Post on 29-Jan-2018
Rational points on elliptic curvesIV Congreso de Jovenes Investigadores
Marc Masdeu
Universitat Autonoma de Barcelona
September 5th, 2017
Marc Masdeu Rational points on elliptic curves September 5th , 2017 1 / 33
Points on a conic
ProblemGiven a homogeneous quadratic equation in 3 variables
C : aX2 + bY 2 + cZ2 + dXY + eXZ + fY Z = 0, a, b, c, d, e, f ∈ Z,
find all solutions (X,Y, Z) with X,Y, Z ∈ Z (⇐⇒ (X : Y : Z) ∈ P2(Q)).
RemarkBy dividing out by Z2, equivalent to finding all the solutions (x, y) to
ax2 + by2 + c+ dxy + ex+ fy = 0, x, y ∈ Q.
May be trivial: for xy = 1, all solutions are (t, 1/t), with t ∈ Qr 0.Sometimes there are no solutions:
I x2 + y2 = −1 has no solutions in R, let alone in Q.I x2 + y2 = 3 has no solutions in Q either. (Why?)
Marc Masdeu Rational points on elliptic curves September 5th , 2017 2 / 33
Points on a conic: x2 + y2 = 1
GoalFind all rational solutions to the equation x2 + y2 = 1.
slope = t
y = t(x+ 1)
x
y
(−1, 0)
P =(
1−t2
1+t2 ,2t
1+t2
) x2 + y2 = 1
x2 + t2(x+ 1)2 = 1
x2 +2t2
1 + t2x+
t2 − 1
1 + t2= 0
(x− x0)(x− x1) = 0 =⇒ x0x1 =t2 − 1
1 + t2
x0 = −1 =⇒ x1 =1− t21 + t2
Marc Masdeu Rational points on elliptic curves September 5th , 2017 3 / 33
Parametrizing cubics
Technique in previous slide works for general conics.Upshot: If a conic has one rational point, then it has infinitely manyand they can be easily parametrized.Consider a cubic equation:
aX3+bX2Y+cXY 2+dY 3+eX2Z+fXY Z+gY 2Z+mXZ2+nY Z2+rZ3 = 0
Sometimes it has no solutions:
X3 + 14Y 3 = 12Z3 (work modulo 7)
I Start with a solution (X,Y, Z) such that gcd(X,Y, Z) = 1.I LHS can take values 0,±1 modulo 7.I RHS can take values 0,±2 modulo 7.I So any solution will satisfy 7 | X and 7 | Z.I But then 7 | Y , which is a contradiction.
Marc Masdeu Rational points on elliptic curves September 5th , 2017 4 / 33
Elliptic curves
DefinitionA cubic E : y2 = x3 + ax+ b is an elliptic curve if∆ = −16(4a3 + 27b2) 6= 0.
We write E(Q) for the set of the rational points of E.I If K ⊇ Q is another field, write E(K) for the set of solutions where we
allow the coordinates to be in K.E(Q) always has a point, namely O = (0 : 1 : 0) (“point at infinity”).
I The curve y2 = x3 − 108 has O as its only point: E(Q) = O.I The curve y2 = x3 − 27 has two rational points: E(Q) = O, (3, 0).I The curve y2 = x3 + 4 has three: E(Q) = O, (0, 2), (0,−2).I The curve y2 = x3 − x+ 1 has infinitely many points:E(Q) = O, (1,−1), (−1,−1), (0, 1), (3, 5), (5,−11), ( 1
4 ,−78 ), (−11
9 , 1727 ), . . ..
Is there any pattern??
Marc Masdeu Rational points on elliptic curves September 5th , 2017 5 / 33
The set E(Q) has a group structure
Given two points P , Q on E(Q), we can produce a third one.
P
Q
P +Q
R
Obviously commutative. . .I . . . but proving associativity can turn into a nightmare!
The point O is the neutral element.To add a point to itself (P = Q), use tangent line instead of cord.
Marc Masdeu Rational points on elliptic curves September 5th , 2017 6 / 33
Mordell–Weil Theorem
Louis Mordell Andre Weil
Theorem (Mordell 1922, Weil 1928)Let K be an algebraic number field (i.e. [K : Q] <∞).Then E(K) is a finitely generated abelian group. So it is of the form
E(K) = (torsion)⊕ Zr
There are algorithms to calculate the torsion.But the rank r is very hard to compute!
Marc Masdeu Rational points on elliptic curves September 5th , 2017 7 / 33
Counting points in finite fields
Consider the curve E : y2 = x3 − x+ 1.Can think of it as an equation modulo 3. In that case it has 7 points:
O, (0, 1), (0, 2), (1, 1), (1, 2), (2, 1), (2, 2)
Or we can think modulo 5, where it has 8 points:
O, (0, 1), (0, 4), (1, 1), (1, 4), (3, 0), (4, 1), (4, 4)
Note that we should expect #E(Fp) to be roughly about p+ 1.Let ap(E) = p+ 1−#E(Fp).p 2 3 5 7 11 13 17 19 23 29 31 37#E(Fp) 3 7 8 12 10 19 14 22 23 37 35 36ap(E) 0 -3 -2 -4 2 -5 4 -2 1 -7 -3 2
Marc Masdeu Rational points on elliptic curves September 5th , 2017 8 / 33
Hasse’s bound
2e4 4e4 6e4 8e4 1e5p
-600
-400
-200
200
400
600
p + 1 #E( p)y = 2 x
Theorem (Hasse, 1933):∣∣∣p+ 1−#E(Fp)
∣∣∣ ≤ 2√p.
How the points distribute inside the above parabola is the statementof the Sato–Tate conjecture:
-1 -0.5 0 0.5 1ap/2 p
500
1000
1500
2000
2500
3000
n
Marc Masdeu Rational points on elliptic curves September 5th , 2017 9 / 33
Birch and Swinnerton-Dyer
Bryan Birch Sir Peter Swinnerton-Dyer
In the late 1950’s, Birch and Swinnerton-Dyer studied the asymptoticbehavior of the quantity
CE(x) =∏p≤x
#E(Fp)p
as x→∞
Marc Masdeu Rational points on elliptic curves September 5th, 2017 10 / 33
Experimental rank data
Plots of CE(x) =∏p≤x
#E(Fp)p against log(x) (doubly-logarithmic axes).
-0.4 -0.2 0 0.2 0.4 0.6loglogx
0.5
1
1.5
2
2.5
3
3.5
logCE(x)slope 0.104slope 1.147slope 2.254slope 3.015
(5077.a1) y2 + y = x3 − 7x+ 6
(389.a1) y2 + y = x3 + x2 − 2x
(37.a1) y2 + y = x3 − x(11.a1) y2 + y = x3 − x2 − 7820x− 263580
Conjecture BSD (Birch–Swinnerton-Dyer)
CE(x) ∝ log(x)r as x→∞.Marc Masdeu Rational points on elliptic curves September 5th, 2017 11 / 33
The L-function of E
One can rephrase the BSD conjecture using L-functions.Recall the “error” in Hasse’s bound
ap(E) = p+ 1−#E(Fp)
Define a function of a complex variable s:
L(E, s)“ = ”∏
p prime
(1− app−s + p1−2s
)−1, <(s) > 3/2.
I Note that L(E, 1)“ = ”∏ p
#E(Fp)“ = ”CE(∞)−1. . .
Conjecture BSD, second version1 The L-function L(E, s) can be analytically continued to all C.2 L(E, s) has a functional equation relating L(E, s) to L(E, 2− s)3
ords=1 L(E, s) = r.
Marc Masdeu Rational points on elliptic curves September 5th, 2017 12 / 33
Plots of L(E, s) restricted to <(s) = 1 (www.lmfdb.org)
y2 + y = x3 − x2 − 7820x− 263580
y2 + y = x3 + x2 − 2x
y2 + y = x3 − x
y2 + y = x3 − 7x+ 6Marc Masdeu Rational points on elliptic curves September 5th, 2017 13 / 33
Remarks on BSD
The correct definition of L(E, s) involves knowing how E behaves“bad primes” (where the reduction of E modulo p is “singular”).
I This is encoded in the conductor N = cond(E).Birch and Swinnerton-Dyer predicted also a formula for the leadingterm of the Taylor expansion of L(E, s) at s = 1.The first two statements of the refined conjecture are aconsequence of extremely deep theorems, known as “modularity”.Conjecture extends to elliptic curves over other number fields.
I In this generality, one doesn’t even know whether L(E, s) can beextended to all C.
The BSD conjecture is one of the CMI Problems of the Millenium.
Theorem (Gross–Zagier 1986 + Kolyvagin 1989)If ords=1 L(E, s) ≤ 1, then ords=1 L(E, s) = r.
If ords=1 L(E, s) = 1, need to produce a point P of infinite order!
Marc Masdeu Rational points on elliptic curves September 5th, 2017 14 / 33
The main tool for BSD: Heegner points (1952)
Kurt Heegner
Heegner points are defined over (extensions of) quadratic fields K.Only available when K = Q(
√D) is imaginary: D < 0.
We will further require the additional condition:I Heegner hypothesis: p | N =⇒ p split in K.
This ensures that ords=1 L(E/K, s) is odd (so ≥ 1).
Marc Masdeu Rational points on elliptic curves September 5th, 2017 15 / 33
Modular forms
For an integer N ≥ 1, set Γ0(N) = (a bc d
)∈ SL2(Z) : N | c.
Γ0(N) acts on the upper-half plane H = z ∈ C : Im(z) > 0:I Via
(a bc d
)· z = az+b
cz+d .A cusp form of level N is a holomorphic map f : H→ C such that:
1 f(γz) = (cz + d)2f(z) for all γ =(a bc d
)∈ Γ0(N).
2 Cuspidal: limz→i∞ f(z) = 0.( 1 1
0 1 ) ∈ Γ0(N) ; have Fourier expansions f(z) =∑∞
n=1 an(f)e2πinz.Given an elliptic curve E, define an for all n ≥ 1 as follows:
ap = ap(E) for all primes p.anm = anam if n and m are coprime.apr = apapr−1 − papr−2 for r ≥ 2 (ommit second term if p | cond(E)).
Modularity Theorem (Wiles, . . . , Breuil–Conrad–Diamond–Taylor 2001)
The function fE(z) =∑n≥1
ane2πinz
is the Fourier expansion of a modular form of level N = cond(E).Marc Masdeu Rational points on elliptic curves September 5th, 2017 16 / 33
Heegner Points (K/Q imaginary quadratic)Modularity =⇒ ∃ modular form fE attached to E.
ωE = 2πifE(z)dz = 2πi∑n≥1
ane2πinzdz.
This is a differential form on H, invariant under Γ0(N).
Given τ ∈ K ∩H, set Jτ =
∫ τ
i∞ωE ∈ C.
Well-defined up to ΛE =∫
γ ωE | γ closed path in Γ0(N)\H
.
Theorem (Eichler–Shimura 1959)There exists a computable complex-analytic group isomorphism
ηWeierstrass : C/ΛE → E(C), ΛE = lattice of rank 2.
Theorem (Shimura, Gross–Zagier, Kolyvagin)1 Pτ = ηWeierstrass(Jτ ) ∈ E(C) has algebraic coordinates.2 PK = Tr(Pτ ) is nontorsion ⇐⇒ ords=1 L(E/K, s) = 1.3 If ords=1 L(E/Q, s) ≤ 1 then BSD holds for E(Q).
Marc Masdeu Rational points on elliptic curves September 5th, 2017 17 / 33
An example: E : y2 + y = x3 − x2 − 10x− 20 (“11a1”)
fE(z) = q − 2q2 − q3 + 2q4 + q5 + 2q6 − 2q7 − 2q9 − 2q10 + · · ·fE is a modular form of level N = 11.Embed K = Q(
√−2)→M2(11) by sending
√−2 7→
(3 −111 −3
). It
identifies OK = Z[√−2] with the maximal order of M2(11).
Such an order fixes the point τ = −3+√−2
11 .Jτ =
∑n≥1
ann e
2πinτ ∼ 0.126920930427956− 0.536079610338652 · i.Pτ = ηWeierstrass(Jτ ) ∼ (−3.00000 + 1.41421 · i, 3.00000 + 4.242640 · i).Pτ is very close to the algebraic point of infinite order
(−3 +√−2, 3 + 3
√−2) ∈ E(K).
Marc Masdeu Rational points on elliptic curves September 5th, 2017 18 / 33
An example of Mark WatkinsLet E be the elliptic curve of conductor NE = 66157667:
E : Y 2 + 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:367770537186677506614005642341827170087932269492285584726218770061653546349271015805365134370326743061141306464500052886704651998399766478840791915307861741507273933802628157325092479708268760217101755385871816780548765478502284415627682847192752681899094962659937870630036760359293577021806237483971074931228416346507852381696883227650072039964481597215995993299744934117106289850389364006552497835877740257534533113775202882210048356163645919345794812074571029660897173224370337701056165735008590640297090298709121506266697266461993201825397369999550868142294312756322177410730532828064759604975369242350993568030726937049911607264109782746847951283794119298941214490794330902986582991229569401523519938742746376107190770204010513818349012786637889254711059455555173810904911927619899031855149292325338589831979737026402711049742594116000380601480839982975557506035851728035645241044229165029649347049289119188596869401159325131363345962579503132339847275422440094553824705189225653677459512863117911721838552934309124508134493366437408093924362039749911907416973504142322111757058584200725022632116164720164998641729522677460525999499077942125820428879526063735692685991018516862938796047597323986537154171248316943796373217191993996993714654629536884396057924790938647656663281596178145722116098216500930333824321806726937018190136190556573208807048355335567078793126656928657859036779350593274598717379730880724034301867739443749841809456715884193720328901461552659882628405842209756757167816662139945081864642108533595989975716259259240152834050940654479617147685922500856944449822045386092122409096978544817218847897640513477806598329177604246380812377739049184475550777341620985976570393037880282764967019552408400730754822676441481715385344001979832232652414888335865567377214360456003296961668177481944809066257442596772347829664126972931904101685281128944780074646796760942430959617022257479874089403564965038885379817866920048929814520268493677507059073765902671638087366488496702836326268574593123245107420348878101763123893347657020275591248824247800594270862052082185973393290009189867677259458080676065098703453539525576975639543700507625640729872340789406314394468400584455920683361976200121834430751233901473228497490561998078486251074993528871318797403348087370426900997556442577081254910572185107856605139877331015042842121106080690743578173268489400499056898312621953947967012358414547752897081097091795744203697684046066255663201242292760126759871266004516377432961917272040217147083563399987612420595275792033855676991823368254862159558450043808051481533297270035287382247038279293223946385070118082306958987268603396924054403103857444058848605587415400517670032631121206127732481340391088277796488544415738156553014768406246154666005139690428085145098272500791416214774673484501826722500527091164944262537169595848931680754096774712860490572746224094031187043204526107239201079603468297522895106598567437015083348797875364162797693968819804139548885751282687152237078260358705230284426203064493684250614282879918107733796207067250003823959412935677624093236047038637365577326399589008804507786011973155927731073034706536557461443806622707622411087809371872157210456836892493613836792026761820382217165481998924123604782787923229739171920575447007099501678380795077013113325989801385729993920818301654424251339564606876820121928372246213399859213282792511168043953443839793901139974194479300297566097664539199384651908436188732428818373302383046388859427937893841888014266685177616605644783704135794931830750265686335934066565240944049448213005591997128985560760260399214278635912634351586762354869354021530746189992899582554597632108309638569296964800046983072736238483149014714600896056552029642747991419063454749142059564274298254654925893866404955146903330024475746163543714996249652420171171054231726336493541586971431778944051481059633738399411418574323811770949729726843612672925000631355659834164200554441315451003433452466204707123811663623662837296862948061758759928631763661985185615801886205770721032006304144867787347058316392295671580091655872087209485913286930128858640442589125454268580397484571921012318872311624898317615607628176460097441336323549031828235965636277950827328087547939511112374216436584203379248450122647406094035171130740663723547675939885959363881135893035102018389444212746146250328348242610673524022378994978392020098814721974502062692815736689229759065822093942795318705345275598989426335235935505605311411301560321192269430861733743544402908586497305353600909431214933202522528717109214492959330016065810287623144179288466664888540622702346704213752456372574449563979215782406566937885352945871994541770838871930542220307771671498466518108722622109421676741544945695403509866953167277628280232464839215003474048896968037544660029755740065581270139083249903212572230417942249795467100700393944310325009677179182109970943346807335014446839612282508824324073679584122851208360459166315484891952299449340025896509298935939357721723543933108743241997387447018395925320167637640328407957069845439501381234605867495003402016724626400855369636521155009147176245904149069225438646928549072337653348704931901764847439772432025275648964681387210234070849306330191790380412396115446240832583481366372132300849060835262136832315311052903367503857437920508931305283143379423930601369154572530677278862066638884250221791647123563828956462530983567929499493346622977494903591722345188975062941907415400740881
Marc Masdeu Rational points on elliptic curves September 5th, 2017 19 / 33
Darmon points (K/Q real quadratic)
Henri Darmon
If K = Q(√D) is real and p ‖ N is inert in K,
then ords=1 L(E/K, s) is odd as well. . .. . . but Heegner points are not available.
I Note that in this case, K ∩H = ∅!In 2001, Darmon gave another analyticconstruction:
I p-adic analytic (instead of complex analytic).I The algebraicity of these points is still open
nowadays.
Marc Masdeu Rational points on elliptic curves September 5th, 2017 20 / 33
Recall: how to construct C
On Q, consider the usual absolute value
|x| =x x ≥ 0,
−x x < 0.
Complete Q with respect to the metric given by | · | (we call this R).I Have decimal expansions, e.g 2.147581534 . . ..I They are really power series
∑n≥n0
an10−n.
Adjoin a root i of an irreducible quadratic to get C.I We also have decimal expansions, e.g.
2.147581534 . . .+ i · 3.6171346234 . . ..
Marc Masdeu Rational points on elliptic curves September 5th, 2017 21 / 33
How to construct its p-adic analogue
On Q, consider the p-adic absolute value |x|p = p−vp(x) on Q, where
vp
(ptm
n
)= t if p - m and p - n, vp(0) = +∞.
I |3|3 = 1/3, |18|3 = 1/9, |2/3|3 = 3, . . .Complete Q with respect to the metric given by | · |p (we call this Qp).
I Have p-adic expansions:∑n≥n0
anpn, an ∈ 0, . . . , p− 1.
Take an irreducible quadratic and adjoin a root α to get Qp2 .I These too have p-adic expansions:∑
n≥n0
(an + αbn)pn, an, bn ∈ 0, . . . , p− 1.
The p-adic analogue to H is Hp = Qp2 rQp.I More like an analogue of H together with the lower half-plane. . .
Marc Masdeu Rational points on elliptic curves September 5th, 2017 22 / 33
Heegner points vs Darmon points
Recall K = Q(√D) real quadratic, p is inert in K.
I Note that X2 −D doesn’t have roots in Qp.I Take Qp2 = Qp(
√D), and note that K → Qp2 .
Recall that the conductor of E is of the form pM with p -M .
Theorem (Tate)There is a p-adic number qE ∈ Q×p , and a p-adic analytic isomorphism
ηTate : Q×p2/qZE → E(Qp2).
LetΓ =
γ =
(a bc d
)∈ SL2(Z[1/p]) |M | c
.
E gives rise to a “rigid analytic” differential (1, 1)-form ωE on H×Hp.I Invariant under Γ, so it “descends” to XΓ = Γ\H×Hp.
Marc Masdeu Rational points on elliptic curves September 5th, 2017 23 / 33
A null-homologous cycle in XΓ = Γ\H×Hp
Start with an embedding ψ : K →M2(Q).ψ induces an action of K× on Hp.Set γ = γψ = ψ(ε2), where O×K = ±1 × 〈ε〉.Let τ = τψ ∈ Hp be the unique fixed point of γ.Note: γ fixes τ , and so does any power of γ.Fact: Γab is finite.
I So if e = #Γab, then γe is a product of commutators.I Assume (for simplicity) that γe = aba−1b−1, for some a, b ∈ Γ.
Consider the 1-cycle in XΓ = Γ\H×Hp:
Θ = (γe∞→∞)× τ,
where∞ ∈ H is any choice of base point.I Note that (γe∞, τ) = γe · (∞, τ), so it is closed.
Turns out that Θ is null-homologous: it is the boundary of a2-chain.
Marc Masdeu Rational points on elliptic curves September 5th, 2017 24 / 33
A 2-chain with boundary Θ
×
×
×
∞
a∞
∞
b∞
a−1τ
τ
τ∞
a∞ ab∞
aba−1∞
aba−1b−1∞
∞
a∞
×τ
×τ
×τ
×τ
×τ
×τ
×b−1τ
×τ
×a−1τ
×a∞
×∞
×∞
×b∞
a∞
ab∞
ab∞
aba−1∞
aba−1∞
aba−1b−1∞
aba−1b−1∞
∞
∞
a∞
∞
a∞
∞
b∞
∞
b∞
b−1τ
τ
a−1τ
τ
a−1τ
τ
b−1τ
τ
b−1τ
τ
+
+ + + +
−
+
−
−
+
−
+
+
−
+
×∞
b−1τ
τ
×a∞
τ
b−1τ
+ ×∞
τ
a−1τ
+ ×b∞
a−1τ
τ
+ = ∞×
b−1τ
τ
a−1τ
a−1b−1τ
τ
a−1τ
b−1a−1τ
b−1τ
+ + + =∞×
b−1a−1τ
a−1b−1τ
∂
Marc Masdeu Rational points on elliptic curves September 5th, 2017 25 / 33
ConjectureRecallΘ = ∂
(× τ − (∞→ a∞)× (b−1τ → τ) + (∞→ b∞)× (a−1τ → τ)
).
ωE = (1, 1) form on H×Hp attached to E.
Technical detail: Can define a “multiplicative integral” for ωE .I Essentially, replace Riemann sums with products.I Its p-adic logarithm recovers the usual integral.
Jψ = ×∫ a∞
∞×∫ b−1τ
τωE −×
∫ b∞
∞×∫ a−1τ
τωE ∈ Q×
p2.
The 2-chain is not unique: it can be changed by any 2-cycle.Jψ is well defined modulo elements in
Λ =
×∫×∫ξωE : ξ ∈ H2 (Γ\H×Hp,Z)
⊂ Q×
p2
Marc Masdeu Rational points on elliptic curves September 5th, 2017 26 / 33
The conjecture
Jψ = ×∫ a∞
∞×∫ b−1τ
τωE −×
∫ b∞
∞×∫ a−1τ
τωE ∈ Q×
p2/Λ.
Λ =
×∫×∫ξωE : ξ ∈ H2 (Γ\H×Hp,Z)
⊂ Q×
p2
Theorem (Bertolini–Darmon)
Λ = qZE .
Note that it makes sense to consider ηTate(Jψ) ∈ E(Qp2).
Conjecture (Darmon)1 Pψ = ηTate(Jψ) ∈ E(Qp2) has algebraic coordinates.2 PK = Tr(Pψ) is nontorsion ⇐⇒ L′(E/K, 1) 6= 0.
Marc Masdeu Rational points on elliptic curves September 5th, 2017 27 / 33
Example: E : y2 + xy + y = x3 + x2 − 10x− 10 (“15a1”)
fE(z) = q − q2 − q3 − q4 + q5 + q6 + 3q8 + q9 − q10 + · · ·fE is a modular form of level N = 15.Embed K = Q(
√13)→M2(Q) by sending
√13 7→
(−3 22 3
). It
identifies OK = Z[1+√
132 ] with the maximal order of M2(Q).
We get γ = ψ(ε2) = ψ(
11−3√
132
)=(
10 −3−3 1
).
The fixed point τ for γ is a root of x2 + 3x− 1 in Qp2 .
We can calculate Jψ very efficiently, obtaining (set β = 1+√
132 ∈ Qp2)
(3β+4)+(4β+1)52+(2β+2)5
3+(2β+4)5
4+(3β+2)5
5+(β+2)5
6+(2β+2)5
7+(β+4)5
8+(2β+4)5
9+· · · .
Pψ = ηTate(Jψ) ∈ E(Qp2) is((3β + 2) + 4β · 5 + 4β · 52 + 4β · 53 + 4β · 54 + · · · , (4β + 4) + 3 · 5 + 4 · 52 + 4 · 53 + 4 · 54 + · · ·
).
This is 5-adically close to the algebraic point of infinite order(1−√
13,−4 + 2√
13)∈ E(K).
Marc Masdeu Rational points on elliptic curves September 5th, 2017 28 / 33
Generalization
F a number field, K/F a quadratic extension.
n+ s = #v | ∞F : v splits in K = rkZO×K/O×F .
K/F is CM ⇐⇒ n+ s = 0.I If n+ s = 1 we call K/F quasi-CM.
S(E,K) =v | N∞F : v not split in K
.
Sign of functional equation for L(E/K, ·) should be (−1)#S(E,K).I From now on, we assume that this is odd.I #S(E,K) = 1 =⇒ split automorphic forms,I #S(E,K) > 1 =⇒ quaternionic automorphic forms.
Fix a place ν ∈ S(E,K).1 If ν = p is finite =⇒ non-archimedean construction.2 If ν is infinite =⇒ archimedean construction.
Marc Masdeu Rational points on elliptic curves September 5th, 2017 29 / 33
Particular cases being generalized
Non-archimedean
I H. Darmon (1999): F = Q, split.
I M. Trifkovic (2006): F = Q(√−d) ( =⇒ K/F quasi-CM), split.
I M. Greenberg (2008): F totally real, quaternionic.
Archimedean
I H. Darmon (2000): F totally real, split.
I J. Gartner (2010): F totally real, quaternionic.
Marc Masdeu Rational points on elliptic curves September 5th, 2017 30 / 33
Overview of the construction
Assume that F has narrow class number 1.I Removed in joint work with X. Guitart and S. Molina.
Find a quaternion algebra B and a v-arithmetic group Γ ⊂ B.Attach to E a cohomology class
ΦE ∈ Hn+s(Γ,Ω1
Hν).
Attach to each embedding ψ : K → B a homology class
Θψ ∈ Hn+s
(Γ,Div0 Hν
).
I Well defined up to the image of Hn+s+1(Γ,Z)δ→ Hn+s(Γ,Div0 Hν).
Cap-product and integration on the coefficients yield an element:
Jψ = ×∫
Θψ
ΦE ∈ K×ν .
Jψ is well-defined up to the lattice L =×∫δ(θ) ΦE : θ ∈ Hn+s+1(Γ,Z)
.
Marc Masdeu Rational points on elliptic curves September 5th, 2017 31 / 33
Conjectures
Conjecture 1 (Oda, Yoshida, Greenberg, Guitart-M-Sengun)There is an isogeny η : 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 = η(Jψ) ∈ E(Kν).
Conjecture 2 (Darmon, Greenberg, Trifkovic, Gartner, G-M-S)1 The local point Pψ is global, defined over E(Hψ).2 For all σ ∈ Gal(Hψ/K), σ(Pψ) = Prec(σ)·ψ (Shimura reciprocity).3 TrHψ/K(Pψ) is nontorsion if and only if L′(E/K, 1) 6= 0.
Marc Masdeu Rational points on elliptic curves September 5th, 2017 32 / 33
Moltes gracies, merces plan, muchas gracias,eskerrik asko, moitas grazas!
Non-archimedean
Archimedean
Ramification
Darmon Points
H∗ H∗
Modularity
E/FK/F quadratic
P?∈ E(Kab)
Marc Masdeu Rational points on elliptic curves September 5th, 2017 33 / 33