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Non-archimedean construction ofelliptic curves and abelian surfaces
ICERM WORKSHOPModular 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
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
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 BF 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 BˆFˆ ãÑ PGL2pRqn ˆ PGL2pCqs ýHn ˆHs3.
R>0
C
H3
R>0
H
R
PGL2(R)PGL2(C)
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Quaternionic automorphic forms of level N (II)
Fix RD0 pnq Ă B Eichler order of level n.
ΓD0 pnq “ RD
0 pnqˆOˆF acts discretely on Hn ˆHs
3.Obtain an orbifold of (real) dimension 2n` 3s:
Y D0 pnq “ ΓD
0 pnqz pHn ˆHs3q .
The cohomology of Y D0 pnq can be computed via
H˚pY D0 pnq,Cq – H˚pΓD
0 pnq,Cq.
Hecke algebra TD “ ZrTq : q - Ds acts on H˚pΓD0 pnq,Zq.
Hn`spΓD0 pnq,Cq “
à
χ
Hn`spΓN0 pnq,Cqχ, χ : TD Ñ C.
Each χ cuts out a field Kχ, s.t. rKχ : Qs “ dimHn`spΓD0 pnq,Cqχ.
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Abelian varieties from cohomology classes
Definitionf 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 DEfF 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, thenD abelian variety Aχ, with dimAχ “ rK : F s and multiplication by K.
Assumption above avoids “fake abelian varieties”, and it is needed inour construction anyway.
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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.
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F “ Q: Cremona’s algorithm for elliptic curves
Eichler–Shimura construction
X0pNq // JacpX0pNqq
ş
–H0pX0pNq,Ω1q
_
H1pX0pNq,ZqHecke// // CΛf – Ef pCq.
1 Compute H1pX0pNq,Zq (modular symbols).2 Find the period lattice Λf by explicitly integrating
Λf “
C
ż
γ2πi
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 .
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F ‰ Q: constructions for elliptic curves
F totally real. rF : Qs “ n, fix σ : F ãÑ R.
S2pΓ0pNqq Q f ; ωf P HnpΓ0pNq,Cq; Λf Ď C.
Conjecture (Oda, Darmon, Gartner)CΛf is isogenous to Ef ˆF Fσ.
Known to hold (when F real quadratic) for base-change of EQ.Exploited in very restricted cases (Dembele, Stein+7).Explicitly computing Λf is hard.
§ No quaternionic computations (except for Voight–Willis?).
F not totally real: no known algorithms. . .TheoremIf F is imaginary quadratic, the lattice Λf is contained in R.
IdeaAllow for non-archimedean constructions.
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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 : Fˆp xqEy Ñ EpFpq,
with qE P Fˆp satisfying jpEq “ q´1E ` 744` 196884qE ` ¨ ¨ ¨ .
Choose a coprime factorization N “ pDm, with D “ discpBF q.Compute qE as a replacement for Λf .Starting data: f P Hn`spΓD
0 pmq,Zqp´new, pDm “ N.
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Non-archimedean path integrals on Hp
Consider Hp “ P1pCpqr 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 ofşτ2τ1ω P Cp.
Get a PGL2pFpq-equivariant pairingş
: Ω1HpˆDiv0 Hp Ñ Cp.
For each Γ Ă PGL2pFpq, get induced pairing (cap product)
H ipΓ,Ω1Hpq ˆHipΓ,Div0 Hpq
ş
// Cp
´
φ,ř
γ γ bDγ
¯
//ř
γ
ż
Dγ
φpγq.
Ω1Hp– space of Cp-valued boundary measures Meas0pP1pFpq,Cpq.
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Measures and integrals
Bruhat-Tits tree of GL2pFpq, |p| “ 2.P1pFpq – EndspT q.Harmonic cocycles HCpAq “
tEpT q fÑ A |
ř
opeq“v fpeq “ 0u
Meas0pP1pFpq, Aq – HCpAq.So replace ω P Ω1
Hpwith
µω P Meas0pP1pFpq,Zq – HCpZq.
P1(Fp)
U ⊂ P1(Fp)µ(U)
v∗
v∗
e∗
T
Coleman integration: if τ1, τ2 P Hp, thenż τ2
τ1
ω “
ż
P1pFpq
logp
ˆ
t´ τ2
t´ τ1
˙
dµωptq “ limÝÑU
ÿ
UPUlogp
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 “ limÝÑU
ź
UPU
ˆ
tU ´ τ2
tU ´ τ1
˙µωpUq
.
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The tpu-arithmetic group Γ
Choose a factorization N “ pDm.BF “ quaternion algebra with RampBq “ tq | Du Y tvn`1, . . . , vru.
Recall also RD0 ppmq Ă RD
0 pmq Ă B.Fix ιp : RD
0 pmq ãÑM2pZpq.Define ΓD
0 ppmq “ RD0 ppmq
ˆOˆF and ΓD0 pmq “ RD
0 pmqˆOˆF .
Let Γ “ RD0 pmqr1ps
ˆOF r1psˆ ιp
ãÑ PGL2pFpq.
ExampleF “ Q and D “ 1, so N “ pM .B “M2pQq.Γ0ppMq “
`
a bc d
˘
P GL2pZq : pM | c(
t˘1u.Γ “
`
a bc d
˘
P GL2pZr1psq : M | c(
t˘1u ãÑ PGL2pQq Ă PGL2pQpq.
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The tpu-arithmetic group Γ
LemmaAssume that h`F “ 1. Then ιp induces bijections
ΓΓD0 pmq – V0pT q, ΓΓD
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. ΓD0 ppmq).
Corollary
MapspE0pT q,Zq – IndΓΓD
0 ppmqZ, MapspVpT q,Zq –
´
IndΓΓD
0 pmqZ¯2.
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Cohomology
Γ “ RD0 pmqr1ps
ˆOF r1psˆ ιp
ãÑ PGL2pFpq.
MapspE0pT q,Zq – IndΓΓD
0 ppmqZ, MapspVpT q,Zq –
´
IndΓΓD
0 pmqZ¯2.
Want to define a cohomology class in Hn`spΓ,Ω1Hpq.
Consider the Γ-equivariant exact sequence
0 // HCpZq //MapspE0pT q,Zq β //MapspVpT q,Zq // 0
ϕ // rv ÞÑř
opeq“v ϕpeqs
So get:
0 Ñ HCpZq Ñ IndΓΓD
0 ppmqZ βÑ
´
IndΓΓD
0 pmqZ¯2Ñ 0
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Cohomology (II)
0 Ñ HCpZq Ñ IndΓΓD
0 ppmqZ βÑ
´
IndΓΓD
0 pmqZ¯2Ñ 0
Taking Γ-cohomology, . . .
Hn`spΓ,HCpZqq Ñ Hn`spΓ, IndΓΓD
0 ppmq,Zq β
Ñ Hn`spΓ, IndΓΓD
0 pmq,Zq2 Ñ ¨ ¨ ¨
. . . and using Shapiro’s lemma:
Hn`spΓ,HCpZqq Ñ Hn`spΓD0 ppmq,Zq
βÑ Hn`spΓD
0 pmq,Zq2 Ñ ¨ ¨ ¨
f P Hn`spΓD0 ppmq,Zq being p-new ô f P Kerpβq.
Pulling back get
ωf P Hn`spΓ,HCpZqq – Hn`spΓ,Ω1
Hpq.
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Holomogy
Consider the Γ-equivariant short exact sequence:
0 Ñ Div0 Hp Ñ DivHpdegÑ ZÑ 0.
Taking Γ-homology yields
Hn`s`1pΓ,ZqδÑ Hn`spΓ,Div0 Hpq Ñ Hn`spΓ,DivHpq Ñ Hn`spΓ,Zq
Λf “
#
ˆ
ż
δpcqωf : c P Hn`s`1pΓ,Zq
+
Ă Cˆp
Conjecture A (Greenberg, Guitart–M.–Sengun)
The multiplicative lattice Λf is homothetic to qZE .
F “ Q: Darmon, Dasgupta–Greenberg, Longo–Rotger–Vigni.F totally real, |p| “ 1, B “M2pF q: Spiess.Open in general.
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Lattice: explicit construction
Start with f P Hn`spΓD0 ppm,Zqqp´new.
Duality yields f P Hn`spΓD0 ppmq,Zqqp´new.
Mayer–Vietoris exact sequence for Γ “ ΓD0 pmq ‹ΓD
0 ppmqΓD0 pmq:
¨ ¨ ¨ Ñ Hn`s`1pΓ,Zqδ1Ñ Hn`spΓ
D0 ppmq,Zq
βÑ Hn`spΓ
D0 pmq,Zq2 Ñ ¨ ¨ ¨
f new at p ùñ βpfq “ 0.§ f “ δ1pcf q, for some cf P Hn`s`1pΓ,Zq.
Conjecture (rephrased)The element
Lf “ż
δpcf qωf .
satisfies (up to a rational multiple) logppqEq “ Lf .
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Algorithms
Only in the cases n` s ď 1.
§ Both H1 and H1: fox calculus (linear algebra for finitely-presentedgroups).
Use explicit presentation + word problem for ΓD0 ppmq and ΓD
0 pmq.
§ John Voight (s “ 0).
§ Aurel Page (s “ 1).
Need the Hecke action on H1pΓD0 ppmq,Zq and H1pΓ
D0 ppmq,Zq.
§ Shapiro’s lemma ùñ enough to work with ΓD0 pmq.
Integration pairing uses the overconvergent method.
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Overconvergent Method
Starting data: cohomology class φ “ ωf P H1pΓ,Ω1
Hpq.
Goal: to compute integralsşτ2τ1φγ , for γ P Γ.
Recall thatż 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
tidµγptq for all γ P Γ.
Note: Γ Ě ΓD0 pmq Ě ΓD
0 ppmq.Technical lemma: All these integrals can be recovered from#
ż
Zp
tidµγptq : γ P ΓD0 ppmq
+
.
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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 ϕp1Zpq induces a projection:
H1pΓD0 ppmq,Dq
ρ // H1pΓD0 ppmq,Zpq.
PΦ
//
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.
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Overconvergent Method (III)
But we wanted to compute the moments of a system of measures. . .
PropositionConsider the map ΓD
0 ppmq Ñ D:
γ ÞÑ”
hptq ÞÑ
ż
Zp
hptqdµγptqı
.
1 It satisfies a cocycle relation ùñ induces a classΨ P H1
´
ΓD0 ppmq,D
¯
.
2 Ψ is a lift of φ.3 Ψ is a Up-eigenclass.
CorollaryThe explicitly computed Φ “ Ψ knows the above integrals.
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Recovering E from Λf
Λf “ xqf y gives us qf?“ qE .
Assume ordppqf q ą 0 (otherwise, replace qf ÞÑ 1qf ).Get
jpqf q “ q´1f ` 744` 196884qf ` ¨ ¨ ¨ P Cˆp .
From N guess the discriminant ∆E .§ Only finitely-many possibilities, ∆E P SpF, 12q.
jpqf q “ c34∆E ; recover c4.
Recognize c4 algebraically.1728∆E “ c3
4 ´ c26 ; recover c6.
Compute the conductor of Ef : Y 2 “ X3 ´ c448X ´
c6864 .
§ If conductor is correct, check aq’s.
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Example curve (joint with X. Guitart and H. Sengun)
F “ Qpαq, pαpxq “ x4 ´ x3 ` 3x´ 1, ∆F “ ´1732.N “ pα´ 2q “ p13.BF 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 “ 113
´
´ 4656377430074α3 ` 10862248656760α2 ´ 14109269950515α ` 4120837170980¯
.
c4 “ 2698473α3 ` 4422064α2 ` 583165α´ 825127.c6 “ 20442856268α3´ 4537434352α2´ 31471481744α` 10479346607.
EF : y2 ``
α3 ` α` 3˘
xy “ x3`
``
´2α3 ` α2 ´ α´ 5˘
x2
``
´56218α3 ´ 92126α2 ´ 12149α` 17192˘
x
´ 23593411α3 ` 5300811α2 ` 36382184α´ 12122562.
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Tables: Imaginary quadratic fields
|∆K | fKpxq NmpNq pDm c4pEq, c6pEq3 r1,´1s 196 p3r ´ 2q7p´6r ` 2q28p1q ´131065r,
474493313 r1,´1s 196 p´3r ` 1q7p6r ´ 4q28p1q ´131065r,
474493314 r1, 0s 130 p3r ´ 2q13p´r ´ 3q10p1q ´264r ` 257,
´6580r ` 25834 r1, 0s 130 p´3r ´ 2q13p´3r ´ 1q10p1q 264r ` 257,
6580r ` 25837 r2,´1s 44 prq2p3r ` 1q22p1q 648r ` 481,
´28836r ` 44477 r2,´1s 44 pr ´ 1q2p3r ´ 4q22p1q ´648r ` 1129,
28836r ´ 243898 r2, 0s 99 pr ` 1q3p´4r ` 1q33p1q 444r ` 25,
14794r ´ 162638 r2, 0s 99 pr ´ 1q3p´4r ´ 1q33p1q ´444r ` 25,
´14794r ´ 162638 r2, 0s 99 p´r ´ 3q11p3q9p1q ´444r ` 25,
´14794r ´ 162638 r2, 0s 99 pr ´ 3q11p3q9p1q 444r ` 25,
14794r ´ 16263
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Tables: cubic p1, 1q fields|∆K | fKpxq NmpNq pDm c4pEq, c6pEq
23 r1, 0,´1s 185 pr2 ` 1q5p3r2 ´ r ` 1q37p1q 643318r2 ´ 1128871r ` 852306,
925824936r2 ´ 1624710823r ` 1226456111
31 r´1, 1, 0s 129 p´r ´ 1q3p´3r2 ´ 2r ´ 1q43p1q ´4787r2 ` 10585r ` 3349,
1268769r2 ´ 371369r ` 424764
44 r1, 1,´1s 121 p2r ´ 1q11pr2 ` 2q11p1q 4097022r2 ´ 6265306r ` 7487000,
14168359144r2 ´ 21861492432r `
26039140708
44 r1, 1,´1s 121 p2r ´ 1q11pr2 ` 2q11p1q 1774r2 ´ 1434r ´ 1304,
´42728r2 ´ 123104r ´ 54300
44 r1, 1,´1s 121 p2r ´ 1q11pr2 ` 2q11p1q 4097022r2 ´ 6265306r ` 7487000,
14168359144r2 ´ 21861492432r `
26039140708
59 r´1, 2, 0s 34 p´r2 ´ 1q2p´r2 ´ 2r ´ 2q17p1q 262r2 ` 513r ` 264,
´2592r2 ` 448r ` 13231
59 r´1, 2, 0s 34 p´r2 ´ 2r ´ 2q17p´r2 ´ 1q2p1q 16393r2 ` 20228r ´ 12524,
4430388r2 ´ 5579252r ` 1619039
59 r´1, 2, 0s 46 p´2r2 ` r ´ 2q23p´r2 ´ 1q2p1q 18969r2 ` 8532r ` 41788,
4216716r2 ` 1911600r ` 9298151
59 r´1, 2, 0s 74 p´r2 ´ 1q2p2r2 ` 2r ` 1q37p1q 33054r2 ` 15049r ` 72776,
9702640r2 ` 4400116r ` 21401723
59 r´1, 2, 0s 88 p´r2 ´ 1q2pr ´ 2q11pr2 ` r ` 1q4 16609r2 ` 7084r ` 37332,
3522136r2 ` 1613876r ` 7760395
59 r´1, 2, 0s 187 p2r2 ` r ` 2q17pr ´ 2q11p1q ´32r2 ´ 848r ` 432,
´7600r2 ` 23368r ´ 8704
76 r´2,´2, 0s 117 p2r2 ´ r ´ 3q13p´r2 ` 2r ` 1q9p1q 48r ` 16,
´128r2 ´ 224r ´ 216
83 r´2, 1,´1s 65 pr ` 1q5p´2r ` 1q13p1q 3089r2 ` 1086r ` 4561,
333604r2 ` 117840r ` 493059
83 r´2, 1,´1s 65 pr ` 1q5p´2r ` 1q13p1q 304r2 ` 112r ` 449,
6616r2 ` 2328r ` 9791
83 r´2, 1,´1s 65 p´2r ` 1q13pr ` 1q5p1q 3089r2 ` 1086r ` 4561,
333604r2 ` 117840r ` 493059
83 r´2, 1,´1s 65 p´2r ` 1q13pr ` 1q5p1q 4499473r2 ` 1589254r ` 6650137,
18573712184r2`6560420272r`27451337687
83 r´2, 1,´1s 106 prq2p2r2 ´ 3r ` 3q53p1q 2329r2 ` 822r ` 3441,
´34264r2 ´ 12104r ´ 50645
87 r1, 2,´1s 123 pr2 ´ r ` 1q3pr2 ` 4q41p1q 1424r2 ` 3792r ` 2384,
´245696r2 ´ 201800r ´ 4144
87 r1, 2,´1s 129 pr2 ´ r ` 1q3p´3r ` 1q43p1q ´752r2 ` 2272r ` 1009,
27496r2 ´ 152144r ´ 63977
87 r1, 2,´1s 129 pr2 ´ r ` 1q3p´3r ` 1q43p1q ´752r2 ` 2272r ` 1009,
27496r2 ´ 152144r ´ 63977
104 r´2,´1, 0s 143 pr2 ` r ´ 1q11p2r ` 1q13p1q 12r2 ` 12r ` 25,
´144r2 ´ 90r ´ 125
107 r´2, 3,´1s 40 p´r2 ´ 1q5pr2 ´ r ` 3q4prq2 3880r2 ´ 984r ` 10473,
405820r2 ´ 105348r ` 1142075
107 r´2, 3,´1s 135 p´r2 ´ 1q5p3q27p1q 16r2 ´ 16r,184r ´ 296
108 r´2, 0, 0s 34 p2r ` 1q17prq2p1q 184r2 ` 212r ` 265,
´5010r2 ´ 6306r ´ 7773
108 r´2, 0, 0s 85 p´r2 ´ 1q5p2r ` 1q17p1q 2224r2 ` 2816r ` 3520,
´229056r2 ´ 288672r ´ 363768
108 r´2, 0, 0s 85 p2r ` 1q17p´r2 ´ 1q5p1q 2224r2 ` 2816r ` 3520,
´229056r2 ´ 288672r ´ 363768
108 r´2, 0, 0s 125 p´r2 ´ 1q5pr2 ´ 2r ´ 1q25p1q 496r2,
22088
108 r´2, 0, 0s 145 p´r2 ´ 1q5pr ` 3q29p1q 144r2 ` 176r ` 240,
3816r2 ` 4752r ` 6088
108 r´2, 0, 0s 155 p´r2 ´ 1q5pr2 ` 3q31p1q 16r2 ` 20r ` 17,
´606r2 ´ 762r ´ 929
116 r´2, 0,´1s 34 p´2r ` 1q17p´r ` 1q2p1q 846760r2 ` 589024r ` 998761,
1781332252r2 ` 1239131712r ` 2101097467
116 r´2, 0,´1s 34 p´r ` 1q2p´2r ` 1q17p1q 4592r2 ` 3192r ` 5417,
274400r2 ` 190876r ` 323659
116 r´2, 0,´1s 38 p´r ` 1q2p2r ` 1q19p1q 82921r2 ` 57626r ` 97746,
54599355r2 ` 37980374r ` 64400978
116 r´2, 0,´1s 38 p2r ` 1q19p´r ` 1q2p1q 1081r2 ` 746r ` 1266,
66555r2 ` 46310r ` 78482
116 r´2, 0,´1s 58 p´r ` 1q2pr2 ` r ´ 3q29p1q 22024r2 ` 15320r ` 25977,
´4956678r2 ´ 3447968r ´ 5846447
135 r´1, 3, 0s 55 pr2 ´ r ` 2q11pr2 ` 1q5p1q 4139r2 ´ 19599r ` 5885,
2077971r2 ´ 1764501r ` 352796
135 r´1, 3, 0s 88 pr2 ´ r ` 2q11p2q8p1q ´1751r2 ´ 1226r ` 577,
´131901r2 ´ 120528r ` 52524
139 r2, 1,´1s 46 pr ´ 3q23p´rq2p1q 22560r2 ` 19560r ` 1033,
´8413992r2 ` 2336724r ` 7421723
139 r2, 1,´1s 57 pr ´ 1q3p´2r ` 1q19p1q 18r2 ` 61r ` 39,296r ` 239
139 r2, 1,´1s 57 p´2r ` 1q19pr ´ 1q3p1q 258r2 ` 541r ` 279,
´17136r2 ´ 9280r ` 3767
140 r´2, 2, 0s 25 pr2 ` 1q5pr ` 1q5p1q 1488r2 ` 992r ` 3968,
110440r2 ` 88352r ` 287144
140 r´2, 2, 0s 70 pr2 ` r ` 1q7pr ` 1q5prq2 139012r2 ` 106502r ` 360441,
´100613641r2 ´ 77548384r ´ 260995189
140 r´2, 2, 0s 95 pr2 ` 1q5pr2 ` 2r ` 3q19p1q 16r2 ` 16r,
´64r2 ` 240r ´ 120
140 r´2, 2, 0s 95 pr2 ` 1q5pr2 ` 2r ` 3q19p1q 64r2 ´ 64r ` 48,
´824r2 ´ 368r ` 616
172 r3,´1,´1s 45 pr ´ 2q5pr2 ´ r ´ 1q9p1q ´1072r2 ´ 80r ` 1872,
´49976r2 ´ 48864r ` 25920
175 r´3, 2,´1s 27 prq3pr2 ´ r ` 2q9p1q ´384r2 ` 816r ´ 416,
5904r2 ´ 32472r ` 31816
199 r´1, 4,´1s 21 p´r2 ` r ´ 2q7pr2 ´ r ` 3q3p1q 98529r2 ` 22348r ´ 12672,
´41881233r2 ` 130193546r ´ 31313977
199 r´1, 4,´1s 21 p´r2 ` r ´ 2q7pr2 ´ r ` 3q3p1q ´112647r2 ´ 62978r ` 24321,
´60304454r2 ´ 96556295r ` 29529884
199 r´1, 4,´1s 33 pr ´ 2q11pr2 ´ r ` 3q3p1q 2802r2 ` 3055r ´ 996,
´398780r2 ` 635911r ´ 139543
199 r´1, 4,´1s 49 p´r2 ` r ´ 2q7p´r2 ´ 3q7p1q 6447r2 ´ 31223r ` 7758,
3699375r2 ´ 3171676r ` 577928
199 r´1, 4,´1s 49 p´r2 ´ 3q7p´r2 ` r ´ 2q7p1q 6447r2 ´ 31223r ` 7758,
3699375r2 ´ 3171676r ` 577928
199 r´1, 4,´1s 77 pr ` 1q7pr ´ 2q11p1q 12952r2 ´ 10791r ` 49899,
2866751r2 ´ 2163173r ` 10872899
199 r´1, 4,´1s 99 pr ´ 2q11pr2 ` 1q9p1q ´120r2 ` 576r ´ 143,
380r2 ` 4776r ´ 1281
200 r2, 2,´1s 14 pr ` 1q2pr2 ´ r ` 1q7p1q ´401r2 ´ 3756r ´ 2274,
182521r2 ´ 243668r ´ 235802
200 r2, 2,´1s 14 pr2 ´ r ` 1q7pr ` 1q2p1q ´241r2 ` 404r ` 366,
5649r2 ` 3068r ´ 394
200 r2, 2,´1s 65 p´r2 ´ r ´ 1q13p´r2 ` r ´ 3q5p1q ´1176r2 ´ 1944r ´ 767,
75636r2 ´ 142236r ´ 124561
204 r´3, 1,´1s 21 pr2 ` r ` 1q7prq3p1q ´48r2 ` 96r ´ 32,
´288r2 ` 1008r ´ 872
204 r´3, 1,´1s 21 pr2 ` r ` 1q7prq3p1q 262r2 ´ 326r ´ 44,
´1784r2 ´ 5128r ` 11612
211 r´3,´2, 0s 21 pr ` 2q7p´rq3p1q 22010896r2 ` 41672992r ` 34877233,
296072400488r2 ` 560550677168r `
469139740087
212 r´2, 4,´1s 35 pr2 ´ r ` 1q7pr2 ´ r ` 3q5p1q 29888r2 ´ 13952r ` 112113,
10054302r2 ´ 4693580r ` 37714701
216 r´2, 3, 0s 34 prq2pr2 ` r ` 5q17p1q 307r2 ` 194r ` 1057,
´11235r2 ´ 6786r ´ 37821
216 r´2, 3, 0s 34 pr2 ` r ` 5q17prq2p1q 307r2 ` 194r ` 1057,
´11235r2 ´ 6786r ´ 37821
|∆K | fKpxq NmpNq pDm c4pEq, c6pEq
216 r´2, 3, 0s 38 p´2r2 ´ 2r ´ 7q19prq2p1q 16r2 ` 81,
´216r2 ´ 192r ´ 601
231 r3, 0,´1s 33 p´r ` 1q3pr2 ´ r ` 2q11p1q 465r2 ´ 1011r ` 1189,
25273r2 ´ 54957r ` 64546
231 r3, 0,´1s 33 pr2 ´ r ` 2q11p´r ` 1q3p1q 465r2 ´ 1011r ` 1189,
25273r2 ´ 54957r ` 64546
231 r3, 0,´1s 51 pr2 ` 1q17prq3p1q ´47r2 ´ 50r ` 145,
938r2 ´ 291r ` 4
239 r´3,´1, 0s 24 pr ` 1q3p2q8p1q 9r2 ` 18r ` 25,
143r2 ` 236r ` 268
239 r´3,´1, 0s 57 p´r2 ´ r ` 1q19pr ` 1q3p1q 1170r2 ` 1953r ` 2098,
108233r2 ` 180929r ` 194227
243 r´3, 0, 0s 10 pr ´ 2q5pr ´ 1q2p1q 27576r2 ` 39771r ` 57360,
11428272r2 ` 16482420r ` 23771763
243 r´3, 0, 0s 10 pr ´ 1q2pr ´ 2q5p1q 27576r2 ` 39771r ` 57360,
11428272r2 ` 16482420r ` 23771763
243 r´3, 0, 0s 22 pr ` 2q11pr ´ 1q2p1q 2002130917752r2 ` 2887572455827r `
4164600133648,
7076846143946804016r2 `
10206578310238918020r `
14720433182250993839
243 r´3, 0, 0s 34 pr2 ` 2q17pr ´ 1q2p1q 10167352r2 ` 14663859r ` 21148944,
´80986535280r2 ´ 116802795708r ´
168458781921
243 r´3, 0, 0s 46 p´2r ` 1q23pr ´ 1q2p1q 19946163r2 ` 28767345r ` 41489691,
222892996797r2 ` 321467328855r `
463636116909
255 r´3, 0,´1s 15 p´r2 ´ 1q5pr ´ 1q3p1q 248r2 ´ 320r ´ 263,
´2556r2 ´ 4104r ` 16523
255 r´3, 0,´1s 15 p´r2 ´ 1q5prq3p1q 19r2 ´ r ` 88,
279r2 ` 908
255 r´3, 0,´1s 51 pr2 ´ 2q17pr ´ 1q3p1q ´32r2 ` 240r ´ 336,
´3416r2 ` 2400r ` 7392
255 r´3, 0,´1s 51 pr2 ´ 2q17pr ´ 1q3p1q 80r2 ´ 80r ´ 128,
288r2 ´ 2088r ` 2888
255 r´3, 0,´1s 65 pr2 ´ r ` 1q13pr ` 1q5p1q 3r2 ´ 105r ` 88,
´909r2 ` 1116r ´ 1576
268 r5,´3,´1s 14 pr2 ´ 2q7pr ´ 1q2p1q ´285113701784r2 ´ 52062773310r `
950706811227,
144006413291532359r2 `
50857254178772568r ´ 433038348784793416
300 r´3,´3,´1s 9 p´r2 ` 2r ` 2q3prq3p1q 26r2 ` 46r ` 4,
504r2 ` 504r ` 460
300 r´3,´3,´1s 9 p´r2 ` 2r ` 2q3prq3p1q 26r2 ` 46r ` 4,
504r2 ` 504r ` 460
300 r´3,´3,´1s 33 pr2 ´ r ´ 1q11prq3p1q 11072r2 ` 17760r ` 12865,
4675808r2 ` 7475664r ` 5398495
300 r´3,´3,´1s 90 p´r2 ` 2r ` 2q3p´r ´ 3q30p1q ´71,´1837
307 r2, 3,´1s 10 pr ´ 1q5p´rq2p1q ´1450479r2 ´ 118958r ` 338681,
´1778021804r2´7601175244r´3506038549
307 r2, 3,´1s 45 pr ´ 1q5pr2 ´ 2r ` 5q9p1q r2 ` 154r ` 81,
´1744r2 ´ 1756r ´ 441
324 r´4,´3, 0s 4 pr ´ 2q2p´r ´ 1q2p1q 345255874728r2 ` 758120909880r `
628931968401,
686899433218582980r2 `
1508309811434747772r `
1251283596457392135
324 r´4,´3, 0s 22 pr2 ´ r ´ 1q11pr ´ 2q2p1q 808464801r2 ` 1775245884r ` 1472731953,
77832295537635r2 ` 170905971571164r `141782435639127
324 r´4,´3, 0s 84 pr2 ` 3r ` 3q7p´r2 ´ 3r ´ 2q12p1q 143742984r2 ` 315634200r ` 261847993,
4700399015844r2 ` 10321245891900r `
8562435635987
327 r´3,´2,´1s 9 prq3pr ` 1q3p1q 13r2 ` 22r ` 25,
144r2 ` 225r ` 242
327 r´3,´2,´1s 15 p´r ` 1q5prq3p1q 1645r2 ´ 2647r ´ 2984,
55543r2 ´ 6268r ´ 298328
327 r´3,´2,´1s 15 prq3p´r ` 1q5p1q 1645r2 ´ 2647r ´ 2984,
55543r2 ´ 6268r ´ 298328
335 r1, 4,´1s 25 pr2 ´ r ` 3q5p´r ` 1q5p1q ´951r2 ` 1190r ` 57,
61922r2 ´ 78025r ´ 346
335 r1, 4,´1s 25 pr2 ´ r ` 3q5p´r ` 1q5p1q 10r2 ´ 11r ` 10,
52r2 ´ 271r ´ 29
335 r1, 4,´1s 65 p´r2 ` 2r ´ 4q13p´r ` 1q5p1q 61r2 ´ 77r ` 247,
101r2 ´ 107r ` 380
351 r´3, 3, 0s 33 pr ´ 2q11prq3p1q 16r2 ` 144r ´ 128,
1824r2 ´ 72r ´ 1160
356 r7, 1,´1s 14 p´r ´ 2q7p12r2 ´ r ` 3
2q2p1q 1577904r2 ` 58258032r ` 83210433,
157810225239r2 ` 783843846012r `
817040026548
356 r7, 1,´1s 26 p´r ` 2q13p´12r2 ` r ´ 5
2q2p1q ´353192r2 ´ 495936r ` 44233,
´560380445r2 ´ 897785708r ´ 94909392
356 r7, 1,´1s 26 p´r ` 2q13p´12r2 ` r ´ 5
2q2p1q 88412r2 ` 1393648r ` 1878333,
112777386r2 ` 1758482408r ` 2367346473
356 r7, 1,´1s 196 prq7pr ´ 3q28p1q 4182384r2 ´ 3886864r ´ 15048991,
´37671142504r2 ´ 30349104360r `
38274580847
364 r´2, 4, 0s 21 pr ´ 1q3p´r ´ 1q7p1q ´368r2 ´ 3712r ` 1840,
´72736r2 ` 343360r ´ 146264
364 r´2, 4, 0s 26 p´r2 ´ 1q13p´rq2p1q ´266582r2 ` 148350r ´ 10479,
´274275343r2 ` 306719520r ´ 83736937
379 r´4, 1,´1s 6 pr ´ 1q3p´r ` 2q2p1q 1418236432r2 ` 1053691808r ` 3254778265,
137488390576232r2 ` 102148264969648r `315528648990403
379 r´4, 1,´1s 6 p´r ` 2q2pr ´ 1q3p1q 1418236432r2 ` 1053691808r ` 3254778265,
137488390576232r2 ` 102148264969648r `315528648990403
379 r´4, 1,´1s 21 pr ´ 1q3pr ` 1q7p1q 15373338r2 ` 11421763r ` 35281005,
´155147444344r2 ´ 115268221468r ´
356055251669
379 r´4, 1,´1s 21 pr ` 1q7pr ´ 1q3p1q 15373338r2 ` 11421763r ` 35281005,
´155147444344r2 ´ 115268221468r ´
356055251669
379 r´4, 1,´1s 27 pr ´ 1q3pr2 ` 1q9p1q 1532208r2 ` 1138368r ` 3516337,
1280550616r2 ` 951396864r ` 2938796535
379 r´4, 1,´1s 34 pr ´ 3q17p´r ` 2q2p1q 90342993r2 ` 67121158r ` 207332433,
2363568298948r2 ` 1756034817652r `
5424265343699
439 r5,´2,´1s 15 p´r ` 1q3pr ´ 2q5p1q ´439r2 ` 1212r ´ 1252,
27743r2 ´ 76494r ` 78935
439 r5,´2,´1s 15 pr ´ 2q5p´r ` 1q3p1q ´439r2 ` 1212r ´ 1252,
27743r2 ´ 76494r ` 78935
440 r´8, 2, 0s 10 p´r2 ´ 2r ´ 5q5p´12r2 ´ r ´ 2q2p1q ´349392832r2 ´ 1512227664r` 3500497481,
´12893566003280r2 ´ 143880769408104r `276285496852283
440 r´8, 2, 0s 10 p´ 12r2 ´ r ´ 2q2p´r
2 ´ 2r ´ 5q5p1q ´349392832r2 ´ 1512227664r` 3500497481,
´12893566003280r2 ´ 143880769408104r `276285496852283
440 r´8, 2, 0s 26 p2r ´ 3q13p´12r2 ´ r ´ 2q2p1q
9532r2 ´ 6046r ` 8769,
´ 4195612
r2 ` 835646r ´ 810505
451 r8,´5,´1s 26 p2r ´ 3q13p´r ` 2q2p1q 34296r2 ` 4776r ´ 189951,
5707476r2 ` 13155804r ´ 1647297
459 r´8, 3, 0s 22 p 12r2 ´ 1
2r ` 1q11p´
12r2 ´ 1
2r ´ 2q2p1q 16r2 ´ 104r ` 121,
´240r2 ` 1260r ´ 1357
459 r´8, 3, 0s 33 p´ 12r2 ´ 1
2r ` 1q11p
12r2 ` 1
2r ` 3q3p1q ´ 19
2r2 ` 15
2r ` 21,
´36r2 ` 96r ´ 37
459 r´8, 3, 0s 33 pr2 ` r ` 5q11p12r2 ` 1
2r ` 3q3p1q
1788292
r2 ` 2705212
r ` 472861,
´83966694r2 ´ 127020222r ´ 444049333
459 r´8, 3, 0s 34 p 12r2 ` 3
2r ´ 3q17p´
12r2 ´ 1
2r ´ 2q2p1q
1252r2 ` 79
2r ´ 59,
282r2 ´ 2430r ` 2691
459 r´8, 3, 0s 44 p 12r2 ´ 1
2r ` 1q11pr ´ 1q4p1q
312r2 ´ 105
2r ` 44,
411r2 ´ 1452r ` 1256
459 r´8, 3, 0s 44 p 12r2 ´ 1
2r ` 1q11pr ´ 1q4p1q
1032r2 ´ 55
2r ´ 60,
237r2 ` 1374r ´ 2984
460 r´3, 5,´1s 6 p´rq3pr ´ 1q2p1q 38808r2 ` 63978r ´ 55637,
28650959r2 ` 29220772r ´ 29738968
460 r´3, 5,´1s 25 p2r2 ´ r ` 10q5p´r2 ´ 4q5p1q 36772r2 ´ 83396r ` 37921,
32322356r2 ´ 98725758r ` 49331449
460 r´3, 5,´1s 26 pr2 ´ r ` 1q13pr ´ 1q2p1q 973808r2 ´ 7106166r ` 4086627,
´8777739333r2`7426503436r´1197128148
515 r´4,´1,´1s 14 p´r ` 2q2pr2 ´ 2r ´ 1q7p1q ´7341361r2 ´ 9117211r ´ 13098483,
´14436506787r2 ´ 17928648161r ´
25757667905
519 r7,´4,´1s 39 p´r2 ` 3q13p´r ` 2q3p1q ´280r2 ´ 960r ´ 751,
54220r2 ` 11272r ´ 242353
547 r´4,´3,´1s 14 p´r ` 1q7pr2 ´ 2r ´ 2q2p1q 14509048r2 ` 24346088r ` 21671521,
200457117220r2 ` 336365736396r `
299413898447
652 r5, 7,´1s 14 p´ 12r2 ` r ` 1
2q7p´
12r2 ` r ´ 9
2q2p1q 18r2 ´ 36r ` 147,
405r2 ´ 648r ` 3294
687 r3, 4,´1s 9 prq3pr ` 1q3p1q ´7r2 ` 38r ` 25,
´18r2 ´ 423r ´ 244
743 r´3, 5, 0s 9 p´r ` 1q3prq3p1q 736r2 ` 416r ` 3913,
´110256r2 ´ 62192r ´ 586373
755 r2, 5,´1s 10 p´2r2 ` 3r ´ 11q5p´rq2p1q ´1634r2 ` 10769r ` 4135,
110372r2 ` 1174880r ` 412903
815 r´9,´7, 0s 9 pr ` 1q3p´r ` 3q3p1q 26678105835217r2 ` 83793885354406r `
76443429630973,
717286463675094140331r2 `
2252941797094015980448r `
2055312234304678362824
1196 r´7, 5,´1s 14 p´rq7p´r ` 1q2p1q ´12r2 ´ 4r ` 25,
´4r2 ´ 134r ` 181
Marc Masdeu Non-archimedean constructions September 28th, 2015 24 / 34
Tables: quartic p2, 1q fields (I)|∆K | fKpxq 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 ´ 62772r3 ´ 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 12r2 ´ r ´ 1q11p1qp1q 12r3 ` 48r2 ` 56r ` 20,
´284r3 ´ 460r2 ´ 472r ´ 1024
1600 r´4, 0,´2, 0s 11 p 12r2 ` r ´ 1q11p1qp1q 276r3 ` 490r2 ` 336r ` 628,
´18172r3 ´ 32652r2 ´ 22424r ´ 40464
1600 r´4, 0,´2, 0s 19 p 12r3 ´ 1
2r2 ´ r ´ 1q19p1qp1q ´44r3 ` 112r2 ´ 56r ` 148,
´1660r3 ` 2572r2 ´ 2056r ` 3136
1600 r´4, 0,´2, 0s 19 p´ 12r3 ´ 1
2r2 ` 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 12r3 ´ 2r ´ 1
2q7pr
3 ´ r2 ´ r ´ 2q20p1q ´2436r3 ´ 3240r2 ´ 2688r ` 1045,
´49029102r3 ´ 65262564r2 ´ 54075240r `21032621
Marc Masdeu Non-archimedean constructions September 28th, 2015 25 / 34
Tables: quartic p2, 1q fields (II)|∆K | fKpxq NmpNq pDm c4pEq, c6pEq
2051 r1, 3,´1,´1s 15 pr3 ´ r2 ` 2q5p1qp´r ` 1q3 ´489r3 ` 1228r2 ´ 1242r ` 18,
46792r3 ´ 100917r2 ` 73440r ` 47160
2068 r1, 3,´2,´1s 7 pr ´ 2q7p1qp1q 26909497r3 ` 20141314r2 ´ 35624307r ´11296953,
247303058576r3 ´ 3168333376r2 ´
656295560992r ´ 182979737393
2068 r1, 3,´2,´1s 13 p´r3 ` r2 ` 2r ´ 1q13p1qp1q 34500648r3 ` 3814392r2 ´ 84122424r ´23737447,
´77408488074r3 ´ 354426093238r2 ´
415474468618r ´ 92161502469
2068 r1, 3,´2,´1s 56 pr ´ 2q7pr3 ´ 2r ` 1q8p1q ´3576591826r3 ´ 1882130113r2 `
6123537074r ` 1835712204,
321001991693952r3 ` 322520099276304r2 ´281263304453488r ´ 100176319060369
2068 r1, 3,´2,´1s 182 pr ´ 2q7p´r3 ` r2 ´ 2q26p1q ´1994707423r3 ´ 282234694r2 `
4755878517r ` 1346474783,
´8733155599162r3 ´ 54136988565986r2 ´
71594660083402r ´ 16347374680241
2092 r´2,´3, 1,´1s 8 prq2p1qpr ´ 1q4 ´3r3 ` 29r2 ´ r ` 75,
231r3 ´ 61r2 ` 497r ´ 287
2096 r2,´2,´2, 0s 28 pr3 ´ r ´ 1q7p1qpr3 ` r2 ´ 2q4 116r3 ´ 390r2 ` 402r ´ 94,
7354r3 ` 222r2 ´ 29620r ` 17640
2116 r´2, 0, 1,´1s 5 pr2 ` 1q5p1qp1q 129712r3 ` 31248r2 ` 168480r ` 209073,
´109612390r3´ 26402860r2´ 142375012r´176669575
2116 r´2, 0, 1,´1s 130 pr2 ` 1q5pr ` 2q26p1q 105064r3 ` 25312r2 ` 136464r ` 169353,
78278092r3 ` 18855232r2 ` 101675032r `126166043
2116 r´2, 0, 1,´1s 130 pr3 ´ r2 ` r ` 1q13pr3 ` rq10p1q 105064r3 ` 25312r2 ` 136464r ` 169353,
78278092r3 ` 18855232r2 ` 101675032r `126166043
2183 r´1, 1, 3,´2s 126 p´r3 ` 2r2 ´ 4rq7pr3 ´ 2r2 ` 4r ` 1q18p1q ´330539r3 ´ 223654r2 ` 72664r ` 52816,
421344240r3 ` 649688112r2 ´ 51218957r ´170790474
2191 r´1, 0, 3,´1s 70 p´r3 ´ 2r ´ 2q5p´2r3 ` r2 ´ 5r ´ 2q14p1q ´928r3 ` 6929r2 ´ 312r ´ 2120,
´885775r3 ` 1164640r2 ` 179150r ´ 336602
2191 r´1, 0, 3,´1s 80 p´r3 ´ 2r ´ 2q5p2q16p1q ´408r3 ` 2689r2 ´ 105r ´ 821,
120899r3 ` 70135r2 ´ 44492r ´ 24989
2243 r´1,´3,´1,´1s 75 pr ´ 1q5p´r3 ` 2r2 ´ r ` 2q15p1q 586900359r3 ` 694528587r2 ` 929522310r `
268803085,
63399246832324r3 ` 75025661482408r2 `
100410590521972r ` 29037147633615
2243 r´1,´3,´1,´1s 75 pr3 ´ r2 ´ 2r ´ 2q5p´r3 ` r2 ` 2r ` 3q15p1q 586900359r3 ` 694528587r2 ` 929522310r `
268803085,
63399246832324r3 ` 75025661482408r2 `
100410590521972r ` 29037147633615
2243 r´1,´3,´1,´1s 105 p´r3 ` 2r2 ` 2q7p´r3 ` 2r2 ´ r ` 2q15p1q 4336158r3`5131353r2`6867535r`1985981,
´22914354769r3 ´ 27116483373r2 ´
36291344215r ´ 10494880213
2243 r´1,´3,´1,´1s 105 pr ´ 1q5pr2 ´ r ` 1q21p1q 920025r3 ` 1088737r2 ` 1457115r ` 421377,
3942374598r3 ` 4665343442r2 `
6243862193r ` 1805625754
2284 r´4, 2, 2,´2s 22 p´r2 ` r ` 1q11p1qp12r3 ´ r2 ` 1q2 ´4322076r3 ` 3371584r2 ´ 4531104r ´
14171719,
´293858698818r3 ` 229234508344r2 ´
308070583688r ´ 963537590781
2327 r´2,´1,´1, 0s 48 pr2 ´ 1q3p2q16p1q 60947675662300r3 ` 95467421346487r2 `
88590894936957r ` 77819621400035,
1595218950381053851625r3 `
2498724287457442364789r2 `
2318740987988175420378r `
2036818157516553727423
2327 r´2,´1,´1, 0s 66 pr2 ´ 1q3p´r3 ` 2q22p1q 24654r3 ` 41044r2 ` 36631r ` 33971,
13602419r3 ` 21481224r2 ` 19830770r `17549287
2327 r´2,´1,´1, 0s 78 pr2 ´ 1q3pr3 ´ r2 ` rq26p1q 1632339r3`2556895r2`2372706r`2084241,
5442997756r3 ` 8525820467r2 `
7911705090r ` 6949764691
2443 r´1,´3, 0, 0s 63 pr3 ´ 2q3p´r ´ 2q21p1q 51601r3 ` 81980r2 ` 123695r ` 33695,
´38870055r3 ´ 59926714r2 ´ 92404714r ´25325630
2443 r´1,´3, 0, 0s 117 p´r3 ` r2 ´ r ` 2q13p´r2 ` 1q9p1q ´26624r3 ´ 78583r2 ` 147974r ` 56321,
41156101r3 ´ 906363r2 ´ 80062921r ´
24803969
2480 r´2,´2, 0, 0s 17 p´r3 ` r2 ` r ` 1q17p1qp1q 8r3 ´ 12r2 ´ 12r ` 17,
212r3 ` 628r2 ´ 818r ´ 887
2480 r´2,´2, 0, 0s 19 p´r2 ` r ´ 1q19p1qp1q ´648r3 ` 524r2 ´ 408r ` 1636,
29224r3 ´ 23272r2 ` 18616r ´ 73216
2608 r´2,´2,´2, 0s 50 p´r ` 1q5pr3 ´ r2 ´ rq10p1q ´18122952r3 ` 23309952r2 ` 6270652r `
28184369,
´178706675384r3 ` 229835084602r2 `
61821736238r ` 277904169213
2696 r1,´3, 0,´1s 24 pr3 ´ 2q3pr3 ´ 3q8p1q 25999152r3 ` 20125515r2 ` 35704342r ´
14654974,
´282591287516r3 ´ 218749239468r2 ´
388079405968r ` 159288610195
2816 r´1,´4,´2, 0s 15 pr2 ´ r ´ 1q5p1qpr3 ´ r2 ´ 2r ´ 1q3 134184108r3 ´ 165313100r2 ´ 203588440r ´
41893502,
2470282983044r3 ´ 3964870336170r2 ´
2128766125800r ´ 223343175430
2859 r´3, 3,´1,´1s 7 p´r3 ` r ´ 1q7p1qp1q ´4976r3 ` 12905r2 ´ 15523r ` 9529,
1469059r3´3794717r2`4539759r´2782843
3119 r´4,´3,´2,´1s 23 p 23r3 ´ r2 ´ 1
3r ´ 1
3q23p1qp1q 16743632r3 ` 25416768r2 ` 30512064r `
26598352,
´406345115512r3 ´ 616830291616r2 ´
740486023984r ´ 645505557528
3188 r2,´4, 1,´1s 24 p´r3 ` r2 ´ r ` 3q3p´r3 ` r2 ´ r ` 4q8p1q 2788172026368r3 ` 1423837175512r2 `
4939120830288r ´ 3691304019543,
´10952993228320557238r3 ´
5593370421245480720r2 ´
19402732864546458324r `
14500836945256233797
3216 r3, 0,´1,´2s 5 p´r ´ 1q5p1qp1q 16r3 ´ 40r2 ` 48r ´ 20,
104r3 ´ 376r2 ` 816r ´ 608
3271 r´1,´1, 3, 0s 110 pr3 ` r2 ` 3r ` 2q5p´r3 ` r2 ´ 2r ` 2q22p1q 228r3 ´ 115r2 ` 220r ` 132,
6359r3 ´ 2608r2 ´ 6398r ´ 1760
3275 r´9, 6, 2,´1s 19 p´ 19r3 ´ 2
9r2 ´ 8
9r ´ 7
3q19p1qp1q
233r3 ´ 20
3r2 ` 34
3r ´ 14,
4969r3 ´ 1690
9r2 ` 1979
9r ´ 263
33275 r´9, 6, 2,´1s 19 pr ´ 2q19p1qp1q 3r3 ` 30r2 ´ 27r ´ 10,
3329r3 ´ 632
9r2 ´ 7136
9r ` 2693
33284 r´2, 0,´1,´1s 6 pr ´ 1q3p1qprq2 2016r3 ` 1720r2 ` 1160r ` 2161,
´290488r3 ´ 248004r2 ´ 169132r ´ 313401
3407 r´3, 1,´2,´1s 84 p 12r3 ´ 2r ` 1
2q7pr
2 ´ rq12p1q28129013
2r3 ` 15057426r2 ` 3048856r `
407549212
,
´125734882980r3 ´ 134611455788r2 ´
27256382584r ´ 182171881573
3475 r´11, 8,´2,´1s 11 p´ 17r3 ´ 2
7r2 ´ 4
7r ` 1
7q11p1qp1q
617r3 ´ 214
7r2 ´ 351
7r ` 905
7,
´ 16327r3 ` 2420
7r2 ` 6940
7r ´ 10751
73475 r´11, 8,´2,´1s 11 p´rq11p1qp1q ´16r3 ´ 8r2 ` 24r ´ 95,
90087r3 ` 5948
7r2 ´ 8124
7r ` 59025
7
Marc Masdeu Non-archimedean constructions September 28th, 2015 26 / 34
Tables: quartic p2, 1q fields (III)|∆K | fKpxq NmpNq pDm c4pEq, c6pEq
3559 r´2,´1, 3,´2s 20 pr2 ´ 1q5p1qp´r2 ` r ´ 2q4 266r3 ´ 251r2 ` 481r ` 402,
´8721r3 ` 6359r2 ´ 17561r ´ 13594
3571 r3, 5,´5,´1s 45 pr2 ´ 3q3pr2 ` r ´ 5q15p1q ´247r3 ` 1481r2 ` 5186r ` 1954,
313052r3 ` 544864r2 ´ 374892r ´ 240173
3632 r2,´2, 0,´2s 13 pr3 ´ r2 ´ r ´ 1q13p1qp1q 110352r3 ` 24580r2 ` 54624r ´ 99300,
124669648r3 ` 27763200r2 ` 61709112r ´112178880
3632 r2,´2, 0,´2s 14 p´r ´ 1q7p1qp´rq2 2474r3 ` 522r2 ` 1234r ´ 2217,
523532r3 ` 116757r2 ` 258994r ´ 471065
3632 r2,´2, 0,´2s 26 pr3 ´ r2 ´ 3q13p1qp´rq2 10028r3 ` 2232r2 ` 4964r ´ 9023,
´3562482r3´793354r2´1763358r`3205557
3723 r´1, 3, 1,´1s 7 pr3 ´ r2 ` 2r ` 2q7p1qp1q 381r3 ´ 208r2 ´ 592r ` 201,
´9752r3 ´ 3598r2 ` 7307r ´ 1656
3723 r´1, 3, 1,´1s 17 pr ´ 2q17p1qp1q 168r3 ´ 1126r2 ´ 1303r ` 504,
´24313r3 ` 42209r2 ` 63347r ´ 22837
3775 r´11, 7, 0,´1s 19 p 38r3 ´ 1
4r2 ` 1
4r ` 3
8q19p1qp1q 36r3 ` 8r2 ` 32r ` 253,
´662r3 ´ 228r2 ´ 448r ´ 5011
3775 r´11, 7, 0,´1s 19 p 38r3 ´ 1
4r2 ` 1
4r ` 27
8q19p1qp1q ´17r3 ` 30r2 ´ 70r ` 8,
4892r3 ´ 607r2 ` 1283r ´ 1633
23888 r3,´6, 0,´2s 3 p 1
2r3 ´ 1
2r2 ´ 1
2r ´ 5
2q3p1qp1q 12362r3 ` 8406r2 ` 22518r ´ 13842,
7035016r3`4781484r2`12812832r´7875996
3899 r´3, 1, 2,´2s 23 pr3 ´ 2r2 ` r ` 1q23p1qp1q ´14r3 ` 14r2 ´ 25r ´ 21,
381r3 ´ 249r2 ` 364r ` 978
3967 r1, 5,´2,´1s 13 p 12r3 ´ 2r ` 1
2q13p1qp1q
33212r3 ` 1456r2 ´ 2668r ´ 1081
2,
´163448r3 ` 28056r2 ` 583644r ` 107371
3967 r1, 5,´2,´1s 17 p 12r3 ´ 2r ` 5
2q17p1qp1q ´ 3537
2r3 ´ 125r2 ` 4948r ` 1841
2,
´99064r3 ´ 306744r2 ´ 273576r ´ 41165
4108 r´2,´2, 0,´1s 52 pr2 ´ r ` 1q13p´r3 ` 2r2 ´ r ` 2q4p1q ´52r3 ` 56r2 ` 316r ` 177,
3676r3 ´ 2050r2 ´ 1438r ` 1283
4192 r´2,´2, 1, 0s 28 pr2 ` r ` 1q7pr2 ` r ` 2q4p1q 68388r3 ´ 97900r2 ´ 25440r ` 47889,
50048814r3 ´ 57380110r2 ´ 27657416r `22526745
4192 r´2,´2, 1, 0s 44 pr3 ` r ´ 1q11pr2 ` r ` 2q4p1q 993568r3 ´ 1182928r2 ´ 521264r ` 485673,
´2157501576r3 ` 964037714r2 `
2148667444r ` 353613881
4204 r´4,´2, 0, 0s 20 p´r ` 1q5p´rq4p1q 145360531282796r3 ` 161931312392192r2 ´390193058066092r ´ 440654493862007,
´1159392135670300645002r3 `
9949620873463783912066r2 ´
2497558503469317783050r ´
17611520739674724691341
4319 r2,´1,´4,´1s 42 prq2p´r3 ` 2r2 ` 3r ´ 1q21p1q 2626337501r3`4156522706r2`229413693r´
2033846625,
694511908654437r3`1099155960247844r2`60666438159866r ´ 537832894958445
4384 r´4, 0, 3,´2s 10 p 12r3 ´ r2 ` 5
2r ´ 2q5p1qp
12r3 ` 1
2r ` 2q2 ´39342r3 ` 91445r2 ` 10340r ´ 83032,
´8399230r3 ´ 58605841r2 ` 42062128r `73787052
4423 r1, 4,´3,´1s 50 p´r ` 2q5p´r2 ´ r ` 2q10p1q ´4642767r3 ´ 1724885r2 ` 13234188r `
2913911,
´19031399895r3 ´ 11910891879r2 `
44594523793r ` 10072542896
4423 r1, 4,´3,´1s 50 p´r ` 2q5p´r2 ´ r ` 2q10p1q 3516856r3`2151917r2´8338704r´1880324,
´9366159063r3 ` 477887546r2 `
34591729866r ` 7408649776
4564 r1,´5, 0,´1s 5 p 12r3 ` r ´ 3
2q5p1qp1q ´280r3 ` 240r2 ` 64r ` 1449,
10942r3 ´ 8954r2 ´ 1978r ´ 55513
4568 r´1,´3, 2,´1s 12 pr2 ` 2q3p1qpr2 ` 3q4 10845937505r3 ` 4588202505r2 `
28221044093r ` 7621698413,
´1760006389370257r3 ´
744542896235865r2 ´ 4579522784006957r ´1236798376628657
4652 r2, 5,´3,´1s 44 p 12r3 ´ 3
2r ` 2q11p´
12r3 ` 3
2r ´ 1q4p1q ´1938032413r3 ` 62742964314r2 `
143570326721r ` 41574563255,
14844318169935843r3 `
50626339684931473r2 `
49275897864569564r ` 11502761970012547
4663 r2,´5, 2,´1s 11 p´2r3 ` r2 ´ 3r ` 9q11p1qp1q 4296r3 ` 1705r2 ` 10968r ´ 6148,
´3722961r3 ´ 1477666r2 ´ 9510026r `
5330364
4775 r´9,´9, 2,´1s 11 p´ 512r3 ` 2
3r2 ´ 5
6r ` 13
4q11p1qp1q
3074r3 ´ 499r2 ´ 2771
2r ´ 2953
4,
´ 690643
r3 ` 1467853
r2 ´ 1827233
r ´ 87279
4775 r´9,´9, 2,´1s 11 p 16r3 ` 1
3r2 ` 1
3r ´ 1
2q11p1qp1q 247r3 ` 539r2 ´ 163r ´ 335,
´41241r3 ´ 13659r2 ´ 21597r ´ 27719
4832 r´2,´4,´1, 0s 17 p´r3 ` r2 ` 2r ` 5q17p1qp1q ´24r3 ` 17r2 ` 12r ` 82,
580r3 ´ 347r2 ´ 524r ´ 1770
4907 r´1,´4,´2,´1s 11 pr3 ´ r2 ´ 3r ´ 3q11p1qp1q ´191405r3 ´ 287504r2 ´ 336559r ´ 76491,
1214356660r3 ` 1824081112r2 `
2135314036r ` 485335595
4944 r´1,´4,´1, 0s 17 pr3 ´ 4q17p1qp1q 316049736r3 ` 586633069r2 ` 772824316r `170272429,
´23749113529508r3 ´ 44081717952580r2 ´58072797643568r ´ 12794882314805
4979 r1,´3,´1,´1s 13 p´r3 ` r2 ` r ` 1q13p1qp1q 32r3 ´ 128r2 ` 144r ´ 32,
´1464r3 ` 3856r2 ´ 1824r ` 240
Marc Masdeu Non-archimedean constructions September 28th, 2015 27 / 34
Surfaces (joint with X. Guitart and H. Sengun)
This all generalizes to higer dimensional (e.g. 2-dim’l) components.The pairing
H1pΓ0pNq,Zq ˆH1pΓ0pNq,Zq Ñ Cˆpyields, by taking bases of the irreducible factors, a lattice Λ Ă pCˆp 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 aMumford–Schottky genus 2 curve.
§ Lp P TbZ Qp.§ Corresponding to a hyperelliptic curve X with JacX “ A.
We can use formulas of Teitelbaum (1988) to recover a Weierstrassequation 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
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 JacXp15 ¨ 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
ˆ
ş
θ1φ1
ş
θ2φ1
ş
θ1φ2
ş
θ2φ2
˙
“
ˆ
A BB D
˙
.
A “ 3 ¨ 114 ` 3 ¨ 115 ` 4 ¨ 116 ` ¨ ¨ ¨ `Op1124q
B “ 4` 7 ¨ 11` 7 ¨ 112 ` 4 ¨ 113 ` ¨ ¨ ¨ `Op1120q
D “ 9 ¨ 114 ` 9 ¨ 115 ` 8 ¨ 116 ` 9 ¨ 117 ` ¨ ¨ ¨ `Op1121q
Marc Masdeu Non-archimedean constructions September 28th, 2015 29 / 34
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 TbQp.
We recover the Igusa–Clebsch invariants
pI2 : I4 : I6 : I10q “ p2584 : ´75356 : 37541976 : 2123453113q
Mestre’s algorithm (together with model reduction) yields thehyperelliptic curve
y2 “ ´x6 ` 4x4 ´ 10x3 ` 16x2 ´ 9
After twisting (by ´1 in this case) we get a curve whose first fewEuler factors match with those obtained by the T-action on J .
Marc Masdeu Non-archimedean constructions September 28th, 2015 30 / 34
Example surface over cubic p1, 1q field
Let F “ Qprq, where r3 ´ r2 ` 2r ´ 3 “ 0.Let p7 “ pr
2 ´ r ` 1q.Let BF be the totally definite quaternion algebra of disc. q3 “ prq.
JacXBpΓ0ppqq 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 ` ¨ ¨ ¨ `Op7300q
` p72 ` 6 ¨ 74 ` 2 ¨ 75 ` 2 ¨ 76 ` ¨ ¨ ¨ `Op7300qq ¨ T2 P TbQ7.
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
Beyond degree 1
F “ quartic totally-complex field, N “ p (for simplicity).
In this setting Γ “ SL2pOF r1psq, which acts on H23.
The relevant groups are now H2pΓ,Div0 Hpq and H2pΓ,Ω1Hpq.
§ H2pSL2pOF q,DivHpq & H2pΓ0ppq,Dq – H2pSL2pOF q, coIndDq.
§ 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
Sharblies and overconvergent leezarbs
Have a short exact sequence of GL2pF q-modules
0 Ñ ∆0 Ñ DivP1pF qdegÑ ZÑ 0, ∆0 “ Div0 P1pF q
Applying the functors p´q bV or Homp´,W q and taking homologyand cohomology yields connecting homomorphims
H2pΓ0ppq, V q
δ
ˆ H2pΓ0ppq,W qX // H0pΓ0ppq, V bW q
H1pΓ0ppq,∆0 bV q ˆ H1pΓ0ppq,Homp∆0,W qq
X //
δ
OO
H0pΓ0ppq, Xq
ev˚
OO
X “ ∆0 bV bHomp∆0,W qevÑ 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,W qq.
Marc Masdeu Non-archimedean constructions September 28th, 2015 33 / 34
Sharblies and overconvergent leezarbs (II)
Reduced to H1pΓ0ppq,∆0 bV q and H1pΓ0ppq,Homp∆0,W qq.
Sharblies were invented by Szczarba and Lee (szczarb-lee) tocompute 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,W qq, we introduce a dualversion of sharblies, the leezarbs.
§ Leezarbs are an acyclic resolution of Homp∆0,W q.§ 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
Thank you !and
Congratulations to the people of CatalunyaWho have realized that, in the course of human events,
it has become necessary to dissolve the political bandswhich have connected them with Spain.
Marc Masdeu Non-archimedean constructions September 28th, 2015 34 / 34