Darmon points for fields of mixed signature

Darmon points for number ﬁelds of mixed signature
London Number Theory Seminar, UCL
February 4th, 2015
Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3
1Universitat de Barcelona
2University of Warwick
3Shefﬁeld University
Marc Masdeu Darmon points 0 / 24

Plan
1 Two Theorems
2 Two Conjectures
3 Two Effective Conjectures
4 Two (or three) Examples

1 Two Theorems
2 Two Conjectures

Two prototypical theorems (I)
Fix an integer N ě 1 (a level/conductor)
Let f P S2pΓ0pNqqnew, f “ q `
ř
ně2 anpfqqn, with anpfq P Z.
f ; ωf “ 2πifpzqdz P Ω1
X0pNq, where X0pNq “ Γ0pNqzH.
Consider the lattice Λf “
!ş
γ ωf
ˇ
ˇ
ˇ γ P H1pX0pNq, Zq
)
.
Theorem 1 (Eichler–Shimura–Manin)
Dη: C{Λf – EpCq, where E{Q is an elliptic curve, condpEq “ N, and
#EpFpq “ p ` 1 ´ appfq for all p N.

Two prototypical theorems (II)
Let K{Q be an imaginary quadratic field.
Suppose that | N ùñ split in K.
§ Heegner hypothesis, to force signpLpE{K, sqq “ ´1.
Given τ P K X H, set Pτ “ η
ˆż τ
i8
ωf
˙
P EpCq.
Theorem 2 (Shimura, Gross–Zagier)
1 Pτ P EpHτ q, where Hτ {K is a ring class field attached to τ.
2 TrpPτ q P EpKq nontorsion ðñ L1pE{K, 1q ‰ 0.
Generalizes to CM fields (F totally real and K{F totally complex).

1 Two Theorems
2 Two Conjectures

Quaternionic automorphic forms of level N
F a number ﬁeld of signature pr, sq.
v1, . . . , vr : F ãÑ R, w1, . . . , ws : F ãÑ C.
Consider a coprime factorization N “ Dn with D squarefree.
Let B{F be the quaternion algebra such that
RampBq “ tq: q | Du Y tvn`1, . . . , vru, pn ď rq.
Fix isomorphisms
B bFvi – M2pRq, i “ 1, . . . , n; B bFwj – M2pCq, j “ 1, . . . , s.
These yield Bˆ{Fˆ ãÑ PGL2pRqn ˆ PGL2pCqs ý Hn ˆ Hs
3.
R>0
C
H3
R>0
H
R
PGL2(R)
PGL2(C)

Quaternionic automorphic forms of level N (II)
Fix RB
0 pnq Ă B Eichler order of level n.
ΓB
0 pnq “ RB
0 pnqˆ{Oˆ
F acts discretely on Hn ˆ Hs
3.
Obtain a manifold of (real) dimension 2n ` 3s:
Y D
0 pnq “ ΓB
0 pnqz pHn
ˆ Hs
3q .
Y D
0 pnq is compact ðñ B is division.
The cohomology of Y D
0 pnq can be computed via
H˚
pY D
0 pnq, Cq – H˚
pΓB
0 pnq, Cq.
Hecke algebra TD “ ZrTq : q Ds acts on H˚pΓB
0 pnq, Zq.
Deﬁnition
f P Hn`spΓB
0 pnq, Cq eigen for TD is rational if appfq P Z, @p P TD.

Elliptic curves from cohomology classes
Conjecture 1 (Taylor, ICM 1994)
Let f P Hn`spΓB
0 pnq, Zq be a new, rational eigenclass.
Then DEf {F of conductor N “ Dn such that
#Ef pOF {pq “ 1 ` |p| ´ appfq @p N.
When F is totally imaginary, instead of Ef we may get an abelian
surface Af {F with QM deﬁned over F (a.k.a. fake elliptic curve).
In fact, fake elliptic curves can only arise when:
§ F is totally imaginary, and
§ N is square-full: p | N ùñ p2
| N.
Our construction automatically rules out this setting.
However: the conjecture above is not effective: it doesn’t give a
candidate Ef .

Algebraic points from quadratic extensions
Suppose E “ Ef is attached to f.
Let K{F be a quadratic extension of F.
§ Assume that N is square-free, coprime to discpK{Fq.
Hasse-Weil L-function of the base change of E to K ( psq ąą 0)
LpE{K, sq “
ź
p|N
`
1 ´ ap|p|´s
˘´1
ˆ
ź
p N
`
1 ´ ap|p|´s
` |p|1´2s
˘´1
.
Modularity of E ùñ
§ Analytic continuation of LpE{K, sq to C.
§ Functional equation relating s Ø 2 ´ s.
Conjecture 2 (very coarse version of BSD conjecture)
Suppose that ords“1 LpE{K, sq “ 1. Then DPK P EpKq of inﬁnite order.
Known results rely on existence of Heegner points.
Without Heegner points, don’t even have candidate for PK.

1 Two Theorems
2 Two Conjectures

Notation for Darmon points
F a number field, K{F a quadratic extension.
n ` s “ #tv | 8F : v splits in Ku “ rkZ Oˆ
K{Oˆ
F .
K{F is CM ðñ n ` s “ 0.
§ If n ` s “ 1 we call K{F quasi-CM.
SpE, Kq “
!
v | N8F : v not split in K
)
.
Sign of functional equation for LpE{K, ¨q should be p´1q#SpE,Kq.
§ From now on, we assume that this is odd.
§ #SpE, Kq “ 1 ùñ split automorphic forms,
§ #SpE, Kq ą 1 ùñ quaternionic automorphic forms.
Fix a place ν P SpE, Kq.
1 If ν “ p is finite ùñ non-archimedean construction.
2 If ν is infinite ùñ archimedean construction.

Previous constructions of Darmon points
Non-archimedean
§ H. Darmon (1999): F “ Q, split.
§ M. Trifkovic (2006): F “ Qp
?
´dq ( ùñ K{F quasi-CM), split.
§ M. Greenberg (2008): F totally real, quaternionic.
Archimedean
§ H. Darmon (2000): F totally real, split.
§ J. Gartner (2010): F totally real, quaternionic.
Generalizations
§ Rotger–Longo–Vigni: lift to Jacobians of Shimura curves.
§ Rotger–Seveso: cycles attached to higher weight modular forms.

Integration Pairing
Let Hν “ the ν-adic upper half plane. That is:
§ The Poincaré upper half-plane H if ν is infinite,
§ The p-adic (upper) half-plane Hp if ν “ p is finite.
As a set, we have (up to taking connected component):
HνpKνq “ P1
pKνq P1
pFνq.
Hν comes equipped with an analytic structure (complex- or rigid-).
When ν is infinite, there is a natural PGL2pRq-equivariant pairing
Ω1
H ˆ Div0
H Ñ C “ Kν,
which sends
pω, pτ2q ´ pτ2qq ÞÑ
ż τ2
τ1
ω P C.
Analogously, Coleman integration gives a natural pairing
Ω1
Hν
ˆ Div0
Hν Ñ Kν.

Rigid one-forms and measures
The assignment
µ ÞÑ ω “
ż
P1pFpq
dz
z ´ t
dµptq
induces an isomorphism Meas0pP1pFpq, Zq – Ω1
Hp,Z.
Inverse given by ω ÞÑ
“
U ÞÑ µpUq “ resApUq ω
‰
.
Theorem (Teitelbaum)
ż τ2
τ1
ω “
ż
P1pFpq
log
ˆ
t ´ τ1
t ´ τ2
˙
dµptq “ limÝÑ
U
ÿ
UPU
log
ˆ
tU ´ τ1
tU ´ τ2
˙
µpUq.
If the residues of ω are all integers, have a multiplicative reﬁnement:
ˆ
ż τ2
τ1
ω “ limÝÑ
U
ź
UPU
ˆ
tU ´ τ2
tU ´ τ1
˙µpUq
P Kˆ
p .

The group Γ
B{F “ quaternion algebra with RampBq “ SpE, Kq tνu.
Write N “ Dn (or N “ pDn if ν “ p).
Fix Eichler order RB
0 pnq Ă B, and a splitting ιν : RB
0 pnq ãÑ M2pFνq.
Let Γ “ RB
0 pnqr1{νsˆ{OF r1{νsˆ ιν
ãÑ PGL2pFνq.
Example (archimedean)
F real quadratic, SpE, Kq “ tνu.
This gives B “ M2pFq.
Γ “ Γ0pNq “
` a b
c d
˘
P GL2pOF q: c P N
(
{Oˆ
F Ă PGL2pOF q.
Example (non-archimedean)
F “ Q, N “ pM, SpE, Kq “ tpu.
This gives B “ M2pQq.
Γ “
` a b
c d
˘
P GL2pZr1{psq: M | c
(
{t˘1u ãÑ PGL2pQq Ă PGL2pQpq.

Overview of the construction
Need to assume that F has narrow class number 1.
We attach to E a cohomology class
ΦE P Hn`s
`
Γ, Ω1
Hν
˘
.
We attach to each embedding ψ: K ãÑ B a homology class
Θψ P Hn`s
`
Γ, Div0
Hν
˘
.
§ Well defined up to the image of Hn`s`1pΓ, Zq
δ
Ñ Hn`spΓ, Div0
Hνq.
Cap-product and integration on the coefficients yield an element:
Jψ “ ˆ
ż
Θψ
ΦE P Kˆ
ν .
Jψ is well-defined up to the lattice L “
!
ˆ
ş
δpθq ΦE : θ P Hn`s`1pΓ, Zq
)
.

Conjectures
Conjecture 1 (Oda, Yoshida, Greenberg, Guitart-M-Sengun)
There is an isogeny η: Kˆ
ν {L Ñ EpKνq.
Dasgupta–Greenberg, Rotger–Longo–Vigni: some non-arch. cases.
Completely open in the archimedean case.
The Darmon point attached to E and ψ: K Ñ B is:
Pψ “ ηpJψq P EpKνq.
Conjecture 2 (Darmon, Greenberg, Trifkovic, Gartner, G-M-S)
1 The local point Pψ is global, deﬁned over EpHψq.
2 For all σ P GalpHψ{Kq, σpPψq “ Precpσq¨ψ (Shimura reciprocity).
3 TrHψ{KpPψq is nontorsion if and only if L1pE{K, 1q ‰ 0.

Cohomology class attached to E
Recall that SpE, Kq and ν determine:
Γ “ ιν
`
RB
0 pnqr1{νsˆ
˘
Ă PGL2pFνq.
Theorem (Darmon, Greenberg, Trifkovic, Gartner, G.–M.–S.)
There exists a unique (up to sign) class
ΦE P Hn`s
`
Γ, Ω1
Hν
˘
such that:
1 TqΦE “ aqΦE for all q N.
2 UqΦE “ aqΦE for all q | N.
3 ΦE is “integrally valued”.
Idea: relate Hn`spΓ, Ω1
Hν
q with H˚pΓB
0 pNq, Cq (as Hecke modules).

Homology classes attached to orders O Ă K
Let ψ: O ãÑ RB
0 pnq be an embedding of an order O of K.
§ Which is optimal: ψpOq “ RB
0 pnq X ψpKq.
Consider the group Oˆ
1 “ tu P Oˆ : NmK{F puq “ 1u.
§ rankpOˆ
1 q “ rankpOˆ
q ´ rankpOˆ
F q “ n ` s.
Choose a basis u1, . . . , un`s P Oˆ
1 for the non-torsion units.
§ ; ∆ψ “ ψpu1q ¨ ¨ ¨ ψpun`sq P Hn`spΓ, Zq.
Kˆ{Fˆ ψ
ãÑ Bˆ{Fˆ ιν
ãÑ PGL2pFνq ý Hν.
§ Let τψ be the (unique) ﬁxed point of Kˆ
on Hν.
Hn`s`1pΓ, Zq
δ // Hn`spΓ, Div0
Hνq // Hn`spΓ, Div Hνq
deg
// Hn`spΓ, Zq
Θψ
? // r∆ψ bτψs // r∆ψs
Fact: r∆ψs is torsion.
§ Can pull back a multiple of r∆ψ bτψs to Θψ P Hn`spΓ, Div0
Hνq.
§ Well deﬁned up to δpHn`s`1pΓ, Zqq.

1 Two Theorems
2 Two Conjectures

Example curve
F “ Qprq with r4 ´ r3 ` 3r ´ 1 “ 0.
F has signature p2, 1q and discriminant ´1732.
N “ pr ´ 2q “ p “ p13.
B{F ramiﬁed only at all inﬁnite real places of F.
There is a rational eigenclass f P S2pΓB
0 pNqq.
From f we compute ωf P H1pΓ, Ω1
Hp
q and Λf “ qZ
f .
qf “ 8 ¨ 13 ` 11 ¨ 132 ` 5 ¨ 133 ` 3 ¨ 134 ` ¨ ¨ ¨ ` Op13100q.
jE “ 1
13
´
´ 4656377430074r3
` 10862248656760r2
´ 14109269950515r ` 4120837170980
¯
.
E{F : y2
`
`
r3
` r ` 3
˘
xy “ x3
`
`
`
´2r3
` r2
´ r ´ 5
˘
x2
`
`
´56218r3
´ 92126r2
´ 12149r ` 17192
˘
x
´ 23593411r3
` 5300811r2
` 36382184r ´ 12122562.

Non-archimedean cubic Darmon point
F “ Qprq, with r3 ´ r2 ´ r ` 2 “ 0.
F has signature p1, 1q and discriminant ´59.
Consider the elliptic curve E{F given by the equation:
E{F : y2
` p´r ´ 1q xy ` p´r ´ 1q y “ x3
´ rx2
` p´r ´ 1q x.
E has conductor NE “
`
r2 ` 2
˘
“ p17q2, where
p17 “
`
´r2
` 2r ` 1
˘
, q2 “ prq .
Consider K “ Fpαq, where α “
?
´3r2 ` 9r ´ 6.
The quaternion algebra B{F has discriminant D “ q2:
B “ Fxi, j, ky, i2
“ ´1, j2
“ r, ij “ ´ji “ k.

Non-archimedean cubic Darmon point (II)
The maximal order of K is generated by wK, a root of the polynomial
x2
` pr ` 1qx `
7r2 ´ r ` 10
16
.
One can embed OK in the Eichler order of level p17 by:
ψ: wK ÞÑ p´r2
` rqi ` p´r ` 2qj ` rk.
We obtain γψ “ 6r2´7
2 ` 2r`3
2 i ` 2r2`3r
2 j ` 5r2´7
2 k, and
τψ “ p12g`8q`p7g`13q17`p12g`10q172
`p2g`9q173
`p4g`2q174
`¨ ¨ ¨
This yields Θψ P H1pΓ, Div0
Hpq.
After integrating we get:
Jψ “ 16`9¨17`15¨172
`16¨173
`12¨174
`2¨175
`¨ ¨ ¨`5¨1720
`Op1721
q,
which corresponds to:
Pψ “ ´108 ˆ
ˆ
r ´ 1,
α ` r2 ` r
2
˙
P EpKq.

Archimedean cubic Darmon point
Let F “ Qprq with r3 ´ r2 ` 1 “ 0.
F signature p1, 1q and discriminant ´23.
Consider the elliptic curve E{F given by the equation:
E{F : y2
` pr ´ 1q xy `
`
r2
´ r
˘
y “ x3
`
`
´r2
´ 1
˘
x2
` r2
x.
E has prime conductor NE “
`
r2 ` 4
˘
of norm 89.
K “ Fpαq, with α2 ` pr ` 1q α ` 2r2 ´ 3r ` 3 “ 0.
§ K has class number 1, thus we expect the point to be deﬁned over K.
SpE, Kq “ tσu, where σ: F ãÑ R is the real embedding of F.
§ Therefore the quaternion algebra B is just M2pFq.
The arithmetic group to consider is
Γ “ Γ0pNEq Ă PGL2pOF q.
Γ acts naturally on the symmetric space H ˆ H3:
H ˆ H3 “ tpz, x, yq: z P H, x P C, y P Rą0u.

Archimedean cubic Darmon point (II)
E ; ωE, an automorphic form with Fourier-Bessel expansion:
ωEpz, x, yq “
ÿ
αPδ´1OF
α0ą0
apδαqpEqe´2πipα0 ¯z`α1x`α2 ¯xq
yH pα1yq ¨
ˆ
´dx^d¯z
dy^d¯z
d¯x^d¯z
˙
Hptq “
ˆ
´
i
2
eiθ
K1p4πρq, K0p4πρq,
i
2
eíθ
K1p4πρq
˙
t “ ρeiθ
.
§ K0 and K1 are hyperbolic Bessel functions of the second kind:
K0pxq “
ż 8
0
e´x coshptq
dt, K1pxq “
ż 8
0
e´x coshptq
coshptqdt.
ωE is a 2-form on Γz pH ˆ H3q.
The cocycle ΦE is defined as (γ P Γ):
ΦEpγq “
ż γÖ
O
ωEpz, x, yq P Ω1
H with O “ p0, 1q P H3.

Archimedean cubic Darmon point (III)
Consider the embedding ψ: K ãÑ M2pFq given by:
α ÞÑ
ˆ
´2r2 ` 3r r ´ 3
r2 ` 4 2r2 ´ 4r ´ 1
˙
Let γψ “ ψpuq, where u is a fundamental norm-one unit of OK.
γψ ﬁxes τψ “ ´0.7181328459824 ` 0.55312763561813i P H.
§ Construct Θ1
ψ “ rγψ bτψs P H1pΓ, Div Hq.
Θ1
ψ is equivalent to a cycle
ř
γi bpsi ´ riq taking values in Div0
H.
Jψ “
ÿ
i
ż si
ri
ΦEpγiq “
ÿ
i
ż γi¨O
O
ż si
ri
ωEpz, x, yq.
We obtain, summing over all ideals pαq of norm up to 400, 000:
Jψ “ 0.0005281284234 ` 0.0013607546066i ; Pψ P EpCq.
Numerically (up to 32 decimal digits) we obtain:
Pψ
?
“ ´10 ˆ
`
r ´ 1, α ´ r2
` 2r
˘
P EpKq.

Conclusion
For all quadratic K{F, proposed an analytic construction of
candidate PK P EpKνq whenever ords“1 LpE{K, sq “ 1.
Heegner points are a very special case of the construction.
The TruthTM covers much more than the CM paradigm.
Along the way, give the first systematic method to find equations
for elliptic curves using automorphic forms over non-totally-real
fields.
We have extensive numerical evidence.
§ However, nothing beyond quasi-CM!

What’s next
Equations for abelian surfaces of GL2-type (in progress).
Computing in H2 and H2, maybe using sharblies?
Higher class numbers (technical and computational difﬁculties).
Reductive groups other than GL2? (Please help!)

Thank you !
Bibliography, code and slides at:
http://www.warwick.ac.uk/mmasdeu/

Cohomology
Γ “ RB
0 pnqr1{psˆ
{OF r1{psˆ ιp
ãÑ PGL2pFpq.
MapspE0pT q, Zq – IndΓ
ΓD
0 ppmq
Z, MapspVpT q, Zq –
´
IndΓ
ΓD
0 pmq
Z
¯2
.
Consider the Γ-equivariant exact sequence
0 // HCpZq // MapspE0pT q, Zq
∆ // MapspVpT q, Zq // 0
ϕ // rv ÞÑ
ř
opeq“v ϕpeqs
So get:
0 Ñ HCpZq Ñ IndΓ
ΓD
0 ppmq
Z
∆
Ñ
´
IndΓ
ΓD
0 pmq
Z
¯2
Ñ 0

Cohomology (II)
0 Ñ HCpZq Ñ IndΓ
ΓD
0 ppmq
Z
∆
Ñ
´
IndΓ
ΓD
0 pmq
Z
¯2
Ñ 0
Taking Γ-cohomology,. . .
Hn`s
pΓ, HCpZqq Ñ Hn`s
pΓ, IndΓ
ΓD
0 ppmq
, Zq
∆
Ñ Hn`s
pΓ, IndΓ
ΓD
0 pmq
, Zq2
Ñ
. . . and using Shapiro’s lemma:
Hn`s
pΓ, HCpZqq Ñ Hn`s
pΓD
0 ppmq, Zq
∆
Ñ Hn`s
pΓD
0 pmq, Zq2
Ñ ¨ ¨ ¨
f P Hn`spΓD
0 ppmq, Zq being p-new ô f P Kerp∆q.
Pulling back get
Φf P Hn`s
pΓ, HCpZqq.

Recovering E from Λf
Λf “ xqf y gives us qf
?
“ qE.
Assume ordppqf q ą 0 (otherwise, replace qf ÞÑ 1{qf ).
Get
jpqf q “ q´1
f ` 744 ` 196884qf ` ¨ ¨ ¨ P Cˆ
p .
From N guess the discriminant ∆E.
§ Only ﬁnitely-many possibilities, ∆E P SpF, 12q.
jpqf q “ c3
4{∆E ; recover c4.
Recognize c4 algebraically.
1728∆E “ c3
4 ´ c2
6 ; recover c6.
Compute the conductor of Ef : Y 2 “ X3 ´ c4
48X ´ c6
864.
§ If conductor is correct, check aq’s.

Overconvergent Method
Starting data: cohomology class Φ P H1pΓ, Ω1
Hp
q.
Goal: to compute integrals
şτ2
τ1
Φγ, for γ P Γ.
Recall that ż τ2
τ1
Φγ “
ż
P1pFpq
logp
ˆ
t ´ τ1
t ´ τ2
˙
dµγptq.
Expand the integrand into power series and change variables.
§ We are reduced to calculating the moments:
ż
Zp
ti
dµγptq for all γ P Γ.
Note: Γ Ě ΓD
0 pmq Ě ΓD
0 ppmq.
Technical lemma: All these integrals can be recovered from
#ż
Zp
ti
dµγptq: γ P ΓD
0 ppmq
+
.

Overconvergent Method (II)
D “ tlocally analytic Zp-valued distributions on Zpu.
§ ϕ P D maps a locally-analytic function h on Zp to ϕphq P Zp.
§ D is naturally a ΓD
0 ppmq-module.
The map ϕ ÞÑ ϕp1Zp q induces a projection:
H1pΓD
0 ppmq, Dq
ρ
// H1pΓD
0 ppmq, Zpq.
P
f
Theorem (Pollack-Stevens, Pollack-Pollack)
There exists a unique Up-eigenclass ˜Φ lifting Φ.
Moreover, ˜Φ is explicitly computable by iterating the Up-operator.

Overconvergent Method (III)
But we wanted to compute the moments of a system of measures. . .
Proposition
Consider the map Ψ: ΓD
0 ppmq Ñ D:
γ ÞÑ
”
hptq ÞÑ
ż
Zp
hptqdµγptq
ı
.
1 Ψ belongs to H1
´
ΓD
0 ppmq, D
¯
.
2 Ψ is a lift of f.
3 Ψ is a Up-eigenclass.
Corollary
The explicitly computed ˜Φ “ Ψ knows the above integrals.

Darmon points for fields of mixed signature

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