greenberg)
There exists a unique class [ΦE ] ∈ H1(Γ,Ω1Hp,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 explicitI Suitable for computation.
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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(ε2), where O×K = ±1 × 〈ε〉.
ψ ; Θψ = [γψ ⊗ τψ] ∈ H1(Γ,DivHp).
Key exact sequence:
H1(Γ,Div0Hp) // H1(Γ,DivHp)deg // H1(Γ,Z)
Θψ ? // Θψ
// torsion
Challenge: pull
a multiple of
Θψ back to Θψ ∈ H1(Γ,Div0Hp).I Requires algorithms adapted to the nature of Γ.
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Conjecture
Recall our data1 Θψ = [
∑γ γ ⊗Dγ ] ∈ H1(Γ,Div0Hp)
2 ΦE ∈ Z1(Γ,Ω1Hp,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.
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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).
F Currently depends on MAGMA.
Overconvergent methods (adapted to D > 1)I Efficient (polynomial time) integration algorithm.I Apart from checking the conjecture, can use the method to actually
finding the points.
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Examples
Please show themthe examples !
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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 ,2527
√37− 70
27
)73 1
(−17
32
√73 + 77
32 ,187128
√73− 1199
128
)88 1
(−17
9 ,1427
√22 + 4
9
)97 1
(− 25
121
√97 + 123
121 ,3752662
√97− 4749
2662
)133 1
(1039 ,
9227
√133− 56
9
)172 1
(−1923
1681 ,1178168921
√43 + 121
1681
)193 1
(1885288
√193 + 25885
288 ,2921753456
√193 + 4056815
3456
)
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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 ,63125
√29− 449
125
)44 1
(− 9
49
√11− 52
49 ,54343
√11 + 557
343
)53 1
(− 37
169
√53 + 184
169 ,5552197
√53− 5633
2197
)92 1
(53346 ,
173252116
√23− 533
92
)137 1
(− 1959
11449
√137 + 242
11449 ,2958092450086
√137− 162481
2450086
)149 1
(− 261
2809
√149 + 2468
2809 ,8091
148877
√149− 101789
148877
)197 1
(− 79135143
209961032
√197 + 977125081
209961032 ,14395473863131075630366936
√197− 9297639417941
537815183468
)D h hD(x)
65 2 x2 +(618516241
√65− 491926
6241
)x− 403782
6241
√65 + 3256777
6241
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p = 11, Curve 33A1
E : y2 + xy = x3 + x2 − 11x
dK h P+
13 1(− 1
2
√13 + 3
2, 12
√13− 7
2
)28 1
(227, 5549
√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, 20651473048625
√43 + 25921
21025
)184 1
(5948821609
, 1092523176523
√46− 29744
21609
)193 1
(94663533349261678412148664608
√193 + 1048806825770477
678412148664608,
14777895792093129931712494688311813553741184
√193 + 30862934493092416035541
12494688311813553741184
)D h hD(x)
40 2 x2 +(
28491681
√10− 6347
1681
)x− 5082
1681
√10 + 16819
1681
85 2 x2 +(
119361
√85− 1022
361
)x− 168
361
√85 + 1549
361
145 4 x4 +(
16901600345383168215321
√145− 1621540207320
83168215321
)x3
+(− 1534717557538
83168215321
√145 + 18972823294799
83168215321
)x2 +
(553340519048983168215321
√145− 66553066916820
83168215321
)x
+− 641491338945683168215321
√145 + 77248348177561
83168215321
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p = 13, Curve 78A1
Exclusive for MEGAFirst examples of Quaternionic Darmon Points.
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Thank you !
Find the slides at: http://www.math.columbia.edu/∼masdeu/
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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.
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Differential forms and measures
Theorem (Teitelbaum?)The assignment
µ 7→ ω =
∫P1(Qp)
dz
z − tdµ(t)
induces an isomorphism Meas0(P1(Qp),Cp) ∼= Ω1Hp .
Theorem (Teitelbaum)∫ τ2
τ1
ω =
∫P1(Qp)
log
(t− τ1t− τ2
)dµ(t).
Proof sketch.∫ τ2
τ1
ω =
∫ τ2
τ1
∫P1(Qp)
dz
z − tdµ(t) =
∫P1(Qp)
log
(t− τ2t− τ1
)dµ(t).
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Continued Fractions
Theorem (Manin)The cusps∞ and γ∞ can always be connected by a chain of adjacentcusps∞ = 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−1i τ
=∑i
(∞− 0)⊗ t−1τi + (0−∞)⊗ tsτi
=∑i
(∞− 0)⊗ (t−1τi − 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