Rational points on elliptic curves

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?)


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


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.


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??


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.


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!


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


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


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.


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!


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).


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).


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).


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


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.


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 . . ..


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. . .


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.


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.


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

∂


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


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.


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).


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.


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.


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)

.


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.


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)


Rational points on elliptic curves

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