Hubble Asteroid Hunter III. Physical properties of newly found asteroids
Numerical experiments with plectic Stark-Heegner points
1. NUMERICAL EXPERIMENTS WITH PLECTIC
STARK–HEEGNER POINTS
LFANT SEMINAR
April 6, 2021
Marc Masdeu
Universitat Autònoma de Barcelona
2. The Hasse-Weil L-function
Let F be a number field.
Let E{F be an elliptic curve of conductor N “ NE.
Let K{F be a quadratic extension of F.
§ Assume for simplicity that N is square-free, coprime to discpK{Fq.
For each prime p of K, appEq “ 1 ` |p| ´ #EpFpq.
Hasse-Weil L-function of the base change of E to K (<psq ą
ą 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 conjecture ùñ
§ Analytic continuation of LpE{K, sq to C.
§ Functional equation relating s Ø 2 ´ s.
Marc Masdeu Plectic Stark–Heegner points 1 / 22
3. The BSD conjecture and Heegner points
Brian Birch Sir P. Swinnerton-Dyer Kurt Heegner
Coarse version of BSD conjecture
ords“1 LpE{K, sq “ rkZ EpKq.
Heegner Points
Only for F totally real and K{F totally complex (CM extension).
Simplest setting: F “ Q (and K{Q imaginary quadratic), and
` | N ùñ ` split in K.
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4. Heegner Points (K{Q imaginary quadratic)
Γ0pNq “ t
` a b
c d
˘
P SL2pZq: N | cu.
Attach to E a modular form:
fEpzq “
ÿ
ně1
ane2πinz
P S2pΓ0pNqq.
Given τ P K X H, set Jτ “
ż τ
8
2πifEpzqdz P C.
Well-defined up to the lattice
ΛE “
!ş
γ 2πifEpzqdz | γ P H1
´
Γ0pNqzH, Z
¯)
.
§ There exists an isogeny η: C{ΛE Ñ EpCq.
§ Set Pτ “ ηpJτ q P EpCq.
Fact: Pτ P EpHτ q, where Hτ {K is a class field attached to τ.
Theorem (Gross–Zagier)
PK “ TrHτ {KpPτ q nontorsion ðñ L1
pE{K, 1q ‰ 0.
Marc Masdeu Plectic Stark–Heegner points 3 / 22
5. Darmon points – history
n “ #tv | 8F : v splits in Ku.
SpE, Kq “
!
v | N8F : v not split in K
)
.
Sign of functional equation for LpE{K, sq should be p´1q#SpE,Kq.
Assume that s “ #SpE, Kq is odd.
Fix a finite place p P SpE, Kq.
§ There is also an archimedean version. . .
Darmon (’99): First construction, with F “ Q and s “ 1.
Trifkovic (’06): F imaginary quadratic, still s “ 1.
Greenberg (’08): F totally real, arbitrary ramification, and s ě 1.
Guitart–M.–Sengun (’14): F of arbitrary signature, arbitrary
ramification, and s ě 1.
Guitart–M.–Molina (’18): Adelic generalization, removing all
restrictions.
Marc Masdeu Plectic Stark–Heegner points 4 / 22
6. Review of Darmon points
Define a quaternion algebra B{F and a group Γ Ă SL2pFpq.
§ The group Γ acts (non-discretely) on Hp.
Attach to E a cohomology class
ΦE P Hn
`
Γ, Meas0
pP1
pFp, Zqq
˘
.
Attach to each embedding ψ: K ãÑ B a homology class
Θψ P Hn
`
Γ, Div0
Hp
˘
.
§ Well defined up to the image of Hn`1pΓ, Zq
δ
Ñ HnpΓ, Div0
Hpq.
§ Here δ is a connecting homomorphism arising from
0 // Div0
Hp
// Div Hp
deg
// Z // 0
Cap-product and integration on the coefficients yield an element:
Jψ “ xΦE, Θψy P Kˆ
p .
Jψ well-defined up to a multiplicative lattice L “ xΦE, δpHn`1pΓ, Zqqy.
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7. Conjectures on Darmon points
Jψ “ xΦE, Θψy P Kˆ
p {L.
Conjecture 1
There is an isogeny ηTate : Kˆ
p {L Ñ EpKpq.
Proven for totally-real fields (Greenberg, Rotger–Longo–Vigni,
Spiess, Gehrmann–Rosso).
The Darmon point attached to E and ψ: K Ñ B is:
Pψ “ ηTatepJψq P EpKpq.
Conjecture 2
1 The local point Pψ is global, and belongs to EpKabq.
2 Pψ is nontorsion if and only if L1pE{K, 1q ‰ 0.
Predicts also the exact number field over which Pψ is defined.
Includes a Shimura reciprocity law like that of Heegner points.
Marc Masdeu Plectic Stark–Heegner points 6 / 22
8. The tpu-arithmetic group Γ
B{F “ quaternion algebra with RampBq “ SpE, Kq r tpu.
Induces a factorization N “ pDm.
Set RB
0 ppmq Ă RB
0 pmq Ă B, Eichler orders of levels pm and m.
Define ΓB
0 ppmq “ RB
0 ppmqˆ
1 and ΓB
0 pmq “ RB
0 pmqˆ
1 .
Set
Γ “
`
RB
0 pmqrp´1
s
˘ˆ
1
.
Fix an embedding ιp : RB
0 pmq ãÑ M2pZpq.
Lemma
ιp induces bijections
Γ{ΓB
0 pmq – V0, Γ{ΓB
0 ppmq – E0
V0 (resp. E0) are the even vertices (resp. edges) of the BT tree.
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9. Integration on Hp
Let µ P Meas0
pP1pFpq, Zq.
Coleman integration on Hp “ P1pCpq r P1pFpq can be defined as:
ż τ2
τ1
ωµ “
ż
P1pFpq
logp
ˆ
t ´ τ2
t ´ τ1
˙
dµptq “ lim
Ý
Ñ
U
ÿ
UPU
logp
ˆ
tU ´ τ2
tU ´ τ1
˙
µpUq.
For Γ Ă PGL2pFpq, induce a pairing
Hi
pΓ, Meas0
pP1pFpq, Zqq ˆ HipΓ, Div0
Hpq
x¨,¨y
// Cp .
Bruhat-Tits tree of GL2pFpq, |p| “ 2.
Hp having the Bruhat-Tits as retract.
Can identify Meas0
pP1pFpq, Zq – HCpZq
“ tc : EpTpq Ñ Z |
ř
opeq“v cpeq “ 0u.
tU is any point in U Ă P1pFpq.
P1(Fp)
U ⊂ P1
(Fp) eU
Marc Masdeu Plectic Stark–Heegner points 8 / 22
10. Plectic conjectures
Jan Nekovář Tony Scholl
“ LprqpE{K, 1q should be related to
CM-points on a r-dimensional
quaternionic Shimura variety.
”
Goal : Construct Q P ^rpEpKqq such that
Q non-torsion ðñ Lprq
pE{K, 1q ‰ 0.
Marc Masdeu Plectic Stark–Heegner points 9 / 22
11. p-adic Plectic invariants
Michele Fornea
Let r ě 1 with same parity as #SpE, Kq.
S “ tp1, . . . , pru Ď SpE, Kq, |pi| “ p.
Let B{F with RampBq “ SpE, Kq r S.
Set ΓS “
`
RB
0 pmqrS´1s
˘ˆ
1
.
FS “
ś
pPS Fp, P1pFSq “
ś
pPS P1pFpq, and HS “
ś
pPS Hp.
Construct ΦE P Hn
pΓS, Meas0
pP1pFSq, Zqq.
§ µ
`
P1
pFpq ˆ USp
˘
“ 0, for all p P S, all USp Ď P1
pFSp q.
Construct Θψ P HnpΓS, Z0pHSqq.
Pairing Meas0
pP1pFSq, Zq ˆ Div0
pHSq Ñ
Â
pPS Kp.
§ Hn
pΓS, Meas0
pP1
pFSq, Zqq ˆ HnpΓS, Z0pHSqq
x¨,¨y
Ñ
Â
pPS Kp.
Plectic invariant attached to E, K and S
J :“ xΦE, Θψy P
Â
pPS Kp.
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12. Cohomology class
Consider ϕE P Hn
pΓB
0 ppSmq, Zq attached to E.
§ Via Eichler–Shimura and Jacquet–Langlands.
Shapiro isomorphism ; ϕ̃E P Hn
pΓS, coIndΓS
ΓB
0 ppSmq
Zq.
coIndΓS
ΓB
0 ppSmq
Z – MapspEpTSq, Zq.
HCSpZq “ tc: EpTSq Ñ Z “harmonic in each variable”u:
0 Ñ HCSpZq Ñ MapspEpTSq, Zq
ν
Ñ
à
pPS
MapspVpTpqˆEpTSp q, Zq Ñ ¨ ¨ ¨
Meas0
pP1pFSq, Zq identified with HCSpZq.
Since ϕE is p-new, have an isomorphism
Hn
pΓS, HCSpZqqE – Hn
pΓS, MapspEpTSq, ZqqE.
Therefore we can define ΦE, unique up to sign.
Marc Masdeu Plectic Stark–Heegner points 11 / 22
13. Homology class
Let ψ: O ãÑ RB
0 pmq be an embedding of an order O of K.
§ Which is optimal: ψpOq “ RB
0 pmq X ψpKq.
Consider the group Oˆ
1 “ tu P Oˆ : NmK{F puq “ 1u.
§ rankpOˆ
1 q “ rankpOˆ
q ´ rankpOˆ
F q “ n.
Choose a basis u1, . . . , un P Oˆ
1 for the non-torsion units.
∆ψ “ ψpu1q ^ ¨ ¨ ¨ ^ ψpunq P HnpΓ, Zq.
Kˆ
1 acts on HS through Kˆ
1
ψ
ãÑ Bˆ
1
À
pPS ιp
ãÑ SL2pFSq.
Let τp, τ̄p be the fixed points of Kˆ
1 acting on Hp.
§ Set D “
Â
pPSpτp ´ τ̄pq P Z0pHSq.
Define Θψ “ r∆ψ bDs P HnpΓS, Z0pHSqq.
Ideally, we’d like to define a class attached to
â
pPS
τp.
Marc Masdeu Plectic Stark–Heegner points 12 / 22
14. Conjectures
Granting BSD + parity conjectures, expect ralgpE{Kq ” r pmod 2q.
Fix embeddings ιp : K ãÑ Kp. Get a regulator map
det: ^r EpKq Ñ ÊpKSq, Q1 ^ ¨ ¨ ¨ ^ Qr ÞÑ detpιpi pQjqq.
Conjecture 1 (algebraicity)
Suppose that ralgpE{Kq ě r. Then:
Dw P ^rEpKq such that ηTatepJq “ detpwq.
ηTatepJq ‰ 0 ùñ ralgpE{Kq “ r.
Marc Masdeu Plectic Stark–Heegner points 13 / 22
15. Conjectures (II)
Write TpEq “ tp P S | appEq “ 1u.
Set ρpE, Sq “ ralgpE{Fq ` |TpEq|.
Bergunde–Gehrmann construct a p-adic L-function attached to
pE, K, Sq.
§ Interpolates central L-values of twists of by characters ramified at S.
§ Vanishes to order at least rpE, K, Sq “ maxtρpE, Sq, ρpEK
, Squ.
Fornea–Gehrmann show that L
prpE,K,Sqq
p
¨
“ J.
Assume that F “ QpjpEqq.
Conjecture 2 (non-vanishing)
If ralgpE{Kq “ r “ maxtρpE, Sq, ρpEK, Squ, then J ‰ 0.
If ralgpE{Kq ă r, then J ‰ 0 (but don’t know arithmetic meaning).
§ Provided that the order of vanishing of Lp allows for it.
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16. Numerical evidence
Joint work with Xevi Guitart and Michele Fornea.
We have restricted to F real quadratic of narrow class number one.
§ Therefore take r “ 2.
For β P F, define K “ Fp
?
βq.
Case 1
We first consider curves E{F where ralgpE{Fq “ 0.
Generically, ralgpE{Kq “ 0 as well.
Expect J to often be nonzero, unrelated to global points.
We have checked that this is the case in the following:
§ F “ Qp
?
13q, E “ 36.1-a2, β “ ´9w ` 8, ´12w ` 17.
§ F “ Qp
?
37q, E “ 36.1-a2 , β “ ´4w ` 9.
For the following two curves, we have observed J » 0 for many β.
§ F “ Qp
?
37q, E “36.1-b1.
§ F “ Qp
?
37q, E “36.1-c1.
Due to the fact that ap1 pEqap2 pEq “ ´1 ùñ extra vanishing of Lp.
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17. Numerical evidence. Case 2
We consider curves E{F where ralgpE{Fq “ 1.
We impose that ap1 pEqap2 pEq “ 1, so maxtρpE, Sq, ρpEK, Squ ą 2.
Generically, ralgpE{Kq “ 2.
In those cases, J should vanish because of an exceptional zero in
the p-adic L-function.
We have checked that this is the case (up to precision p6) in the
following:
§ F “ Qp
?
13q, E “ 225.1-b2, β “ ´3w ´ 1, ´12w ` 17.
§ F “ Qp
?
37q, E “ 63.1-a2, β “ ´4w ` 9.
§ F “ Qp
?
37q, E “ 63.1-b1, β “ ´4w ` 9.
§ F “ Qp
?
37q, E “ 63.2-a1, β “ ´3w ` 5.
§ F “ Qp
?
37q, E “ 63.2-b1, β “ ´3w ` 5.
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18. Numerical evidence. Case 3
We consider curves E{F where ralgpE{Fq “ 1.
We impose that ap1 pEqap2 pEq “ ´1, so maxtρpE, Sq, ρpEK, Squ “ 2.
Generically, ralgpE{Kq “ 2.
In those cases, J should be nonzero and related to global points.
We have checked that this is the case in the following:
§ F “ Qp
?
13q, E “ 153.2-e2, β “ ´9w ` 8.
§ F “ Qp
?
13q, E “ 207.1-c1, β “ ´9w ´ 4, ´9w ` 8.
§ F “ Qp
?
37q, E “ 63.1-d1, β “ ´4w ` 9.
§ F “ Qp
?
37q, E “ 63.2-d1, β “ ´3w ` 5
§ F “ Qp
?
37q, E “ 99.2-c1, β “ ´8w ` 17, ´16w ` 9, ´20w ` 29,
´9w ` 14, ´12w ` 29, ´32w ` 41, ´12w ´ 7, ´35w ` 17.
In one of the examples, we obtain what seems to be zero. We
expect that this is due to the low working precision. . .
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19. A pretty example
F “ Qp
?
13q, w “ 1`
?
13
2 ,
E{F : y2`xy`y “ x3`wx2`pw ` 1q x`2,
K “ Fp
?
βq, with β “ 62 ´ 21w.
EpKq bQ “ xP, Qy, with
P “ p3 ´ w, 4 ´ wq and Q “ p8 ´ 25
9 w, p´23
27 w ` 17
6 q
?
β ` 25
18w ´ 9
2q.
We may compute
logE1
pP1 ´ P̄1q blogE2
pQ2 ´ Q̄2q ´ logE1
pQ1 ´ Q̄1q blogE2
pP2 ´ P̄2q P Qp2 bQp2 .
Projecting Qp2 bQp2 Ñ Qp, get 2 ¨ 32 ` 36 ` 2 ¨ 37 ` 39 ` Op310q.
This matches our computation of J “ 2 ¨ 32 ` 36 ` Op37q.
1
https://www.lmfdb.org/EllipticCurve/2.2.37.1/63.2/d/1
Marc Masdeu Plectic Stark–Heegner points 18 / 22
20. Computation of the cohomology class
Assume, for concreteness, that r “ 2.
We start with ϕE P H1
pΓ0pp1p2q, Zq.
Shapiro isomorphism yields an isomorphism
H1
pΓ0pp1p2q, Zq – H1
pΓS, coInd Zq.
§ ; rϕ̃Es P H1
pΓS, coInd Zq.
§ The exact cocycle representative depends on a choice of coset
representatives for ΓS{Γ0pp1p2q.
Have a long-exact sequence
H1
pΓS, HCpZqq Ñ H1
pΓS, coInd Zq
ν
Ñ
à
pPS
H1
pΓS, MapspVpTpqˆEpTSp q, Zqq
ϕE is p-new ; rΦEs P H1
pΓS, HCpZqq lifting rϕ̃Es.
When r “ 1, one can choose appropriate coset representatives
(called radial), which ensure that ΦE “ ϕ̃E.
We don’t know whether there are coset representatives that allow for
that in our setting.
Marc Masdeu Plectic Stark–Heegner points 19 / 22
21. Lifting to H1
pΓS, HCpZqq
We know that Dφ : EpTSq Ñ Z such that ϕ̃E ´ Bφ P Z1pΓS, HCpZqq.
First, compute νpϕ̃Eq “ Bpf1, f2q,
f1 : VpTp1 q ˆ EpTp2 q Ñ Z, f2 : EpTp1 q ˆ VpTp2 q Ñ Z.
For each pv, eq P VpTp1 q ˆ EpTp2 q, pick γ P ΓS such that
γpv, eq “ pv0, e˚q, with v0 P tv˚, v̂˚u.
f1pv, eq ´ f1pv0, e˚q “ ν1pϕ̃Epγqqpv0, e˚q.
Analogously, f2pe, vq ´ f2pe˚, v0q “ ν2pϕ̃Epγqqpe˚, v0q.
Hence the four values f1pv˚, e˚q, f1pv̂˚, e˚q, f2pv˚, e˚q, f2pv̂˚, e˚q
determine all the remaining ones.
Knowing the functions f1 and f2 to some fixed radius allows to find φ
such that νpφq “ pf1, f2q, by solving a linear system of equations.
Marc Masdeu Plectic Stark–Heegner points 20 / 22
22. Linear algebra
To compute φ we need to solve a system of:
§ 2pp`1qppd
´1q
p´1
pd
`pd´1
´2
p´2 “ Opp2d´1
q equations, in
§
pp`1q2
ppd
´1q2
pp´1q2 “ Opp2d
q unknowns.
p “ 3, d “ 7: get 12, 740, 008 equations in 19, 114, 384 unknowns.
Luckily, it’s sparse: only p ` 1 unknowns involved in each equation.
We implemented a custom row reduction, avoiding division and
choosing pivots that maintain sparsity.
Takes „ 60 hours using 16 CPUs to compute f1 and f2.
Solve the system in „ 2 hours (non-parallel), using „ 300GB RAM.
Integration takes „ 10 hours using 64 CPUs.
Marc Masdeu Plectic Stark–Heegner points 21 / 22
23. Further work
So far we can compute invariants attached to differences τp ´ τ̄p.
§ Fornea–Gehrmann: refined invariants attached to τp, more akin to
Darmon points. Effective computation?
The Riemann sums algorithm runs in exponential time in the
precision.
§ Need an overconvergent method to compute the invariants in
polynomial time.
More experiments are needed in other settings (imaginary quadratic,
mixed signature).
To compute plectic Heegner points, need fundamental domains for
Bruhat–Tits trees acted on by groups attached to totally definite
quaternion algebras (work in progress).
Marc Masdeu Plectic Stark–Heegner points 22 / 22