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```EULER SYSTEMS
DAVID LOEFFLER / SARAH ZERBES
These are the notes from my talk at the Harris conference at MSRI in December 2014.
1. A conjecture of Perrin-Riou
Let M be a motive over Q which is not the trivial motive. Denote by L(M, s) the L-function of M .
Assumption 1.1. L(M, s) has analytic continuation to C and satisfies a functional equation.
Denote by Mp the p-adic realisation of M , so Mp is a finite-dimensional Qp -vector space with a continuous
action of GQ = Gal(Q/Q). Denote by Hf1 (Q, Mp ) the Bloch-Kato Selmer group of Mp , which is a
subgroup of H 1 (Q, Mp ), cut out by local conditions
• the unramified local condition at ` 6= p;
• the Hf1 -local condition at p.
The Bloch-Kato conjecture predicts the size of Hf1 (Q, Mp∗ (1)) in terms of the L-function of M :
Conjecture 1.2. (Bloch-Kato)
ords=0 L(M, s) = dim Hf1 (Q, Mp∗ (1)) − dim H 0 (Q, Mp∗ (1)),
together with an explicit formula for the leading term up to a p-adic unit.
Definition. The motive M is effective if all Hodge numbers (= steps in the Hodge filtration of MdR )
are ≥ 0.
Example. The trivial motive M = Q and the motive M (f ) attached to a modular form of weight ≥ 2
are effective, but M = Q(1) is not effective.
Fact 1.3. If M is effective and M 6= Q, then
• the Hf1 =local condition at p is the relaxed local condition unless the local L-factor vanishes;
• ords=0 L(M, s) = dim(MBetti )c=1 , where c denotes complex conjugation
⇒ the Bloch-Kato conjecture predicts that
dim Hf1 (Q, Mp∗ (1)) = dim(MBetti )c=1 =: d+ .
Euler systems are a tool for proving special cases of the Bloch-Kato conjecture for effective motives:
Conjecture 1.4. (Perrin-Riou [PR98]) If M is effective, there exists a non-trivial system of elements
(zm )m≥1 ,
+
zm ∈
d
^
H 1 Q(µm )+ , Mp∗ (1) ,
such that
Q(µ )+
coresQ(µm`
+ (zm` )
m)
(
zm
=
P` (σ`−1 )zm
if `|m or Mp∗ (1) is ramified at `
otherwise,
where P` (X) = det(1 − Xσ`−1 |Mp ) and σ`−1 denotes the arithmetic Frobenius at `.
Definition. Such a system of elements (zm )m≥1 is called a rank d+ Euler system for Mp .
Remark. If M is defined over a number field K, then an Euler system for M should have classes over all
the abelian extensions of K.
1
Theorem 1.5. (Kolyvagin, Rubin [Rub00], Perrin-Riou [PR98]) If such a rank d+ Euler system exists
and z1 6= 0, then (under some technical hypotheses) the Selmer group Hf1 (Q, Mp∗ (1)) is d+ -dimensional,
as prediced by the Bloch-Kato conjecture.
Remark. In proving this theorem, it is very important that the Hf1 -local condition at p is the relaxed
one.
Examples. (rank d+ = 1 Euler systems)
• M = Q (cyclotomic units)
• M = M (f ), where f is a modular form of weight ≥ 2 (Kato’s Euler system, see [Kat04])
• M = Q over an imaginary quadratic field (eliptic units)
Remark. There exist other Euler systems attached to self-dual motives (i.e. M ∼
= M ∗ (1))) satisfying a
Problem. There are no non-trivial examples for d+ > 1! (Unless one assumes the ‘prectic conjecture’ of
Nekovar-Scholl, which gves access to certian settings related to totally real fields.)
2. A new Euler system
Theorem 2.1. (Lei-LZ [LLZ14], Kings-LZ [KLZ14]) Let f, g be modular forms of weights k+2, k 0 +2 ≥ 2,
and let 0 ≤ j ≤ min{k, k 0 }. Then there exists a rank 1 Euler system for M = M (f ) ⊗ M (g)(1 + j),
related to the p-adic L-value Lp (f, g, 1 + j).
The existence of this Euler system does not fit the setting of Perrin-Riou’s conjecture:
• d+ (M ) = 2, but M is not effective: its Hodge numbers are
−1 − j,
, k 0 − j,
k − j,
k + k 0 + 1 − j;
• ords=0 L(M, s) = ords=1+j L(f, g, s) = 1;
• the Hf1 -local condition at p is not the relaxed one;
• the Euler system classes take values in the Bloch-Kato Selmer group.
We therefore need a generalisation of Perrin-Riou’s conjecture for motives which are not effective. We
first generalize the notion of effective:
Definition. Let r ≥ 0. Then M is r-critical if the Archimedean Γ-factor L∞ (M, s) has a pole at s = 0
of order r, and L∞ (M ∗ (1), 0) 6= ∞.
Remarks.
(1) 0-critical is precisely Deligne’s definition of critical;
(2) if M is r-critical, then ords=0 L(M, s) ≥ r (and this lower bound shold be sharp);
(3) M is r-critical if and only if d+ − r Hodge numbers are < 0
⇒ M is effective if and only if it is d+ -critical.
Conjecture 2.2. If M is r-critical, there exists a non-trivial rank r Euler system (zm )m≥1 ,
zm ∈
r
^
Hf1 Q(µm )+ , Mp∗ (1) .
Note. The Euler system in Theorem 2.1 is an example of an Euler system for a 1-critical motive.
Remarks.
• Conjecture 2.2 reduces to Perrin-Riou’s conjecture when r = d+ . (Note that in this
case the Hf1 -local condition is relaxed, so the condition that the Euler system take values in
Hf1 Q(µm )+ , Mp∗ (1) is pretty much automatic.)
• A rank 0 Euler system should be thought of as a p-adic L-function. I will elaborate on this later.
Theorem 2.3. (LZ) If M is 1-critical and (zm )m≥1 is a rank 1 Euler system with z1 6= 0, then (under
some technical hypotheses) Hf1 (Q, Mp∗ (1)) is 1-dimensional, as predicted by the Bloch-Kato conjecture.
Remark. For proving this theorem, we need to adapt the Euler system machine to take into account the
non-relaxed local condition at p.
Here are some examples of 1-critical motives:
2
(1) M = M (f ) ⊗ M (g)(1 + j), where f, g are modular forms of weights k + 2, k 0 + 2 ≥ 2 and
0 ≤ j ≤ min{k, k 0 };
(2) M = MAsai (f )(1 + j), where f is a quadratic Hilbert modular form of weights (k + 2, k 0 + 2) and
0 ≤ j ≤ min{k, k 0 };
(3) M = MSpin (F )(1 + j), where F is a genus 2 Siegel modular form of weights (k + 3, k 0 + 3),
k ≥ k 0 ≥ 0 and 0 ≤ j ≤ k 0 ;
(4) M (1), where M ⊂ h2 (Sh(U (2, 1))) is a rank 3 motive over an imaginary quadratic field.
Remark. There are lots more examples: if M is any motive with distinct Hodge numbers, then some
twist of it will be 1-critical.
For the examples above, Conjecture 2.2 predicts the existence of a rank
 1 Euler system: case (1) is
Antonio Lei

Theorem 2.1, and cases (2), (3) and (4) are joint work in progress with Francesco Lemma .


Chris Skinner
In other words, we can construct some exmaples of Euler systems for 1-critical motives. However, the
interesting cases are those twists of the motives which are 0-critical. We can get at those using p-adic
deformation.
3. Euler systems in p-adic families
Definition. Let A be a complete local Zp -algebra, and let X = Spf(A). A family of motivic Galois
representations is a finite free A-module V with an A-linear action of GQ such that for a Zariski dense
set Xcl ⊂ X, we have
for some motive Mx .
Vx ∼
= Mx,p
The key example is that of the cyclotomic deformation of a motivic Galois representation:
Example. Let A = Λ(Z×
p ) and V = Mp ⊗ A, where GQ acts on A via multiplication by the canonical
×
,→
A
. Note that V specializes to Mp (j) for all j.
character GQ → Z×
p
In order to vary a rank r Euler system in a p-adic family, we need to make an auxiliary choice called an
r-refinement.
Definition. An r-refinement of V is an A-direct summand W ⊂ V which is GQp -stable together with a
Zariski-dense set of points Xcl,W ⊂ Xcl , such that ∀x ∈ Xcl,W ,
• Mx is r-critical,
• Wx has all Hodge-Tate weights ≥ 1, and Vx /Wx has all Hodge-Tate weights ≤ 0 (Panchishkin
condition).
Conjecture 3.1. Let V be a family of motivic Galois representations, and let W ⊂ V be an r-refinement.
Then there exists a non-trivial rank r Euler system (zm )m≥1 ,
zm ∈
r
^
H 1 Q(µm )+ , V
such that
• locp (zm ) ∈ im H 1 (Q(µm )+ , W ) → H 1 (Q(µm )+ , V ) ,
• ∀x ∈ Xcl,W , the specialisation of the Euler system at x agrees with the Euler system from
Conjecture 2.2.
Remark. ∀x ∈ Xcl,W (generically), one has
Hf1 Q(µm )+ , Vx = im H 1 Q(µm )+ , Wx → H 1 Q(µm )+ , Vx ,
Vr 1
so zm,x ∈
Hf as required.
3
Example: Rankin-Selberg convolutions
Let f , g be Hida families. Then there exists a 3-parameter of GQ -representations
×
ˆ (g)∗ ⊗Λ(Z
ˆ
V (f )∗ ⊗V
p)
(two Hida parameters and one cyclotomic parameter) interpolating Mp (fk )∗ ⊗Mp (gk0 )∗ (1+j) for varying
k, k 0 , j. We have the following refinements:
2-refinement:
1-refinement:
0-refinements:
W2 = {0},
×
ˆ + V (g)∗ ⊗Λ(Z
ˆ
W1 = F + V (f )∗ ⊗F
p ),
+
∗ˆ
∗ˆ
×
W0a = F V (f ) ⊗V (g) ⊗Λ(Zp ),
×
ˆ + V (g)∗ ⊗Λ(Z
ˆ
W0b = V (f )∗ ⊗F
p ),
Xcl,W2 = {k, k 0 ≥ 0, j ≤ −1}
Xcl,W1 = {0 ≤ j ≤ min{k, k 0 }}
Xcl,W0a = {k 0 + 1 ≤ j ≤ k}
Xcl,W0b = {k + 1 ≤ j ≤ k 0 }
Here, F + V (f ) denotes the rank 1 submodule as constructed by Wiles in [Wil88].
Theorem 3.2. (Kings-LZ, [KLZ14])
(1) Our Euler systems from Theorem 2.1 for M (fk )∗ ⊗ M (gk0 )∗ (1 + j) interpolate along W1 ;
(2) (Explicit
( reciprocity law) There exist rank-lowering operators corresponding to the inclusions
W0a
W1 ,→
mapping the rank 1 Euler system to Hida’s two 2-variable p-adic L-functions;
W0b
(3) if a rank 2 Euler system exist, then it maps to our Euler system under the rank-lowering operator.
Remarks.
• The rank-lowering operators are multi-variable analogues of Perrin-Riou’s regulator
map, as constructed in [LZ14].
• (KLZ, in progress) The Hida families can be replaced by Coleman families. In this case, the
subrepresentations are replaced by sub-(ϕ, Γ)-modules over the Roba ring.
In general, we expect there to be a hierachy of Euler systems, governed by inclusion of refinements, and
related to each other via rank-lowering operators: if W 0 and W are r0 -, resp. r-refinements of V , then
Conjecture 3.1 predicts the existence of rank r0 , resp. rank r Euler systems. If W 0 ⊂ W , then these
Euler systems should be related to each other via a rank-lowering operator
r
^
0
H
1
+
Q(µm ) , V
-
r
^
H 1 Q(µm )+ , V .
References
[Kat04] Kazuya Kato, P -adic Hodge theory and values of zeta functions of modular forms, Ast´
erisque 295 (2004), ix,
117–290, Cohomologies p-adiques et applications arithm´
etiques. III. MR 2104361
[KLZ14] Guido Kings, David Loeffler, and Sarah Zerbes, Rankin-Selberg Euler systems and p-adic interpolation, preprint,
2014.
[LLZ14] Antonio Lei, David Loeffler, and Sarah Livia Zerbes, Euler systems for Rankin–Selberg convolutions of modular
forms, Ann. of Math. 180 (2014), no. 2, 653–771.
[LZ14] David Loeffler and Sarah Livia Zerbes, Iwasawa theory and p-adic L-functions for Z2p -extensions, Int. J. Number
Theory (to appear) (2014).