ICERM Workshop on "Modular Forms and Curves of Low Genus: Computational Aspects"

1. Non-archimedean construction of
elliptic curves and abelian surfaces
ICERM WORKSHOP
Modular Forms and Curves of Low Genus:
Computational Aspects
Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3
1Universitat de Barcelona
2University of Warwick
3University of Sheffield
September 28th, 2015
Marc Masdeu Non-archimedean constructions September 28th
, 2015 0 / 34

2. Modular Forms and Curves of Low Genus:
Computational Aspects
Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3
1Universitat de Barcelona
2University of Warwick
3University of Sheffield
September 28th, 2015
Marc Masdeu Non-archimedean constructions September 28th
, 2015 1 / 34

3. Quaternionic automorphic forms of level N
F a number field of signature pr, sq, and fix N Ă OF .
Choose factorization N “ Dn, with D square free.
Fix embeddings v1, . . . , vr : F ãÑ R, w1, . . . , ws : F ãÑ C.
Let B{F be a 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)
Marc Masdeu Non-archimedean constructions September 28th
, 2015 2 / 34

4. Quaternionic automorphic forms of level N (II)
Fix RD
0 pnq Ă B Eichler order of level n.
ΓD
0 pnq “ RD
0 pnqˆ{Oˆ
F acts discretely on Hn ˆ Hs
3.
Obtain an orbifold of (real) dimension 2n ` 3s:
Y D
0 pnq “ ΓD
0 pnqz pHn
ˆ Hs
3q .
The cohomology of Y D
0 pnq can be computed via
H˚
pY D
0 pnq, Cq – H˚
pΓD
0 pnq, Cq.
Hecke algebra TD “ ZrTq : q Ds acts on H˚pΓD
0 pnq, Zq.
Hn`s
pΓD
0 pnq, Cq “
à
χ
Hn`s
pΓN
0 pnq, Cqχ
, χ: TD
Ñ C.
Each χ cuts out a field Kχ, s.t. rKχ : Qs “ dim Hn`spΓD
0 pnq, Cqχ.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 3 / 34

5. Abelian varieties from cohomology classes
Definition
f P Hn`spΓD
0 pnq, Cqχ eigen for TD is rational if appfq P Z, @p P TD.
If r “ 0, then assume N is not square-full: Dp N.
Conjecture (Taylor, ICM 1994)
1 f P Hn`spΓD
0 pnq, Zq a new, rational eigenclass.
Then DEf {F of conductor N “ Dn attached to f. i.e. such that
#Ef pOF {pq “ 1 ` |p| ´ appfq @p N.
2 More generally, if χ: TD Ñ C is nontrivial, cutting out a field K, then
D abelian variety Aχ, with dim Aχ “ rK : Fs and multiplication by K.
Assumption above avoids “fake abelian varieties”, and it is needed in
our construction anyway.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 4 / 34

6. Goals of this talk
In this talk we will:
1 Review known explicit forms of this conjecture.
§ Cremona’s algorithm for F “ Q.
§ Generalizations to totally real fields.
2 Propose a new, non-archimedean, conjectural construction.
§ (joint work with X. Guitart and H. Sengun)
3 Explain some computational details.
4 Illustrate with examples.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 5 / 34

7. F “ Q: Cremona’s algorithm for elliptic curves
Eichler–Shimura construction
X0pNq // JacpX0pNqq
ş
–
H0
pX0pNq,Ω1
q
_
H1pX0pNq,Zq
Hecke// // C{Λf – Ef pCq.
1 Compute H1pX0pNq, Zq (modular symbols).
2 Find the period lattice Λf by explicitly integrating
Λf “
Cż
γ
2πi
ÿ
ně1
anpfqe2πinz
: γ P H1
´
X0pNq, Z
¯
G
.
3 Compute c4pΛf q, c6pΛf q P C by evaluating Eistenstein series.
4 Recognize c4pΛf q, c6pΛf q as integers ; Ef : Y 2 “ X3 ´ c4
48X ´ c6
864.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 6 / 34

8. F ‰ Q: constructions for elliptic curves
F totally real. rF : Qs “ n, fix σ: F ãÑ R.
S2pΓ0pNqq Q f ; ˜ωf P Hn
pΓ0pNq, Cq ; Λf Ď C.
Conjecture (Oda, Darmon, Gartner)
C{Λf is isogenous to Ef ˆF Fσ.
Known to hold (when F real quadratic) for base-change of E{Q.
Exploited in very restricted cases (Demb´el´e, Stein+7).
Explicitly computing Λf is hard.
§ No quaternionic computations (except for Voight–Willis?).
F not totally real: no known algorithms. . .
Theorem
If F is imaginary quadratic, the lattice Λf is contained in R.
Idea
Allow for non-archimedean constructions.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 7 / 34

9. Non-archimedean construction
From now on: fix p N.
Denote by ¯Fp “ alg. closure of the p-completion of F.
Theorem (Tate uniformization)
There exists a rigid-analytic, Galois-equivariant isomorphism
η: ¯Fˆ
p {xqEy Ñ Ep ¯Fpq,
with qE P Fˆ
p satisfying jpEq “ q´1
E ` 744 ` 196884qE ` ¨ ¨ ¨ .
Choose a coprime factorization N “ pDm, with D “ discpB{Fq.
Compute qE as a replacement for Λf .
Starting data: f P Hn`spΓD
0 pmq, Zqp´new, pDm “ N.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 8 / 34

10. Non-archimedean path integrals on Hp
Consider Hp “ P1pCpq P1pFpq.
It is a p-adic analogue to H:
§ It has a rigid-analytic structure.
§ Action of PGL2pFpq by fractional linear transformations.
§ Rigid-analytic 1-forms ω P Ω1
Hp
.
§ Coleman integration ; make sense of
şτ2
τ1
ω P Cp.
Get a PGL2pFpq-equivariant pairing
ş
: Ω1
Hp
ˆ Div0
Hp Ñ Cp.
For each Γ Ă PGL2pFpq, get induced pairing (cap product)
HipΓ, Ω1
Hp
q ˆ HipΓ, Div0
Hpq
ş
// Cp
´
φ,
ř
γ γ bDγ
¯
//
ř
γ
ż
Dγ
φpγq.
Ω1
Hp
– space of Cp-valued boundary measures Meas0pP1pFpq, Cpq.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 9 / 34

11. Measures and integrals
Bruhat-Tits tree of GL2pFpq, |p| “ 2.
P1pFpq – EndspT q.
Harmonic cocycles HCpAq “
tEpT q
f
Ñ A |
ř
opeq“v fpeq “ 0u
Meas0pP1pFpq, Aq – HCpAq.
So replace ω P Ω1
Hp
with
µω P Meas0pP1pFpq, Zq – HCpZq.
P1
(Fp)
U ⊂ P1
(Fp)
µ(U)
v∗
ˆv∗
e∗
T
Coleman integration: if τ1, τ2 P Hp, then
ż τ2
τ1
ω “
ż
P1pFpq
logp
ˆ
t ´ τ2
t ´ τ1
˙
dµωptq “ limÝÑ
U
ÿ
UPU
logp
ˆ
tU ´ τ2
tU ´ τ1
˙
µωpUq.
Multiplicative refinement (assume µωpUq P Z, @U):
ˆ
ż τ2
τ1
ω “ ˆ
ż
P1pFpq
ˆ
t ´ τ2
t ´ τ1
˙
dµωptq “ limÝÑ
U
ź
UPU
ˆ
tU ´ τ2
tU ´ τ1
˙µωpUq
.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 10 / 34

12. The tpu-arithmetic group Γ
Choose a factorization N “ pDm.
B{F “ quaternion algebra with RampBq “ tq | Du Y tvn`1, . . . , vru.
Recall also RD
0 ppmq Ă RD
0 pmq Ă B.
Fix ιp : RD
0 pmq ãÑ M2pZpq.
Define ΓD
0 ppmq “ RD
0 ppmqˆ{Oˆ
F and ΓD
0 pmq “ RD
0 pmqˆ{Oˆ
F .
Let Γ “ RD
0 pmqr1{psˆ{OF r1{psˆ ιp
ãÑ PGL2pFpq.
Example
F “ Q and D “ 1, so N “ pM.
B “ M2pQq.
Γ0ppMq “
` a b
c d
˘
P GL2pZq: pM | c
(
{t˘1u.
Γ “
` a b
c d
˘
P GL2pZr1{psq: M | c
(
{t˘1u ãÑ PGL2pQq Ă PGL2pQpq.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 11 / 34

13. The tpu-arithmetic group Γ
Lemma
Assume that h`
F “ 1. Then ιp induces bijections
Γ{ΓD
0 pmq – V0pT q, Γ{ΓD
0 ppmq – E0pT q
V0 “ V0pT q (resp. E0 “ E0pT q) are the even vertices (resp. edges) of T .
Proof.
1 Strong approximation ùñ Γ acts transitively on E0 and V0.
2 Stabilizer of vertex v˚ (resp. edge e˚) is ΓD
0 pmq (resp. ΓD
0 ppmq).
Corollary
MapspE0pT q, Zq – IndΓ
ΓD
0 ppmq
Z, MapspVpT q, Zq –
´
IndΓ
ΓD
0 pmq
Z
¯2
.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 12 / 34

14. Cohomology
Γ “ RD
0 pmqr1{psˆ
{OF r1{psˆ ιp
ãÑ PGL2pFpq.
MapspE0pT q, Zq – IndΓ
ΓD
0 ppmq
Z, MapspVpT q, Zq –
´
IndΓ
ΓD
0 pmq
Z
¯2
.
Want to define a cohomology class in Hn`spΓ, Ω1
Hp
q.
Consider the Γ-equivariant exact sequence
0 // HCpZq // MapspE0pT q, Zq
β
// MapspVpT q, Zq // 0
ϕ // rv ÞÑ
ř
opeq“v ϕpeqs
So get:
0 Ñ HCpZq Ñ IndΓ
ΓD
0 ppmq
Z
β
Ñ
´
IndΓ
ΓD
0 pmq
Z
¯2
Ñ 0
Marc Masdeu Non-archimedean constructions September 28th
, 2015 13 / 34

15. Cohomology (II)
0 Ñ HCpZq Ñ IndΓ
ΓD
0 ppmq
Z
β
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
β
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
β
Ñ Hn`s
pΓD
0 pmq, Zq2
Ñ ¨ ¨ ¨
f P Hn`spΓD
0 ppmq, Zq being p-new ô f P Kerpβq.
Pulling back get
ωf P Hn`s
pΓ, HCpZqq – Hn`s
pΓ, Ω1
Hp
q.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 14 / 34

16. Holomogy
Consider the Γ-equivariant short exact sequence:
0 Ñ Div0
Hp Ñ Div Hp
deg
Ñ Z Ñ 0.
Taking Γ-homology yields
Hn`s`1pΓ, Zq
δ
Ñ Hn`spΓ, Div0
Hpq Ñ Hn`spΓ, Div Hpq Ñ Hn`spΓ, Zq
Λf “
#
ˆ
ż
δpcq
ωf : c P Hn`s`1pΓ, Zq
+
Ă Cˆ
p
Conjecture A (Greenberg, Guitart–M.–Sengun)
The multiplicative lattice Λf is homothetic to qZ
E.
F “ Q: Darmon, Dasgupta–Greenberg, Longo–Rotger–Vigni.
F totally real, |p| “ 1, B “ M2pFq: Spiess.
Open in general.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 15 / 34

17. Lattice: explicit construction
Start with f P Hn`spΓD
0 ppm, Zqqp´new.
Duality yields ˆf P Hn`spΓD
0 ppmq, Zqqp´new.
Mayer–Vietoris exact sequence for Γ “ ΓD
0 pmq ‹ΓD
0 ppmq
{ΓD
0 pmq:
¨ ¨ ¨ Ñ Hn`s`1pΓ, Zq
δ1
Ñ Hn`spΓD
0 ppmq, Zq
β
Ñ Hn`spΓD
0 pmq, Zq2
Ñ ¨ ¨ ¨
ˆf new at p ùñ βp ˆfq “ 0.
§ ˆf “ δ1
pcf q, for some cf P Hn`s`1pΓ, Zq.
Conjecture (rephrased)
The element
Lf “
ż
δpcf q
ωf .
satisfies (up to a rational multiple) logppqEq “ Lf .
Marc Masdeu Non-archimedean constructions September 28th
, 2015 16 / 34

18. Algorithms
Only in the cases n ` s ď 1.
§ Both H1
and H1: fox calculus (linear algebra for finitely-presented
groups).
Use explicit presentation + word problem for ΓD
0 ppmq and ΓD
0 pmq.
§ John Voight (s “ 0).
§ Aurel Page (s “ 1).
Need the Hecke action on H1pΓD
0 ppmq, Zq and H1pΓD
0 ppmq, Zq.
§ Shapiro’s lemma ùñ enough to work with ΓD
0 pmq.
Integration pairing uses the overconvergent method.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 17 / 34

19. Overconvergent Method
Starting data: cohomology class φ “ ωf 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
+
.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 18 / 34

20. 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
Φ
//
P
φ
Theorem (Pollack-Stevens, Pollack-Pollack)
There exists a unique Up-eigenclass Φ lifting φ.
Moreover, Φ is explicitly computable by iterating the Up-operator.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 19 / 34

21. Overconvergent Method (III)
But we wanted to compute the moments of a system of measures. . .
Proposition
Consider the map ΓD
0 ppmq Ñ D:
γ ÞÑ
”
hptq ÞÑ
ż
Zp
hptqdµγptq
ı
.
1 It satisfies a cocycle relation ùñ induces a class
Ψ P H1
´
ΓD
0 ppmq, D
¯
.
2 Ψ is a lift of φ.
3 Ψ is a Up-eigenclass.
Corollary
The explicitly computed Φ “ Ψ knows the above integrals.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 20 / 34

22. 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.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 21 / 34

23. Example curve (joint with X. Guitart and H. Sengun)
F “ Qpαq, pαpxq “ x4 ´ x3 ` 3x ´ 1, ∆F “ ´1732.
N “ pα ´ 2q “ p13.
B{F ramified only at all infinite real places of F.
There is a rational eigenclass f P S2pΓ0p1, Nqq.
From f we compute ωf P H1pΓ, HCpZqq and Λf .
qf
?
“ qE “ 8 ¨ 13 ` 11 ¨ 132 ` 5 ¨ 133 ` 3 ¨ 134 ` ¨ ¨ ¨ ` Op13100q.
jE “ 1
13
´
´ 4656377430074α3
` 10862248656760α2
´ 14109269950515α ` 4120837170980
¯
.
c4 “ 2698473α3 ` 4422064α2 ` 583165α ´ 825127.
c6 “ 20442856268α3 ´ 4537434352α2 ´ 31471481744α ` 10479346607.
E{F : y2
`
`
α3
` α ` 3
˘
xy “ x3
`
`
`
´2α3
` α2
´ α ´ 5
˘
x2
`
`
´56218α3
´ 92126α2
´ 12149α ` 17192
˘
x
´ 23593411α3
` 5300811α2
` 36382184α ´ 12122562.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 22 / 34

24. Tables: Imaginary quadratic fields
|∆K | fK pxq NmpNq pDm c4pEq, c6pEq
3 r1, ´1s 196 p3r ´ 2q7p´6r ` 2q28p1q ´131065r,
47449331
3 r1, ´1s 196 p´3r ` 1q7p6r ´ 4q28p1q ´131065r,
47449331
4 r1, 0s 130 p3r ´ 2q13p´r ´ 3q10p1q ´264r ` 257,
´6580r ` 2583
4 r1, 0s 130 p´3r ´ 2q13p´3r ´ 1q10p1q 264r ` 257,
6580r ` 2583
7 r2, ´1s 44 prq2p3r ` 1q22p1q 648r ` 481,
´28836r ` 4447
7 r2, ´1s 44 pr ´ 1q2p3r ´ 4q22p1q ´648r ` 1129,
28836r ´ 24389
8 r2, 0s 99 pr ` 1q3p´4r ` 1q33p1q 444r ` 25,
14794r ´ 16263
8 r2, 0s 99 pr ´ 1q3p´4r ´ 1q33p1q ´444r ` 25,
´14794r ´ 16263
8 r2, 0s 99 p´r ´ 3q11p3q9p1q ´444r ` 25,
´14794r ´ 16263
8 r2, 0s 99 pr ´ 3q11p3q9p1q 444r ` 25,
14794r ´ 16263
Marc Masdeu Non-archimedean constructions September 28th
, 2015 23 / 34

25. Tables: cubic p1, 1q fields
|∆K | fK pxq NmpNq pDm c4 pEq, c6 pEq
23 r1, 0, ´1s 185 pr2
` 1q5 p3r2
´ r ` 1q37 p1q 643318r2
´ 1128871r ` 852306,
925824936r2
´ 1624710823r ` 1226456111
31 r´1, 1, 0s 129 p´r ´ 1q3 p´3r2
´ 2r ´ 1q43 p1q ´4787r2
` 10585r ` 3349,
1268769r2
´ 371369r ` 424764
44 r1, 1, ´1s 121 p2r ´ 1q11 pr2
` 2q11 p1q 4097022r2
´ 6265306r ` 7487000,
14168359144r2
´ 21861492432r `
26039140708
44 r1, 1, ´1s 121 p2r ´ 1q11 pr2
` 2q11 p1q 1774r2
´ 1434r ´ 1304,
´42728r2
´ 123104r ´ 54300
44 r1, 1, ´1s 121 p2r ´ 1q11 pr2
` 2q11 p1q 4097022r2
´ 6265306r ` 7487000,
14168359144r2
´ 21861492432r `
26039140708
59 r´1, 2, 0s 34 p´r2
´ 1q2 p´r2
´ 2r ´ 2q17 p1q 262r2
` 513r ` 264,
´2592r2
` 448r ` 13231
59 r´1, 2, 0s 34 p´r2
´ 2r ´ 2q17 p´r2
´ 1q2 p1q 16393r2
` 20228r ´ 12524,
4430388r2
´ 5579252r ` 1619039
59 r´1, 2, 0s 46 p´2r2
` r ´ 2q23 p´r2
´ 1q2 p1q 18969r2
` 8532r ` 41788,
4216716r2
` 1911600r ` 9298151
59 r´1, 2, 0s 74 p´r2
´ 1q2 p2r2
` 2r ` 1q37 p1q 33054r2
` 15049r ` 72776,
9702640r2
` 4400116r ` 21401723
59 r´1, 2, 0s 88 p´r2
´ 1q2 pr ´ 2q11 pr2
` r ` 1q4 16609r2
` 7084r ` 37332,
3522136r2
` 1613876r ` 7760395
59 r´1, 2, 0s 187 p2r2
` r ` 2q17 pr ´ 2q11 p1q ´32r2
´ 848r ` 432,
´7600r2
` 23368r ´ 8704
76 r´2, ´2, 0s 117 p2r2
´ r ´ 3q13 p´r2
` 2r ` 1q9 p1q 48r ` 16,
´128r2
´ 224r ´ 216
83 r´2, 1, ´1s 65 pr ` 1q5 p´2r ` 1q13 p1q 3089r2
` 1086r ` 4561,
333604r2
` 117840r ` 493059
83 r´2, 1, ´1s 65 pr ` 1q5 p´2r ` 1q13 p1q 304r2
` 112r ` 449,
6616r2
` 2328r ` 9791
83 r´2, 1, ´1s 65 p´2r ` 1q13 pr ` 1q5 p1q 3089r2
` 1086r ` 4561,
333604r2
` 117840r ` 493059
83 r´2, 1, ´1s 65 p´2r ` 1q13 pr ` 1q5 p1q 4499473r2
` 1589254r ` 6650137,
18573712184r2
` 6560420272r ` 27451337687
83 r´2, 1, ´1s 106 prq2 p2r2
´ 3r ` 3q53 p1q 2329r2
` 822r ` 3441,
´34264r2
´ 12104r ´ 50645
87 r1, 2, ´1s 123 pr2
´ r ` 1q3 pr2
` 4q41 p1q 1424r2
` 3792r ` 2384,
´245696r2
´ 201800r ´ 4144
87 r1, 2, ´1s 129 pr2
´ r ` 1q3 p´3r ` 1q43 p1q ´752r2
` 2272r ` 1009,
27496r2
´ 152144r ´ 63977
87 r1, 2, ´1s 129 pr2
´ r ` 1q3 p´3r ` 1q43 p1q ´752r2
` 2272r ` 1009,
27496r2
´ 152144r ´ 63977
104 r´2, ´1, 0s 143 pr2
` r ´ 1q11 p2r ` 1q13 p1q 12r2
` 12r ` 25,
´144r2
´ 90r ´ 125
107 r´2, 3, ´1s 40 p´r2
´ 1q5 pr2
´ r ` 3q4 prq2 3880r2
´ 984r ` 10473,
405820r2
´ 105348r ` 1142075
107 r´2, 3, ´1s 135 p´r2
´ 1q5 p3q27 p1q 16r2
´ 16r,
184r ´ 296
108 r´2, 0, 0s 34 p2r ` 1q17 prq2 p1q 184r2
` 212r ` 265,
´5010r2
´ 6306r ´ 7773
108 r´2, 0, 0s 85 p´r2
´ 1q5 p2r ` 1q17 p1q 2224r2
` 2816r ` 3520,
´229056r2
´ 288672r ´ 363768
108 r´2, 0, 0s 85 p2r ` 1q17 p´r2
´ 1q5 p1q 2224r2
` 2816r ` 3520,
´229056r2
´ 288672r ´ 363768
108 r´2, 0, 0s 125 p´r2
´ 1q5 pr2
´ 2r ´ 1q25 p1q 496r2
,
22088
108 r´2, 0, 0s 145 p´r2
´ 1q5 pr ` 3q29 p1q 144r2
` 176r ` 240,
3816r2
` 4752r ` 6088
108 r´2, 0, 0s 155 p´r2
´ 1q5 pr2
` 3q31 p1q 16r2
` 20r ` 17,
´606r2
´ 762r ´ 929
116 r´2, 0, ´1s 34 p´2r ` 1q17 p´r ` 1q2 p1q 846760r2
` 589024r ` 998761,
1781332252r2
` 1239131712r ` 2101097467
116 r´2, 0, ´1s 34 p´r ` 1q2 p´2r ` 1q17 p1q 4592r2
` 3192r ` 5417,
274400r2
` 190876r ` 323659
116 r´2, 0, ´1s 38 p´r ` 1q2 p2r ` 1q19 p1q 82921r2
` 57626r ` 97746,
54599355r2
` 37980374r ` 64400978
116 r´2, 0, ´1s 38 p2r ` 1q19 p´r ` 1q2 p1q 1081r2
` 746r ` 1266,
66555r2
` 46310r ` 78482
116 r´2, 0, ´1s 58 p´r ` 1q2 pr2
` r ´ 3q29 p1q 22024r2
` 15320r ` 25977,
´4956678r2
´ 3447968r ´ 5846447
135 r´1, 3, 0s 55 pr2
´ r ` 2q11 pr2
` 1q5 p1q 4139r2
´ 19599r ` 5885,
2077971r2
´ 1764501r ` 352796
135 r´1, 3, 0s 88 pr2
´ r ` 2q11 p2q8 p1q ´1751r2
´ 1226r ` 577,
´131901r2
´ 120528r ` 52524
139 r2, 1, ´1s 46 pr ´ 3q23 p´rq2 p1q 22560r2
` 19560r ` 1033,
´8413992r2
` 2336724r ` 7421723
139 r2, 1, ´1s 57 pr ´ 1q3 p´2r ` 1q19 p1q 18r2
` 61r ` 39,
296r ` 239
139 r2, 1, ´1s 57 p´2r ` 1q19 pr ´ 1q3 p1q 258r2
` 541r ` 279,
´17136r2
´ 9280r ` 3767
140 r´2, 2, 0s 25 pr2
` 1q5 pr ` 1q5 p1q 1488r2
` 992r ` 3968,
110440r2
` 88352r ` 287144
140 r´2, 2, 0s 70 pr2
` r ` 1q7 pr ` 1q5 prq2 139012r2
` 106502r ` 360441,
´100613641r2
´ 77548384r ´ 260995189
140 r´2, 2, 0s 95 pr2
` 1q5 pr2
` 2r ` 3q19 p1q 16r2
` 16r,
´64r2
` 240r ´ 120
140 r´2, 2, 0s 95 pr2
` 1q5 pr2
` 2r ` 3q19 p1q 64r2
´ 64r ` 48,
´824r2
´ 368r ` 616
172 r3, ´1, ´1s 45 pr ´ 2q5 pr2
´ r ´ 1q9 p1q ´1072r2
´ 80r ` 1872,
´49976r2
´ 48864r ` 25920
175 r´3, 2, ´1s 27 prq3 pr2
´ r ` 2q9 p1q ´384r2
` 816r ´ 416,
5904r2
´ 32472r ` 31816
199 r´1, 4, ´1s 21 p´r2
` r ´ 2q7 pr2
´ r ` 3q3 p1q 98529r2
` 22348r ´ 12672,
´41881233r2
` 130193546r ´ 31313977
199 r´1, 4, ´1s 21 p´r2
` r ´ 2q7 pr2
´ r ` 3q3 p1q ´112647r2
´ 62978r ` 24321,
´60304454r2
´ 96556295r ` 29529884
199 r´1, 4, ´1s 33 pr ´ 2q11 pr2
´ r ` 3q3 p1q 2802r2
` 3055r ´ 996,
´398780r2
` 635911r ´ 139543
199 r´1, 4, ´1s 49 p´r2
` r ´ 2q7 p´r2
´ 3q7 p1q 6447r2
´ 31223r ` 7758,
3699375r2
´ 3171676r ` 577928
199 r´1, 4, ´1s 49 p´r2
´ 3q7 p´r2
` r ´ 2q7 p1q 6447r2
´ 31223r ` 7758,
3699375r2
´ 3171676r ` 577928
199 r´1, 4, ´1s 77 pr ` 1q7 pr ´ 2q11 p1q 12952r2
´ 10791r ` 49899,
2866751r2
´ 2163173r ` 10872899
199 r´1, 4, ´1s 99 pr ´ 2q11 pr2
` 1q9 p1q ´120r2
` 576r ´ 143,
380r2
` 4776r ´ 1281
200 r2, 2, ´1s 14 pr ` 1q2 pr2
´ r ` 1q7 p1q ´401r2
´ 3756r ´ 2274,
182521r2
´ 243668r ´ 235802
200 r2, 2, ´1s 14 pr2
´ r ` 1q7 pr ` 1q2 p1q ´241r2
` 404r ` 366,
5649r2
` 3068r ´ 394
200 r2, 2, ´1s 65 p´r2
´ r ´ 1q13 p´r2
` r ´ 3q5 p1q ´1176r2
´ 1944r ´ 767,
75636r2
´ 142236r ´ 124561
204 r´3, 1, ´1s 21 pr2
` r ` 1q7 prq3 p1q ´48r2
` 96r ´ 32,
´288r2
` 1008r ´ 872
204 r´3, 1, ´1s 21 pr2
` r ` 1q7 prq3 p1q 262r2
´ 326r ´ 44,
´1784r2
´ 5128r ` 11612
211 r´3, ´2, 0s 21 pr ` 2q7 p´rq3 p1q 22010896r2
` 41672992r ` 34877233,
296072400488r2
` 560550677168r `
469139740087
212 r´2, 4, ´1s 35 pr2
´ r ` 1q7 pr2
´ r ` 3q5 p1q 29888r2
´ 13952r ` 112113,
10054302r2
´ 4693580r ` 37714701
216 r´2, 3, 0s 34 prq2 pr2
` r ` 5q17 p1q 307r2
` 194r ` 1057,
´11235r2
´ 6786r ´ 37821
216 r´2, 3, 0s 34 pr2
` r ` 5q17 prq2 p1q 307r2
` 194r ` 1057,
´11235r2
´ 6786r ´ 37821
|∆K | fK pxq NmpNq pDm c4 pEq, c6 pEq
216 r´2, 3, 0s 38 p´2r2
´ 2r ´ 7q19 prq2 p1q 16r2
` 81,
´216r2
´ 192r ´ 601
231 r3, 0, ´1s 33 p´r ` 1q3 pr2
´ r ` 2q11 p1q 465r2
´ 1011r ` 1189,
25273r2
´ 54957r ` 64546
231 r3, 0, ´1s 33 pr2
´ r ` 2q11 p´r ` 1q3 p1q 465r2
´ 1011r ` 1189,
25273r2
´ 54957r ` 64546
231 r3, 0, ´1s 51 pr2
` 1q17 prq3 p1q ´47r2
´ 50r ` 145,
938r2
´ 291r ` 4
239 r´3, ´1, 0s 24 pr ` 1q3 p2q8 p1q 9r2
` 18r ` 25,
143r2
` 236r ` 268
239 r´3, ´1, 0s 57 p´r2
´ r ` 1q19 pr ` 1q3 p1q 1170r2
` 1953r ` 2098,
108233r2
` 180929r ` 194227
243 r´3, 0, 0s 10 pr ´ 2q5 pr ´ 1q2 p1q 27576r2
` 39771r ` 57360,
11428272r2
` 16482420r ` 23771763
243 r´3, 0, 0s 10 pr ´ 1q2 pr ´ 2q5 p1q 27576r2
` 39771r ` 57360,
11428272r2
` 16482420r ` 23771763
243 r´3, 0, 0s 22 pr ` 2q11 pr ´ 1q2 p1q 2002130917752r2
` 2887572455827r `
4164600133648,
7076846143946804016r2
`
10206578310238918020r `
14720433182250993839
243 r´3, 0, 0s 34 pr2
` 2q17 pr ´ 1q2 p1q 10167352r2
` 14663859r ` 21148944,
´80986535280r2
´ 116802795708r ´
168458781921
243 r´3, 0, 0s 46 p´2r ` 1q23 pr ´ 1q2 p1q 19946163r2
` 28767345r ` 41489691,
222892996797r2
` 321467328855r `
463636116909
255 r´3, 0, ´1s 15 p´r2
´ 1q5 pr ´ 1q3 p1q 248r2
´ 320r ´ 263,
´2556r2
´ 4104r ` 16523
255 r´3, 0, ´1s 15 p´r2
´ 1q5 prq3 p1q 19r2
´ r ` 88,
279r2
` 908
255 r´3, 0, ´1s 51 pr2
´ 2q17 pr ´ 1q3 p1q ´32r2
` 240r ´ 336,
´3416r2
` 2400r ` 7392
255 r´3, 0, ´1s 51 pr2
´ 2q17 pr ´ 1q3 p1q 80r2
´ 80r ´ 128,
288r2
´ 2088r ` 2888
255 r´3, 0, ´1s 65 pr2
´ r ` 1q13 pr ` 1q5 p1q 3r2
´ 105r ` 88,
´909r2
` 1116r ´ 1576
268 r5, ´3, ´1s 14 pr2
´ 2q7 pr ´ 1q2 p1q ´285113701784r2
´ 52062773310r `
950706811227,
144006413291532359r2
`
50857254178772568r ´ 433038348784793416
300 r´3, ´3, ´1s 9 p´r2
` 2r ` 2q3 prq3 p1q 26r2
` 46r ` 4,
504r2
` 504r ` 460
300 r´3, ´3, ´1s 9 p´r2
` 2r ` 2q3 prq3 p1q 26r2
` 46r ` 4,
504r2
` 504r ` 460
300 r´3, ´3, ´1s 33 pr2
´ r ´ 1q11 prq3 p1q 11072r2
` 17760r ` 12865,
4675808r2
` 7475664r ` 5398495
300 r´3, ´3, ´1s 90 p´r2
` 2r ` 2q3 p´r ´ 3q30 p1q ´71,
´1837
307 r2, 3, ´1s 10 pr ´ 1q5 p´rq2 p1q ´1450479r2
´ 118958r ` 338681,
´1778021804r2
´ 7601175244r ´ 3506038549
307 r2, 3, ´1s 45 pr ´ 1q5 pr2
´ 2r ` 5q9 p1q r2
` 154r ` 81,
´1744r2
´ 1756r ´ 441
324 r´4, ´3, 0s 4 pr ´ 2q2 p´r ´ 1q2 p1q 345255874728r2
` 758120909880r `
628931968401,
686899433218582980r2
`
1508309811434747772r `
1251283596457392135
324 r´4, ´3, 0s 22 pr2
´ r ´ 1q11 pr ´ 2q2 p1q 808464801r2
` 1775245884r ` 1472731953,
77832295537635r2
` 170905971571164r `
141782435639127
324 r´4, ´3, 0s 84 pr2
` 3r ` 3q7 p´r2
´ 3r ´ 2q12 p1q 143742984r2
` 315634200r ` 261847993,
4700399015844r2
` 10321245891900r `
8562435635987
327 r´3, ´2, ´1s 9 prq3 pr ` 1q3 p1q 13r2
` 22r ` 25,
144r2
` 225r ` 242
327 r´3, ´2, ´1s 15 p´r ` 1q5 prq3 p1q 1645r2
´ 2647r ´ 2984,
55543r2
´ 6268r ´ 298328
327 r´3, ´2, ´1s 15 prq3 p´r ` 1q5 p1q 1645r2
´ 2647r ´ 2984,
55543r2
´ 6268r ´ 298328
335 r1, 4, ´1s 25 pr2
´ r ` 3q5 p´r ` 1q5 p1q ´951r2
` 1190r ` 57,
61922r2
´ 78025r ´ 346
335 r1, 4, ´1s 25 pr2
´ r ` 3q5 p´r ` 1q5 p1q 10r2
´ 11r ` 10,
52r2
´ 271r ´ 29
335 r1, 4, ´1s 65 p´r2
` 2r ´ 4q13 p´r ` 1q5 p1q 61r2
´ 77r ` 247,
101r2
´ 107r ` 380
351 r´3, 3, 0s 33 pr ´ 2q11 prq3 p1q 16r2
` 144r ´ 128,
1824r2
´ 72r ´ 1160
356 r7, 1, ´1s 14 p´r ´ 2q7 p 1
2
r2
´ r ` 3
2
q2 p1q 1577904r2
` 58258032r ` 83210433,
157810225239r2
` 783843846012r `
817040026548
356 r7, 1, ´1s 26 p´r ` 2q13 p´ 1
2
r2
` r ´ 5
2
q2 p1q ´353192r2
´ 495936r ` 44233,
´560380445r2
´ 897785708r ´ 94909392
356 r7, 1, ´1s 26 p´r ` 2q13 p´ 1
2
r2
` r ´ 5
2
q2 p1q 88412r2
` 1393648r ` 1878333,
112777386r2
` 1758482408r ` 2367346473
356 r7, 1, ´1s 196 prq7 pr ´ 3q28 p1q 4182384r2
´ 3886864r ´ 15048991,
´37671142504r2
´ 30349104360r `
38274580847
364 r´2, 4, 0s 21 pr ´ 1q3 p´r ´ 1q7 p1q ´368r2
´ 3712r ` 1840,
´72736r2
` 343360r ´ 146264
364 r´2, 4, 0s 26 p´r2
´ 1q13 p´rq2 p1q ´266582r2
` 148350r ´ 10479,
´274275343r2
` 306719520r ´ 83736937
379 r´4, 1, ´1s 6 pr ´ 1q3 p´r ` 2q2 p1q 1418236432r2
` 1053691808r ` 3254778265,
137488390576232r2
` 102148264969648r `
315528648990403
379 r´4, 1, ´1s 6 p´r ` 2q2 pr ´ 1q3 p1q 1418236432r2
` 1053691808r ` 3254778265,
137488390576232r2
` 102148264969648r `
315528648990403
379 r´4, 1, ´1s 21 pr ´ 1q3 pr ` 1q7 p1q 15373338r2
` 11421763r ` 35281005,
´155147444344r2
´ 115268221468r ´
356055251669
379 r´4, 1, ´1s 21 pr ` 1q7 pr ´ 1q3 p1q 15373338r2
` 11421763r ` 35281005,
´155147444344r2
´ 115268221468r ´
356055251669
379 r´4, 1, ´1s 27 pr ´ 1q3 pr2
` 1q9 p1q 1532208r2
` 1138368r ` 3516337,
1280550616r2
` 951396864r ` 2938796535
379 r´4, 1, ´1s 34 pr ´ 3q17 p´r ` 2q2 p1q 90342993r2
` 67121158r ` 207332433,
2363568298948r2
` 1756034817652r `
5424265343699
439 r5, ´2, ´1s 15 p´r ` 1q3 pr ´ 2q5 p1q ´439r2
` 1212r ´ 1252,
27743r2
´ 76494r ` 78935
439 r5, ´2, ´1s 15 pr ´ 2q5 p´r ` 1q3 p1q ´439r2
` 1212r ´ 1252,
27743r2
´ 76494r ` 78935
440 r´8, 2, 0s 10 p´r2
´ 2r ´ 5q5 p´ 1
2
r2
´ r ´ 2q2 p1q ´349392832r2
´ 1512227664r ` 3500497481,
´12893566003280r2
´ 143880769408104r `
276285496852283
440 r´8, 2, 0s 10 p´ 1
2
r2
´ r ´ 2q2 p´r2
´ 2r ´ 5q5 p1q ´349392832r2
´ 1512227664r ` 3500497481,
´12893566003280r2
´ 143880769408104r `
276285496852283
440 r´8, 2, 0s 26 p2r ´ 3q13 p´ 1
2
r2
´ r ´ 2q2 p1q 953
2
r2
´ 6046r ` 8769,
´ 419561
2
r2
` 835646r ´ 810505
451 r8, ´5, ´1s 26 p2r ´ 3q13 p´r ` 2q2 p1q 34296r2
` 4776r ´ 189951,
5707476r2
` 13155804r ´ 1647297
459 r´8, 3, 0s 22 p 1
2
r2
´ 1
2
r ` 1q11 p´ 1
2
r2
´ 1
2
r ´ 2q2 p1q 16r2
´ 104r ` 121,
´240r2
` 1260r ´ 1357
459 r´8, 3, 0s 33 p´ 1
2
r2
´ 1
2
r ` 1q11 p 1
2
r2
` 1
2
r ` 3q3 p1q ´ 19
2
r2
` 15
2
r ` 21,
´36r2
` 96r ´ 37
459 r´8, 3, 0s 33 pr2
` r ` 5q11 p 1
2
r2
` 1
2
r ` 3q3 p1q 178829
2
r2
` 270521
2
r ` 472861,
´83966694r2
´ 127020222r ´ 444049333
459 r´8, 3, 0s 34 p 1
2
r2
` 3
2
r ´ 3q17 p´ 1
2
r2
´ 1
2
r ´ 2q2 p1q 125
2
r2
` 79
2
r ´ 59,
282r2
´ 2430r ` 2691
459 r´8, 3, 0s 44 p 1
2
r2
´ 1
2
r ` 1q11 pr ´ 1q4 p1q 31
2
r2
´ 105
2
r ` 44,
411r2
´ 1452r ` 1256
459 r´8, 3, 0s 44 p 1
2
r2
´ 1
2
r ` 1q11 pr ´ 1q4 p1q 103
2
r2
´ 55
2
r ´ 60,
237r2
` 1374r ´ 2984
460 r´3, 5, ´1s 6 p´rq3 pr ´ 1q2 p1q 38808r2
` 63978r ´ 55637,
28650959r2
` 29220772r ´ 29738968
460 r´3, 5, ´1s 25 p2r2
´ r ` 10q5 p´r2
´ 4q5 p1q 36772r2
´ 83396r ` 37921,
32322356r2
´ 98725758r ` 49331449
460 r´3, 5, ´1s 26 pr2
´ r ` 1q13 pr ´ 1q2 p1q 973808r2
´ 7106166r ` 4086627,
´8777739333r2
` 7426503436r ´ 1197128148
515 r´4, ´1, ´1s 14 p´r ` 2q2 pr2
´ 2r ´ 1q7 p1q ´7341361r2
´ 9117211r ´ 13098483,
´14436506787r2
´ 17928648161r ´
25757667905
519 r7, ´4, ´1s 39 p´r2
` 3q13 p´r ` 2q3 p1q ´280r2
´ 960r ´ 751,
54220r2
` 11272r ´ 242353
547 r´4, ´3, ´1s 14 p´r ` 1q7 pr2
´ 2r ´ 2q2 p1q 14509048r2
` 24346088r ` 21671521,
200457117220r2
` 336365736396r `
299413898447
652 r5, 7, ´1s 14 p´ 1
2
r2
` r ` 1
2
q7 p´ 1
2
r2
` r ´ 9
2
q2 p1q 18r2
´ 36r ` 147,
405r2
´ 648r ` 3294
687 r3, 4, ´1s 9 prq3 pr ` 1q3 p1q ´7r2
` 38r ` 25,
´18r2
´ 423r ´ 244
743 r´3, 5, 0s 9 p´r ` 1q3 prq3 p1q 736r2
` 416r ` 3913,
´110256r2
´ 62192r ´ 586373
755 r2, 5, ´1s 10 p´2r2
` 3r ´ 11q5 p´rq2 p1q ´1634r2
` 10769r ` 4135,
110372r2
` 1174880r ` 412903
815 r´9, ´7, 0s 9 pr ` 1q3 p´r ` 3q3 p1q 26678105835217r2
` 83793885354406r `
76443429630973,
717286463675094140331r2
`
2252941797094015980448r `
2055312234304678362824
1196 r´7, 5, ´1s 14 p´rq7 p´r ` 1q2 p1q ´12r2
´ 4r ` 25,
´4r2
´ 134r ` 181
Marc Masdeu Non-archimedean constructions September 28th
, 2015 24 / 34

26. Tables: quartic p2, 1q fields (I)
|∆K | fK pxq NmpNq pDm c4pEq, c6pEq
643 r1, ´2, 0, ´1s 175 pr3
´ r2
´ r ´ 1q7p2r3
´ r2
´ 2q25p1q ´1783r3
` 1032r2
` 522r ` 3831,
116369r3
´ 62909r2
´ 30125r ´ 248439
688 r´1, ´2, 0, 0s 11 p´r3
` r2
` r ` 2q11p1qp1q 200r3
` 284r2
` 376r ` 136,
´5184r3
´ 7280r2
´ 10024r ´ 3672
688 r´1, ´2, 0, 0s 19 p2r3
´ 3q19p1qp1q 552r3
` 764r2
` 1064r ` 392,
´11536r3
´ 16160r2
´ 22584r ´ 8312
731 r´1, 0, 2, ´1s 80 pr2
` 1q5p1qp2q16 ´848r3
` 1529r2
` 456r ´ 420,
45471r3
´ 164824r2
` 11648r ` 72230
775 r´1, ´3, 0, ´1s 176 p´r3
` r2
` 1q11p2q16p1q ´ 6277
2
r3
´ 2939r2
´ 5696r ´ 3239
2
,
´528578r3
´ 495324r2
´ 959488r ´ 272875
976 r´1, 0, 3, ´2s 44 pr ´ 2q11p1qpr3
´ r2
` r ` 2q4 ´42r3
´ 21r2
` 20r ` 10,
´10860r3
´ 12344r2
` 6618r ` 4899
976 r´1, 0, 3, ´2s 65 pr3
´ 2r2
` 4rq13p1qpr3
´ 2r2
` 3r ` 1q5 72r3
` 20r2
´ 40r ´ 4,
´1456r3
` 3800r2
´ 176r ´ 1200
1107 r´1, ´2, 0, ´1s 99 pr ´ 1q3p´2r ` 1q33p1q 105488r3
` 90125r2
` 152590r ` 66821,
120373437r3
` 96189249r2
` 171765105r `
67816591
1156 r1, ´1, ´2, ´1s 19 pr3
´ r2
´ 2r ´ 3q19p1qp1q ´816481030r3
´882631565r2
´203810962r`
392346684,
´68032828897760r3
´ 73544780430596r2
´
16982427384164r ` 32692074898043
1156 r1, ´1, ´2, ´1s 19 pr ` 2q19p1qp1q ´384131503r3
´ 415253582r2
´ 95887519r `
184588047,
82379129020040r3
` 89053403394404r2
`
20563566138596r ´ 39585957243581
1192 r´1, 1, 2, ´1s 38 pr2
` 2q19p1qpr3
´ r2
` 2rq2 9504r3
` 11111r2
´ 4762r ´ 5690,
´2387028r3
` 7298060r2
` 2454128r ´
3005365
1255 r´1, ´3, ´1, 0s 170 pr3
´ r ´ 2q2p´2r3
` 2r2
` 3q85p1q 517916r3
` 904037r2
` 1060116r ` 296716,
´1433064139r3
´ 2501458160r2
´
2933309166r ´ 820990264
1423 r´1, ´2, 1, ´1s 98 pr ´ 1q2p2r3
´ r2
` 2r ´ 2q49p1q 39690531r3
` 20246442r2
` 70104884r `
26465314,
702653466524r3
` 356968363314r2
`
1240909503739r ` 466012978440
1423 r´1, ´2, 1, ´1s 98 pr3
´ r2
` 2r ´ 1q7pr3
´ 2r2
` 2r ´ 1q14p1q 54577r3
` 27699r2
` 96525r ` 36260,
1735232r3
` 881975r2
` 3066920r ` 1151600
1588 r2, 0, ´3, ´1s 56 p´r3
` r2
` 3r ` 1q7pr3
´ r2
´ 3rq8p1q 94560r3
` 111816r2
´ 39672r ´ 86639,
747493992r3
` 883740564r2
´ 313920684r ´
685060489
1588 r2, 0, ´3, ´1s 152 pr3
´ 3r ´ 1q19pr3
´ r2
´ 3rq8p1q 3496200r3
` 4469800r2
´ 803168r ´ 2816543,
26973722420r3
` 32247663708r2
´
10621228512r ´ 24308855297
1600 r´4, 0, ´2, 0s 11 p 1
2
r2
´ r ´ 1q11p1qp1q 12r3
` 48r2
` 56r ` 20,
´284r3
´ 460r2
´ 472r ´ 1024
1600 r´4, 0, ´2, 0s 11 p 1
2
r2
` r ´ 1q11p1qp1q 276r3
` 490r2
` 336r ` 628,
´18172r3
´ 32652r2
´ 22424r ´ 40464
1600 r´4, 0, ´2, 0s 19 p 1
2
r3
´ 1
2
r2
´ r ´ 1q19p1qp1q ´44r3
` 112r2
´ 56r ` 148,
´1660r3
` 2572r2
´ 2056r ` 3136
1600 r´4, 0, ´2, 0s 19 p´ 1
2
r3
´ 1
2
r2
` r ´ 1q19p1qp1q 44r3
` 112r2
` 56r ` 148,
1660r3
` 2572r2
` 2056r ` 3136
1732 r´1, 3, 0, ´1s 13 pr ´ 2q13p1qp1q 3455801r3
` 1359008r2
´ 3314187r ` 836393,
7590438778r3
´ 14215787438r2
´
23508658710r ` 9402560739
1732 r´1, 3, 0, ´1s 182 pr3
´ r ` 3q7pr2
´ r ´ 2q26p1q ´17184648r3
´ 14365296r2
` 9302744r ´
813151,
93038140030r3
´ 219828160822r2
´
331159079722r ` 135298016971
1823 r´2, 3, 0, ´1s 114 p´r3
` r ´ 3q3pr3
` r2
` 2q38p1q 233810r3
´ 9696r2
´ 336273r ` 159951,
´70457084r3
´ 403468159r2
´ 171041003r `
342434077
1879 r1, ´3, ´2, ´1s 140 p 1
2
r3
´ 2r ´ 1
2
q7pr3
´ r2
´ r ´ 2q20p1q ´2436r3
´ 3240r2
´ 2688r ` 1045,
´49029102r3
´ 65262564r2
´ 54075240r `
21032621
Marc Masdeu Non-archimedean constructions September 28th
, 2015 25 / 34

27. Tables: quartic p2, 1q fields (II)
|∆K | fK pxq NmpNq pDm c4 pEq, c6 pEq
2051 r1, 3, ´1, ´1s 15 pr3
´ r2
` 2q5 p1qp´r ` 1q3 ´489r3
` 1228r2
´ 1242r ` 18,
46792r3
´ 100917r2
` 73440r ` 47160
2068 r1, 3, ´2, ´1s 7 pr ´ 2q7 p1qp1q 26909497r3
` 20141314r2
´ 35624307r ´
11296953,
247303058576r3
´ 3168333376r2
´
656295560992r ´ 182979737393
2068 r1, 3, ´2, ´1s 13 p´r3
` r2
` 2r ´ 1q13 p1qp1q 34500648r3
` 3814392r2
´ 84122424r ´
23737447,
´77408488074r3
´ 354426093238r2
´
415474468618r ´ 92161502469
2068 r1, 3, ´2, ´1s 56 pr ´ 2q7 pr3
´ 2r ` 1q8 p1q ´3576591826r3
´ 1882130113r2
`
6123537074r ` 1835712204,
321001991693952r3
` 322520099276304r2
´
281263304453488r ´ 100176319060369
2068 r1, 3, ´2, ´1s 182 pr ´ 2q7 p´r3
` r2
´ 2q26 p1q ´1994707423r3
´ 282234694r2
`
4755878517r ` 1346474783,
´8733155599162r3
´ 54136988565986r2
´
71594660083402r ´ 16347374680241
2092 r´2, ´3, 1, ´1s 8 prq2 p1qpr ´ 1q4 ´3r3
` 29r2
´ r ` 75,
231r3
´ 61r2
` 497r ´ 287
2096 r2, ´2, ´2, 0s 28 pr3
´ r ´ 1q7 p1qpr3
` r2
´ 2q4 116r3
´ 390r2
` 402r ´ 94,
7354r3
` 222r2
´ 29620r ` 17640
2116 r´2, 0, 1, ´1s 5 pr2
` 1q5 p1qp1q 129712r3
` 31248r2
` 168480r ` 209073,
´109612390r3
´ 26402860r2
´ 142375012r ´
176669575
2116 r´2, 0, 1, ´1s 130 pr2
` 1q5 pr ` 2q26 p1q 105064r3
` 25312r2
` 136464r ` 169353,
78278092r3
` 18855232r2
` 101675032r `
126166043
2116 r´2, 0, 1, ´1s 130 pr3
´ r2
` r ` 1q13 pr3
` rq10 p1q 105064r3
` 25312r2
` 136464r ` 169353,
78278092r3
` 18855232r2
` 101675032r `
126166043
2183 r´1, 1, 3, ´2s 126 p´r3
` 2r2
´ 4rq7 pr3
´ 2r2
` 4r ` 1q18 p1q ´330539r3
´ 223654r2
` 72664r ` 52816,
421344240r3
` 649688112r2
´ 51218957r ´
170790474
2191 r´1, 0, 3, ´1s 70 p´r3
´ 2r ´ 2q5 p´2r3
` r2
´ 5r ´ 2q14 p1q ´928r3
` 6929r2
´ 312r ´ 2120,
´885775r3
` 1164640r2
` 179150r ´ 336602
2191 r´1, 0, 3, ´1s 80 p´r3
´ 2r ´ 2q5 p2q16 p1q ´408r3
` 2689r2
´ 105r ´ 821,
120899r3
` 70135r2
´ 44492r ´ 24989
2243 r´1, ´3, ´1, ´1s 75 pr ´ 1q5 p´r3
` 2r2
´ r ` 2q15 p1q 586900359r3
` 694528587r2
` 929522310r `
268803085,
63399246832324r3
` 75025661482408r2
`
100410590521972r ` 29037147633615
2243 r´1, ´3, ´1, ´1s 75 pr3
´ r2
´ 2r ´ 2q5 p´r3
` r2
` 2r ` 3q15 p1q 586900359r3
` 694528587r2
` 929522310r `
268803085,
63399246832324r3
` 75025661482408r2
`
100410590521972r ` 29037147633615
2243 r´1, ´3, ´1, ´1s 105 p´r3
` 2r2
` 2q7 p´r3
` 2r2
´ r ` 2q15 p1q 4336158r3
`5131353r2
`6867535r `1985981,
´22914354769r3
´ 27116483373r2
´
36291344215r ´ 10494880213
2243 r´1, ´3, ´1, ´1s 105 pr ´ 1q5 pr2
´ r ` 1q21 p1q 920025r3
` 1088737r2
` 1457115r ` 421377,
3942374598r3
` 4665343442r2
`
6243862193r ` 1805625754
2284 r´4, 2, 2, ´2s 22 p´r2
` r ` 1q11 p1qp 1
2
r3
´ r2
` 1q2 ´4322076r3
` 3371584r2
´ 4531104r ´
14171719,
´293858698818r3
` 229234508344r2
´
308070583688r ´ 963537590781
2327 r´2, ´1, ´1, 0s 48 pr2
´ 1q3 p2q16 p1q 60947675662300r3
` 95467421346487r2
`
88590894936957r ` 77819621400035,
1595218950381053851625r3
`
2498724287457442364789r2
`
2318740987988175420378r `
2036818157516553727423
2327 r´2, ´1, ´1, 0s 66 pr2
´ 1q3 p´r3
` 2q22 p1q 24654r3
` 41044r2
` 36631r ` 33971,
13602419r3
` 21481224r2
` 19830770r `
17549287
2327 r´2, ´1, ´1, 0s 78 pr2
´ 1q3 pr3
´ r2
` rq26 p1q 1632339r3
`2556895r2
`2372706r `2084241,
5442997756r3
` 8525820467r2
`
7911705090r ` 6949764691
2443 r´1, ´3, 0, 0s 63 pr3
´ 2q3 p´r ´ 2q21 p1q 51601r3
` 81980r2
` 123695r ` 33695,
´38870055r3
´ 59926714r2
´ 92404714r ´
25325630
2443 r´1, ´3, 0, 0s 117 p´r3
` r2
´ r ` 2q13 p´r2
` 1q9 p1q ´26624r3
´ 78583r2
` 147974r ` 56321,
41156101r3
´ 906363r2
´ 80062921r ´
24803969
2480 r´2, ´2, 0, 0s 17 p´r3
` r2
` r ` 1q17 p1qp1q 8r3
´ 12r2
´ 12r ` 17,
212r3
` 628r2
´ 818r ´ 887
2480 r´2, ´2, 0, 0s 19 p´r2
` r ´ 1q19 p1qp1q ´648r3
` 524r2
´ 408r ` 1636,
29224r3
´ 23272r2
` 18616r ´ 73216
2608 r´2, ´2, ´2, 0s 50 p´r ` 1q5 pr3
´ r2
´ rq10 p1q ´18122952r3
` 23309952r2
` 6270652r `
28184369,
´178706675384r3
` 229835084602r2
`
61821736238r ` 277904169213
2696 r1, ´3, 0, ´1s 24 pr3
´ 2q3 pr3
´ 3q8 p1q 25999152r3
` 20125515r2
` 35704342r ´
14654974,
´282591287516r3
´ 218749239468r2
´
388079405968r ` 159288610195
2816 r´1, ´4, ´2, 0s 15 pr2
´ r ´ 1q5 p1qpr3
´ r2
´ 2r ´ 1q3 134184108r3
´ 165313100r2
´ 203588440r ´
41893502,
2470282983044r3
´ 3964870336170r2
´
2128766125800r ´ 223343175430
2859 r´3, 3, ´1, ´1s 7 p´r3
` r ´ 1q7 p1qp1q ´4976r3
` 12905r2
´ 15523r ` 9529,
1469059r3
´ 3794717r2
` 4539759r ´ 2782843
3119 r´4, ´3, ´2, ´1s 23 p 2
3
r3
´ r2
´ 1
3
r ´ 1
3
q23 p1qp1q 16743632r3
` 25416768r2
` 30512064r `
26598352,
´406345115512r3
´ 616830291616r2
´
740486023984r ´ 645505557528
3188 r2, ´4, 1, ´1s 24 p´r3
` r2
´ r ` 3q3 p´r3
` r2
´ r ` 4q8 p1q 2788172026368r3
` 1423837175512r2
`
4939120830288r ´ 3691304019543,
´10952993228320557238r3
´
5593370421245480720r2
´
19402732864546458324r `
14500836945256233797
3216 r3, 0, ´1, ´2s 5 p´r ´ 1q5 p1qp1q 16r3
´ 40r2
` 48r ´ 20,
104r3
´ 376r2
` 816r ´ 608
3271 r´1, ´1, 3, 0s 110 pr3
` r2
` 3r ` 2q5 p´r3
` r2
´ 2r ` 2q22 p1q 228r3
´ 115r2
` 220r ` 132,
6359r3
´ 2608r2
´ 6398r ´ 1760
3275 r´9, 6, 2, ´1s 19 p´ 1
9
r3
´ 2
9
r2
´ 8
9
r ´ 7
3
q19 p1qp1q 23
3
r3
´ 20
3
r2
` 34
3
r ´ 14,
496
9
r3
´ 1690
9
r2
` 1979
9
r ´ 263
3
3275 r´9, 6, 2, ´1s 19 pr ´ 2q19 p1qp1q 3r3
` 30r2
´ 27r ´ 10,
332
9
r3
´ 632
9
r2
´ 7136
9
r ` 2693
3
3284 r´2, 0, ´1, ´1s 6 pr ´ 1q3 p1qprq2 2016r3
` 1720r2
` 1160r ` 2161,
´290488r3
´ 248004r2
´ 169132r ´ 313401
3407 r´3, 1, ´2, ´1s 84 p 1
2
r3
´ 2r ` 1
2
q7 pr2
´ rq12 p1q 28129013
2
r3
` 15057426r2
` 3048856r `
40754921
2
,
´125734882980r3
´ 134611455788r2
´
27256382584r ´ 182171881573
3475 r´11, 8, ´2, ´1s 11 p´ 1
7
r3
´ 2
7
r2
´ 4
7
r ` 1
7
q11 p1qp1q 61
7
r3
´ 214
7
r2
´ 351
7
r ` 905
7
,
´ 1632
7
r3
` 2420
7
r2
` 6940
7
r ´ 10751
7
3475 r´11, 8, ´2, ´1s 11 p´rq11 p1qp1q ´16r3
´ 8r2
` 24r ´ 95,
9008
7
r3
` 5948
7
r2
´ 8124
7
r ` 59025
7
Marc Masdeu Non-archimedean constructions September 28th
, 2015 26 / 34

28. Tables: quartic p2, 1q fields (III)
|∆K | fK pxq NmpNq pDm c4 pEq, c6 pEq
3559 r´2, ´1, 3, ´2s 20 pr2
´ 1q5 p1qp´r2
` r ´ 2q4 266r3
´ 251r2
` 481r ` 402,
´8721r3
` 6359r2
´ 17561r ´ 13594
3571 r3, 5, ´5, ´1s 45 pr2
´ 3q3 pr2
` r ´ 5q15 p1q ´247r3
` 1481r2
` 5186r ` 1954,
313052r3
` 544864r2
´ 374892r ´ 240173
3632 r2, ´2, 0, ´2s 13 pr3
´ r2
´ r ´ 1q13 p1qp1q 110352r3
` 24580r2
` 54624r ´ 99300,
124669648r3
` 27763200r2
` 61709112r ´
112178880
3632 r2, ´2, 0, ´2s 14 p´r ´ 1q7 p1qp´rq2 2474r3
` 522r2
` 1234r ´ 2217,
523532r3
` 116757r2
` 258994r ´ 471065
3632 r2, ´2, 0, ´2s 26 pr3
´ r2
´ 3q13 p1qp´rq2 10028r3
` 2232r2
` 4964r ´ 9023,
´3562482r3
´793354r2
´1763358r`3205557
3723 r´1, 3, 1, ´1s 7 pr3
´ r2
` 2r ` 2q7 p1qp1q 381r3
´ 208r2
´ 592r ` 201,
´9752r3
´ 3598r2
` 7307r ´ 1656
3723 r´1, 3, 1, ´1s 17 pr ´ 2q17 p1qp1q 168r3
´ 1126r2
´ 1303r ` 504,
´24313r3
` 42209r2
` 63347r ´ 22837
3775 r´11, 7, 0, ´1s 19 p 3
8
r3
´ 1
4
r2
` 1
4
r ` 3
8
q19 p1qp1q 36r3
` 8r2
` 32r ` 253,
´662r3
´ 228r2
´ 448r ´ 5011
3775 r´11, 7, 0, ´1s 19 p 3
8
r3
´ 1
4
r2
` 1
4
r ` 27
8
q19 p1qp1q ´17r3
` 30r2
´ 70r ` 8,
489
2
r3
´ 607r2
` 1283r ´ 1633
2
3888 r3, ´6, 0, ´2s 3 p 1
2
r3
´ 1
2
r2
´ 1
2
r ´ 5
2
q3 p1qp1q 12362r3
` 8406r2
` 22518r ´ 13842,
7035016r3
`4781484r2
`12812832r´7875996
3899 r´3, 1, 2, ´2s 23 pr3
´ 2r2
` r ` 1q23 p1qp1q ´14r3
` 14r2
´ 25r ´ 21,
381r3
´ 249r2
` 364r ` 978
3967 r1, 5, ´2, ´1s 13 p 1
2
r3
´ 2r ` 1
2
q13 p1qp1q 3321
2
r3
` 1456r2
´ 2668r ´ 1081
2
,
´163448r3
` 28056r2
` 583644r ` 107371
3967 r1, 5, ´2, ´1s 17 p 1
2
r3
´ 2r ` 5
2
q17 p1qp1q ´ 3537
2
r3
´ 125r2
` 4948r ` 1841
2
,
´99064r3
´ 306744r2
´ 273576r ´ 41165
4108 r´2, ´2, 0, ´1s 52 pr2
´ r ` 1q13 p´r3
` 2r2
´ r ` 2q4 p1q ´52r3
` 56r2
` 316r ` 177,
3676r3
´ 2050r2
´ 1438r ` 1283
4192 r´2, ´2, 1, 0s 28 pr2
` r ` 1q7 pr2
` r ` 2q4 p1q 68388r3
´ 97900r2
´ 25440r ` 47889,
50048814r3
´ 57380110r2
´ 27657416r `
22526745
4192 r´2, ´2, 1, 0s 44 pr3
` r ´ 1q11 pr2
` r ` 2q4 p1q 993568r3
´ 1182928r2
´ 521264r ` 485673,
´2157501576r3
` 964037714r2
`
2148667444r ` 353613881
4204 r´4, ´2, 0, 0s 20 p´r ` 1q5 p´rq4 p1q 145360531282796r3
` 161931312392192r2
´
390193058066092r ´ 440654493862007,
´1159392135670300645002r3
`
9949620873463783912066r2
´
2497558503469317783050r ´
17611520739674724691341
4319 r2, ´1, ´4, ´1s 42 prq2 p´r3
` 2r2
` 3r ´ 1q21 p1q 2626337501r3
`4156522706r2
`229413693r´
2033846625,
694511908654437r3
` 1099155960247844r2
`
60666438159866r ´ 537832894958445
4384 r´4, 0, 3, ´2s 10 p 1
2
r3
´ r2
` 5
2
r ´ 2q5 p1qp 1
2
r3
` 1
2
r ` 2q2 ´39342r3
` 91445r2
` 10340r ´ 83032,
´8399230r3
´ 58605841r2
` 42062128r `
73787052
4423 r1, 4, ´3, ´1s 50 p´r ` 2q5 p´r2
´ r ` 2q10 p1q ´4642767r3
´ 1724885r2
` 13234188r `
2913911,
´19031399895r3
´ 11910891879r2
`
44594523793r ` 10072542896
4423 r1, 4, ´3, ´1s 50 p´r ` 2q5 p´r2
´ r ` 2q10 p1q 3516856r3
`2151917r2
´8338704r´1880324,
´9366159063r3
` 477887546r2
`
34591729866r ` 7408649776
4564 r1, ´5, 0, ´1s 5 p 1
2
r3
` r ´ 3
2
q5 p1qp1q ´280r3
` 240r2
` 64r ` 1449,
10942r3
´ 8954r2
´ 1978r ´ 55513
4568 r´1, ´3, 2, ´1s 12 pr2
` 2q3 p1qpr2
` 3q4 10845937505r3
` 4588202505r2
`
28221044093r ` 7621698413,
´1760006389370257r3
´
744542896235865r2
´ 4579522784006957r ´
1236798376628657
4652 r2, 5, ´3, ´1s 44 p 1
2
r3
´ 3
2
r ` 2q11 p´ 1
2
r3
` 3
2
r ´ 1q4 p1q ´1938032413r3
` 62742964314r2
`
143570326721r ` 41574563255,
14844318169935843r3
`
50626339684931473r2
`
49275897864569564r ` 11502761970012547
4663 r2, ´5, 2, ´1s 11 p´2r3
` r2
´ 3r ` 9q11 p1qp1q 4296r3
` 1705r2
` 10968r ´ 6148,
´3722961r3
´ 1477666r2
´ 9510026r `
5330364
4775 r´9, ´9, 2, ´1s 11 p´ 5
12
r3
` 2
3
r2
´ 5
6
r ` 13
4
q11 p1qp1q 307
4
r3
´ 499r2
´ 2771
2
r ´ 2953
4
,
´ 69064
3
r3
` 146785
3
r2
´ 182723
3
r ´ 87279
4775 r´9, ´9, 2, ´1s 11 p 1
6
r3
` 1
3
r2
` 1
3
r ´ 1
2
q11 p1qp1q 247r3
` 539r2
´ 163r ´ 335,
´41241r3
´ 13659r2
´ 21597r ´ 27719
4832 r´2, ´4, ´1, 0s 17 p´r3
` r2
` 2r ` 5q17 p1qp1q ´24r3
` 17r2
` 12r ` 82,
580r3
´ 347r2
´ 524r ´ 1770
4907 r´1, ´4, ´2, ´1s 11 pr3
´ r2
´ 3r ´ 3q11 p1qp1q ´191405r3
´ 287504r2
´ 336559r ´ 76491,
1214356660r3
` 1824081112r2
`
2135314036r ` 485335595
4944 r´1, ´4, ´1, 0s 17 pr3
´ 4q17 p1qp1q 316049736r3
` 586633069r2
` 772824316r `
170272429,
´23749113529508r3
´ 44081717952580r2
´
58072797643568r ´ 12794882314805
4979 r1, ´3, ´1, ´1s 13 p´r3
` r2
` r ` 1q13 p1qp1q 32r3
´ 128r2
` 144r ´ 32,
´1464r3
` 3856r2
´ 1824r ` 240
Marc Masdeu Non-archimedean constructions September 28th
, 2015 27 / 34

29. Surfaces (joint with X. Guitart and H. Sengun)
This all generalizes to higer dimensional (e.g. 2-dim’l) components.
The pairing
H1
pΓ0pNq, Zq ˆ H1pΓ0pNq, Zq Ñ Cˆ
p
yields, by taking bases of the irreducible factors, a lattice Λ Ă pCˆ
p q2.
Should correspond to the Cp-points of an abelian surface A split at p.
From the lattice Λ one can compute the p-adic L-invariant Lp of a
Mumford–Schottky genus 2 curve.
§ Lp P T bZ Qp.
§ Corresponding to a hyperelliptic curve X with Jac X “ A.
We can use formulas of Teitelbaum (1988) to recover a Weierstrass
equation for X from Lp.
From this equation ; approximate Igusa invariants of X.
Algebraic recognition algorithms ; actual Igusa invariants.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 28 / 34

30. Toy example: an abelian surface over F “ Q
Consider the Shimura curve Xp15 ¨ 11q, which has genus g “ 9.
One of the factors J of Jac Xp15 ¨ 11q is two-dimensional.
T2 acts on J with characteristic polynomial P2pxq “ x2 ` 2x ´ 1.
We compute a basis tφ1, φ2u of H1pΓ0p15 ¨ 11q, ZqT2“P2 and a
“pseudo-dual basis” tθ1, θ2u of H1pΓ0p15 ¨ 11q, ZqT2“P2 .
The integration pairing yields a symmetric matrix
ˆş
θ1
φ1
ş
θ2
φ1
ş
θ1
φ2
ş
θ2
φ2
˙
“
ˆ
A B
B D
˙
.
A “ 3 ¨ 114
` 3 ¨ 115
` 4 ¨ 116
` ¨ ¨ ¨ ` Op1124
q
B “ 4 ` 7 ¨ 11 ` 7 ¨ 112
` 4 ¨ 113
` ¨ ¨ ¨ ` Op1120
q
D “ 9 ¨ 114
` 9 ¨ 115
` 8 ¨ 116
` 9 ¨ 117
` ¨ ¨ ¨ ` Op1121
q
Marc Masdeu Non-archimedean constructions September 28th
, 2015 29 / 34

31. Toy example: an abelian surface over F “ Q (II)
This allows to recover the 11-adic L-invariant:
L11pJ2q “ 3 ¨ 11 ` 8 ¨ 112
` 3 ¨ 114
` 9 ¨ 115
` ¨ ¨ ¨
` p7 ¨ 112
` 3 ¨ 113
` 5 ¨ 114
` 2 ¨ 115
` ¨ ¨ ¨ q ¨ T2 P T bQp.
We recover the Igusa–Clebsch invariants
pI2 : I4 : I6 : I10q “ p2584 : ´75356 : 37541976 : 212
34
53
113
q
Mestre’s algorithm (together with model reduction) yields the
hyperelliptic curve
y2
“ ´x6
` 4x4
´ 10x3
` 16x2
´ 9
After twisting (by ´1 in this case) we get a curve whose first few
Euler factors match with those obtained by the T-action on J.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 30 / 34

32. Example surface over cubic p1, 1q field
Let F “ Qprq, where r3 ´ r2 ` 2r ´ 3 “ 0.
Let p7 “ pr2 ´ r ` 1q.
Let B{F be the totally definite quaternion algebra of disc. q3 “ prq.
Jac XBpΓ0ppqq has a two-dimensional factor J.
T2 acts on J with characteristic polynomial P2pxq “ x2 ` x ´ 10.
A similar calculation as before recovers the 7-adic L-invariant:
L7pJq “ 4 ¨ 7 ` 2 ¨ 72
` 5 ¨ 73
` 3 ¨ 74
` 3 ¨ 75
` ¨ ¨ ¨ ` Op7300
q
` p72
` 6 ¨ 74
` 2 ¨ 75
` 2 ¨ 76
` ¨ ¨ ¨ ` Op7300
qq ¨ T2 P T bQ7.
Sadly, haven’t yet been able to recover Igusa–Clebsch invariants.
§ (We have about a dozen more examples over cubic and quartic
fields. . . )
Marc Masdeu Non-archimedean constructions September 28th
, 2015 31 / 34

33. Beyond degree 1
F “ quartic totally-complex field, N “ p (for simplicity).
In this setting Γ “ SL2pOF r1{psq, which acts on H2
3.
The relevant groups are now H2pΓ, Div0
Hpq and H2pΓ, Ω1
Hp
q.
§ H2pSL2pOF q, Div Hpq H2
pΓ0ppq, Dq – H2
pSL2pOF q, coInd Dq.
§ The algorithms of J. Voight and A. Page do not extend to this situation.
What did the modular symbols algorithm teach us?
§ Exploit the cusps. . .
§ . . . use sharblies!
Marc Masdeu Non-archimedean constructions September 28th
, 2015 32 / 34

34. Sharblies and overconvergent leezarbs
Have a short exact sequence of GL2pFq-modules
0 Ñ ∆0
Ñ Div P1
pFq
deg
Ñ Z Ñ 0, ∆0
“ Div0
P1
pFq
Applying the functors p´q bV or Homp´, Wq and taking homology
and cohomology yields connecting homomorphims
H2pΓ0ppq, V q
δ
ˆ H2pΓ0ppq, Wq
X // H0pΓ0ppq, V bWq
H1pΓ0ppq, ∆0 bV q ˆ H1pΓ0ppq, Homp∆0, Wqq
X //
δ
OO
H0pΓ0ppq, Xq
ev˚
OO
X “ ∆0
bV bHomp∆0
, Wq
ev
Ñ V bW, γ bv bφ ÞÑ v bφpγq.
This diagram is “compatible”:
θ X δpφq “ ev˚pδθ X φq.
Reduced to H1pΓ0ppq, ∆0 bV q and H1pΓ0ppq, Homp∆0, Wqq.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 33 / 34

35. Sharblies and overconvergent leezarbs (II)
Reduced to H1pΓ0ppq, ∆0 bV q and H1pΓ0ppq, Homp∆0, Wqq.
Sharblies were invented by Szczarba and Lee (szczarb-lee) to
compute with H1pΓ0ppq, ∆0 bV q.
§ Natural generalization of modular symbols to higher rank groups.
‹ Ash–Rudolph, Ash–Gunnells, . . .
§ Used to compute structure as hecke modules.
‹ Ash, Gunnells, Hajir, Jones, McConnell, Yasaki, . . .
§ They give an acyclic a resolution of ∆0
bV .
§ The analogue of continued fractions algorithm is “sharbly reduction”.
In order to compute H1pΓ0ppq, Homp∆0, Wqq, we introduce a dual
version of sharblies, the leezarbs.
§ Leezarbs are an acyclic resolution of Homp∆0
, Wq.
§ Can reuse the sharbly reduction algorithm to work with leezarbs.
§ This is work in progress with X. Guitart and A. Page, stay tuned!
Marc Masdeu Non-archimedean constructions September 28th
, 2015 34 / 34

36. Thank you !
and
Congratulations to the people of Catalunya
Who have realized that, in the course of human events,
it has become necessary to dissolve the political bands
which have connected them with Spain.
Marc Masdeu Non-archimedean constructions September 28th
, 2015 34 / 34