Arithmetic of Euler Systems

100 Pages·2015·1.59 MB·English

Arithmetic of Euler Systems
August 2015
Centro de Ciencias Pedro Pascual
Benasque, Spain
Organizers
X. Guitart and M. Masdeu
Scientific Advisers
D. Loeffler and S. Zerbes

Foreword
The main aim of the workshop was to give an introduction to the recent developments in the
area, in particular the work of Bertolini-Darmon-Rotger, Lei-Loeffler-Zerbes and Kings-Loeffler-
Zerbes on Euler systems for Rankin convolutions, at a level accessible to graduate students and
other younger researchers.
The instructional part of the workshop consisted of 12 lectures giving an account of the above
works, together with a selection of more advanced talks on related areas of current research.
Students and postdocs volunteered to give talks themselves in the introductory lecture series;
which were allocated and coordinated by David Loeffler and Sarah Zerbes, the scientific advisers.
iii
Acknowledgements
The organizers wish to thank the participants for all the work they put in preparing the talks
and typesetting the first version of these notes.
WewishtothankalsotheCentrodeCienciasPedroPascual foritshospitalityandforproviding
us with exceptionally inspiring setting.
Finally, the organizers wish to thank Joan-Carles Lario for his help and guidance, without
which this workshop wouldn’t have been possible.
iv
v
Lectures
1 Introduction to Modular Curves 2
Yukako Kezuka
2 Hida Theory 5
Chris Williams
3 Siegel units and Eisenstein classes 15
Antonio Cauchi
4 Definition of the global classes 26
Guhan Venkat
5 Compatibility in p-adic families 29
Francesc Fit´e
6 Norm compatibility relations 37
Vivek Pal
7 p-adic Hodge theory and Bloch–Kato theory 47
Bruno Joyal
8 p-adic Eichler–Shimura Isomorphisms 57
Yitao Wu
9 (Modified) syntomic and FP cohomology 61
Lennart Gehrmann
10 Evaluation of the regulators 66
Netan Dogra
11 Proofs of the explicit reciprocity laws 77
Giovanni Rosso
12 Applications: bounding Selmer and Sha 84
Jack Lamplugh
vi
Arithmetic of Euler Systems
1
Introduction to Modular Curves
Talk by Yukako Kezuka
[email protected]
Notes by Alex Torzewski
[email protected]
Let H = {τ ∈ C: Im(τ) > 0} denote the complex upper half plane and Γ ⊆ SL (Z) be a
2
conguence subgroup. By the modular curve associated to Γ we refer to the quotient Γ\H where
the action of SL (Z) on H is given by linear fractional transformations.
2
For τ ∈H, let Λ =Z+Zτ and E be the elliptic curve C/Λ . If Γ=Γ (N), then Γ (N)\H
τ τ τ 1 1
parametrises elliptic curves with a marked point of exact order N. Explicitly there is a bijection
Γ (N)·τ ↔[E ,P],
1 τ
betweencosetsandequivalenceclassesofpairs[E ,P]wherewemayassumeP = 1 +Λ ∈E [N].
τ N τ τ
We will rephrase the definition of a pair [E ,P] so that it makes sense with C replaced by any
τ
scheme S.
Definition 1.1. Let S be any scheme. Then an elliptic curve over S is a scheme E with a proper
flatmorphismπ: E →S whosefibresaresmoothgenus1curveswithachoiceofsectionO: S →E.
Definition 1.2. Let Y (N) be the smooth Z[1/N]-scheme representing the functor on Z[1/N]-
1
schemes
 
Isomorphism classes of pairs (E,P)
 
F: S (cid:55)→ where E is an elliptic curve /S and .
 P ∈E(S) a point of exact order N. 
By “Y (N) represents F”, we mean F(·) is isormorphic to Hom(·,Y (N)) as functors. If S =
1 1
C there is a natural bijection φ: Γ (N)\H → Y (N)(C) which is an analytic isomorphism and
1 1
(Y (N),φ) is a model for Γ (N)\H.
1 1
Remark 1.3. [LLZ14, 2.1.4] The cusp Γ (N)·∞ of Y (N)(C) is usually not defined over Q[[q]] but
1 1
rather over Q(µN)[[q]] since it corresponds to the pair (Gm/qZ,ζN) where ζN =e2Nπi.
ThisleadstothefollowingalternativedefinitionofY (N). AchoiceofP ∈E(S)ofexactorder
1
N amounts to giving a closed immersion
ι: (Z/NZ) →E
S
of group schemes, where (Z/NZ) denotes the constant group scheme of Z/NZ over S. Similarly
S
wecanuseamodelforY (N)(C)whichparameterisespairs(E,ι),whereι: (µ ) (cid:44)→E isaclosed
1 N S
immersion. The corresponding smooth scheme is denoted Y (N), and thus we obtain a model
µ
(Y (N),φ ) for Y (N)(C). Here φ is defined by τ (cid:55)→ (E ,ι ) for τ ∈ H and ι denotes the
µ N 1 µ τ τ τ
embedding defined by ι(ζ )= 1 +Λ . The cusp Γ (N) is defined over Q[[q]] with respect to this
N N τ 1
model.
Definition 1.4. Let L ≥ 3 and let Y(L) be the smooth Z[1/L]-scheme representing the functor
sending,
(cid:26) (cid:27)
E is an elliptic curve /S
S (cid:55)→ (E,e ,e )/∼ : .
1 2 and e ,e generate E[L].
1 2
There is a left action of GL (Z/LZ)/{±1} on Y(L) given by the action on e ,e given by,
2 1 2
(cid:18) e(cid:48) (cid:19) (cid:18) a b (cid:19) (cid:18) e (cid:19)
1 = · 1 .
e(cid:48) c d e
2 2
2
Yukako Kezuka Introduction to Modular Curves
Remark 1.5. If S =C we have an isomorphism of analytic spaces
(Z/LZ)××Γ(L)\H→∼ Y(L)(C)
(cid:18) (cid:19)
a 0
(a,τ)(cid:55)→ ·ν(τ),
0 1
where ν is the canonical map ν: H → Y(L)(C),τ (cid:55)→ (E ,τ/L,1/L). This in particular tells us
τ
that Y(L) is not geometrically connected.
Definition 1.6. Let L≥3, M,N ≥1 and M,N | L. Set Y(M,N)=G \Y(L) where
M,N
(cid:26)(cid:18) (cid:19) (cid:27)
a b (a,b)≡(1,0) (mod M),
G = ∈GL (Z/LZ) : .
M,N c d 2 (c,d)≡(0,1) (mod N).
If M +N ≥ 5, Y(M,N) represents the functor of triples (E,e ,e ) where e has order M, e
1 2 1 2
order N and together they generate a subgroup of order MN. In order to define Hecke operators
on K (Y(M,N)) and ther duals, we need also:
2
Definition 1.7. Let A ≥ 1,L ≥ 3, M and N s.t. M | L and AN | L. Define Y(M,N(A)) to be
the quotient of Y(L) by the subgroup G ≤GL (Z/LZ) given by
M,N(A) 2
(cid:26)(cid:18) (cid:19) (cid:27)
a b a≡1 (mod M), b≡0 (mod M)
G = ∈GL (Z/LZ) : .
M,N(A) c d 2 c≡0 (mod NA), d≡1 (mod N).
Similarly define Y(M(A),N) using G given by
M(A),N
(cid:26)(cid:18) (cid:19) (cid:27)
a b a≡1 (mod M), b≡0 (mod MA)
G = ∈GL (Z/LZ) : .
M(A),N c d 2 c≡0 (mod N), d≡1 (mod N).
The Z[1/L]-scheme Y(M,N(A)) (resp. Y(M(A),N) represents the functor which sends
 
C is a cyclic subgroup of order NA (resp. MA),
 
s(cid:55)→ (E,e ,e ,C)/∼ : e ∈C, (resp. e ∈C) and the product (cid:104)e (cid:105)C .
1 2 2 1 1
 (resp. (cid:104)e (cid:105)C) as subgroups of E is a direct sum. 
2
There is an isomorphism
φ : Y(M,N(A))→Y(M(A),N),
A
(E,e ,e ,C)(cid:55)→(E(cid:48),e(cid:48),e(cid:48),C(cid:48)).
1 2 1 2
Given by letting E(cid:48) be the quotient of E by NC, a cyclic subgroup of order A. Then e(cid:48) is defined
1
to be the image of e which by the disjointness of (cid:104)e (cid:105) and C is necessarily of order M. Define e(cid:48)
1 1 2
to be the image of A−1e in E(cid:48). There is necessarily such a point as C is cyclic of order NA and
2
e(cid:48) is independent of this choice. Lastly, set C(cid:48) to be the image of A−1(cid:104)e (cid:105) in E(cid:48). This is a cyclic
2 1
subgroup of E(cid:48) of order MA.
1 Hecke Operators
TheHeckeoperatorsT(n)onK (Y(M,N))andtheirdualsT(cid:48)(n)onH1(Y(M,N)(C),Z)forn≥1,
2
(n,M)=1 are defined as follows:
• For n=1, T(1)=T(cid:48)(1)=id,
• For n = p, p (cid:45) M let π : Y(M,N(p)) → Y(M,N),π : Y(M(p),N) → Y(M,N) be the
1 2
projections defined by forgetting C. Then set T(p)=(π ) ◦(φ−1)∗◦(π )∗ and T(cid:48)(p) to be
2 ∗ p 1
(π ) ◦(φ )∗◦(π )∗.
1 ∗ p 2
• For n=pe, p(cid:45)M (and e≥2) we set,

T(p)e if p | N
 (cid:32) (cid:33)∗
T(pe)= 1 0 .
T(p)T(pe−1)+T(pe−2) 0p p p if p(cid:45)N
For T(cid:48)(pe) the formula is identical in T(cid:48).
3
Yukako Kezuka Introduction to Modular Curves
• If n = (cid:81) pe(p), where e(p) ≥ 0 and p ranges over all prime numbers not dividing M, we
p
define T(n) and T(cid:48)(n) multiplicatively using the above definitions.
Theorem 1.8. If p is a prime p(cid:45)MN, then Y(M,N) has a smooth model over Z .
p
Proof. PickL∈NsuchthatM,N |L(cid:32) . SinceY(M,N)isaquotientofY(L)=Y(L,L)itisenough
to check that Y(L) has a smooth model over Z[1/L], so if we take K = lcm(M,N) then L is
invertible in Z . The functional criterion for smoothness shows that Y (L) is smooth over Z[1/L]
p 1
if and only if for all local Z[1/L]-algebras A and nilpotent ideals I, the map Y(L)(A)→Y(L)(A(cid:48))
is surjective, where A(cid:48) =A/I.
Now,letAbealocalZ[1/L]-algebraandI ⊆Aanilpotentideal,andwriteA(cid:48) forthequotient.
Take (E(cid:48),e(cid:48),e(cid:48))∈Y(L)(A(cid:48)), the A(cid:48) valued points of the modular curve. Let E/A be any lifting of
1 2
E(cid:48)/A(cid:48) obtained by lifting the coefficients to A. Note ∆(E) ∈ A× as its image in A(cid:48) is ∆(E(cid:48)) and
thus a unit.
It remains to check that there exist lifts of e(cid:48) and e(cid:48) to points of E[L]. This is equivalent to
1 2
checking that E[L] is smooth. But [L]:E →E is smooth and E[L] is obtained by composing [L]
with the structure map E →SpecA which is also smooth. The composition of smooth morphisms
is smooth, and so its kernel E[L] → SpecA is also. Hence we have a lift (E,e ,e ) ∈ Y(L)(A) of
1 2
(E(cid:48),e(cid:48),e(cid:48)) as required.
1 2
4