Analytic p-adic L-functions for GL2: a summary
David Hansen
May 11, 2013
One of the most subtle and important invariants of a modular form f is its L-function L(s, f ). For
holomorphic modular forms, the basic analytic properties of their L-functions were established by Hecke,
with Maass treating the real-analytic forms; their construction gave L(s, f ) as a Mellin transform of f . In
the early 70s, the existence of p-adic L-functions associated with classical modular forms emerged. These
functions were constructed and characterized in varying degrees of generality by Mazur-Swinnerton-Dyer,
Manin, Višik, Amice-Vélu, Mazur-Tate-Teiltelbaum, Stevens, Stevens-Pollack and Bellaïche (MSD74, Man73,
AV75, Viš76, MTT86, Ste94, PS11, Bel12). In the last twenty years, these objects have come to play an
increasingly important role in modern number theory, especially as their role in relation to Iwasawa theory
and eigenvarieties has become clear.
For GL2 over an arbitrary number field, where the construction of classical L-functions has been understood since the appearance of Jacquet and Langlands’s seminal book, this situation is far more fragmentary.
Manin, Panchishkin, Kurcanov, Haran, Ash-Ginzburg, Januszewski, and Dimitrov, among others, have given
constructions of p-adic L-functions associated with cohomological cuspidal automorphic representations of
GL2 /F under various more or less restrictive hypothesis.
In the article (Han13) we give a canonical construction of p-adic L-functions associated with cohomological
cuspidal automorphic representations of GL2 (AF ). The L-functions we construct enjoy good interpolation
and growth properties, and they deform naturally into many-variable functions over eigenvarieties. Our
construction is very much a p-adic analogue of the Hecke-Jacquet-Langlands integral. Moreover, we make
almost no explicit reference to the fine geometry of locally symmetric spaces or the theory of modular symbols;
every map we use admits a succinct adelic description. In this note we explain the main results of (Han13).
Let F/Q be a number field of degree d = r1 + 2r2 with ring of integers OF and discriminant ∆F . Fix a
∼
rational prime p, algebraic closures Qp and Q ⊂ C, and an isomorphism ι : C → Qp . Write Σ for the set of
embeddings σ : F #→ C. The map
∼
Σ = Hom(F, C) → Hom(F, Qp )
σ #→ ι ◦ σ
!
determines a natural partition Σ = v|p Σv with σ ∈ Σv if ι ◦ σ induces the valuation v; we write σ|v as
shorthand for σ ∈ Σv . Let L ⊂ Qp be the finite extension of Qp generated by (i ◦ σ)(F ) for all σ ∈ Σ.
Let π be a cohomological cuspidal automorphic representation of GL2 (AF ) of cohomological weight (κ, w),
nebentype χπ and conductor n. Here w is an integer, and κ = (κσ )σ∈Σ is a d-tuple of nonnegative integers
with κσ ≡ w mod 2 and κc◦σ = κσ ; π has weight (κ, w) if the Langlands parameter of πσ satisfies
%
"# $
# $κσ +1
κσ +1
w
w
z
z
(zz)− 2 ,
(zz)− 2
φ(πσ )|C× = diag
|z|
|z|
for each archimedian place. (When π is associated with a modular elliptic curve, we have κσ = 0 for all
σ ∈ Σ.) To simplify matters in this introductory discussion, we assume n + (p) = OF . We set κ+ = supσ∈Σ κσ
and κ− = inf σ∈Σ κσ . We normalize the L-function of π such that it admits an Euler product
&
&
1
...
L(s, π) =
1 − av (π)(Nv)−s + χπ ((v )(Nv)w+1−2s
v|n
v!n
1
and a functional equation under s #→ w + 2 − s. By a fundamental theorem of Shimura, Harder, and Hida, the
numbers av (π) and χπ ((v ) generate a finite algebraic extension Q(π)/Q. Furthermore, there is a collection
×
of 2r1 nonzero complex numbers Ω$π , indexed by characters ) : π0 (F∞
) → {±1} and each well-defined up to
×
multiplication by an element of Q(π) , such that
idj
−
Λ(j, π ⊗ ψ)
∈ Q(π, ψ)
τ (ψ)Ω$π
−
for every critical integer w−κ
+ 1 ≤ j ≤ w+κ
+ 1 and every finite-order Hecke character ψ of F . Here
2
2
Λ(s, π) is the completed L-function of L(s, π), τ is a Gauss sum, ) = sgn(ψNj−1
F/Q ), and Q(π, ψ) is the field
generated by adjoining the values of ψ to Q(π).
To interpolate these L-values p-adically, we need to choose a refinement of π. This is simply a choice, for
each v|p, of a nonzero root αv of the Hecke polynomial X 2 − av (π)X + χπ ((v )(Nv)w+1 . We write (π, α) to
denote the data of π together with a chosen refinement α = (αv )v|p . We will define critical and noncritical
refinements below (Definition 1.2). If gp denotes the number of places of F lying over p, then π has at most
2gp distinct refinements, and we say π is regular if it has exactly 2gp distinct refinements; this is equivalent
to the vth Hecke polynomial having nonzero discriminant for each v|p.1 Let Q(π, α) be the field generated
by adjoining each αv to Q(π).
Let GF be the rigid analytic space over Qp whose R-points, for R a Noetherian commutative Qp -Banach
×
algebra, parametrize p-adically continuous characters A×
→ R× unramified away from the primes diF /F
viding p and ∞. Explicitly, set
#
$×
'
(p)
F
·
F
·
O
ΓF = A×
/
∞,+
F
F
which fits into a short exact sequence
×
1 → (OF ⊗Z Zp )× /OF,+
→ ΓF → Cl+
F → 1.
Then GF is the !
rigid generic fiber of Spf (Zp [[ΓF ]], m) (here m is any ideal of definition). There is a natural
$
partition GF =
"
× GF .
$∈π0 (F∞ )
Theorem 1.1. Let (π, α) be a non-critically refined cohomological representation of weight (κ, w), and let
E be the compositum of L and Qp (π, α). Then there exists a rigid analytic function Lp (π, α) ∈ O(GF ) ⊗Qp E
satisfying the following properties:
i) Canonicity: The restriction L$p (π, α) = Lp (π, α)|GF$ is canonically defined up to multiplication by an
element of E × .
ii) Interpolation: For all χ ∈ GF (Qp ) of the form ψ · Nj with ψ of finite order and
we have
Λ(j + 1, π ⊗ ψ −1 )
Lp (π, α)(ψ · Nj ) = ∆jF idj cp (π, α, j, ψ)
,
τ (ψ −1 )Ω$π
(
where Ω$π is a suitable period, ) = sgn(ψNj ), and cp = v|p cv with
cv (π, α, j, ψ) =
#
(Nv)j+1
αv
$fv (ψv )
w−κ−
2
≤j≤
w+κ−
2
j
−1 −1
w−j
(1 − α−1
)
v ψv ((v )(Nv) )(1 − αv ψv ((v )χπ ((v )(Nv)
(here ψv ((v ) = 0 if ψv is ramified, and fv (ψv ) denotes the conductor exponent of ψv ).
)
+
*
σ
iii) Growth: The function Lp (π, α) has growth of order h = ev vp (αv ) − σ|v w−κ
.
2
v|p
1 When
p is split completely in F (so in particular when F = Q), every π should be regular, but in general this can fail.
2
,
iv) Variation: When F is totally real and π is regular, the function Lp (π, α) spreads out locally to a neighborhood of xπ,α in the GL2 /F -eigenvariety X = X (n). Precisely, there is a smooth admissible affinoid
open U ⊂ X containing xπ,α , global functions Lp ∈ O(U × G ) and b ∈ O(U )× , and a Zariski-dense set
of points U nc ⊂ U containing xπ,α and corresponding to non-critically refined eigenforms (πx , αx ) such
that
Lp (x) = b(x)Lp (πx , αx ) ∈ O(G ) ⊗ k(x)
for any x ∈ U nc . The germ of U at xπ,α is unique.
v) Unicity: When F is totally real, (π, α) is regular, and the Leopoldt conjecture is true for F at p, conditions
ii)-iv) determine L$p (π, α) uniquely up to multiplication by an element of E × .
We turn to a description of our method. The group ΓF is naturally a locally p-adic analytic group, so
by a famous principle due to Amice, rigid analytic functions on GF are in bijection with locally analytic
distributions on ΓF via the Amice transform map
D(ΓF ) → O(GF )
µ #→ Aµ (χ) = µ(χ).
We aim to associate with (π, α) a canonical distribution µ(π, α) ∈ D(ΓF ) ⊗Qp E; Lp (π, α) is the Amice
transform of µ(π, α).
In order to carry this out, consider the left GL2 (F )-module
Lκ,w (R) = ⊗σ∈Σ symκσ (R2 ) ⊗ det
w−κσ
2
,
where R is any ring such that the phrase “GL2 (F ) acts on the σth factor through the embedding σ : F #→ R”
is meaningful; we only care about R = C and R = L. Let Y1 (m) be the usual
locally symmetric quotient
(
of GL2 (F∞ ) with Γ1 (m)-level structure, and set Y = Y1 (np), where p = OF ⊃p⊃(p) p (so p = (p) if p is
unramified in F ). Let T(n) denote the Q-algebra of formal polynomials in the Hecke operators Tv for v ! np
and Uv for v|p, and let mπ,α ⊂ T(n) ⊗Q Q(π, α) be the maximal ideal generated by Uv − αv for v|p and
Tv − av (π) for v ! np. Setting q = r1 + r2 = d − r2 , the module
Vπ,α = Hcq (Y1 (np), Lκ,w (C))[mπ,α ]
×
has rank 2r1 over C, and in fact is free of rank one over the group ring S ⊗ C, where S = Q[π0 (F∞
)] is the
ring of “Hecke operators at infinity”. On the other hand, Lκ,w makes equally good sense over any extension
of L. Generalizing a gorgeous idea of Glenn Stevens from the case F = Q (Ste94), we define a module Dκ,w
of L-valued locally analytic distributions on Op = OF ⊗Z Zp together with a natural surjection
ϕκ,w : Dκ,w → Lκ,w (L),
equivariant for the action of a certain monoid containing the Iwahori subgroup of GL2 (Op ) and thereby
inducing a T(n)- and S -equivariant degree-preserving map
iκ,w : Hc∗ (Y1 (np), Dκ,w ) → Hc∗ (Y1 (np), Lκ,w (L)).
Definition 1.2. The refined form (π, α) is...
+
* )
*
σ
i. extremely noncritical if h(π, α) = v|p ev vp (αv ) − σ|v w−κ
< 1 + κ− .
* 2w−κσ
ii. numerically noncritical if ev vp (αv ) < 1 + minσ|v κσ + σ|v 2 for all v|p.
iii. noncritical if the map iκ,w becomes an isomorphism after localizing the source and target at mπ,α ,
and critical otherwise.
iv. weakly noncritical if
(Hcq (Y1 (np), Dκ,w ) ⊗L E) [mπ,α ]
is free of rank one over the group ring S ⊗ E.
3
v. Galois-noncritical if π is regular, a Galois representation ρπ : GF → GL2 (E) associated with π
fv
exists and satisfies strong local-global compatibility at all places, and the subspace Dcrys (ρπ |GFv )ϕv =αv ⊂
Dcrys (ρπ |GFv ) is in general position with respect to the σ-Hodge filtration for all σ|v|p.
We abbreviate these conditions as x.nc., n.nc., etc. Note that ordinary forms have h(π, α) = 0, so are
extremely noncritical. Note also that ρπ exists and satisfies the requisite hypotheses when F is totally
real (the existence of ρπ and its local-global compatibility in this case is due to Deligne, Carayol, Wiles,
Blasius-Rogawski, Taylor, Saito, and Skinner) or when π is a base change from a totally real field. Of course
we conjecture that ρπ exists for any F and behaves exactly as in the totally real case, so the definition of
Galois-noncriticality should make sense in general.
Theorem 1.3. For any refined form (π, α), x.nc implies n.nc implies nc implies w.nc. Furthermore,
n.nc implies g.nc.
Conjecture 1.4. A refined regular form (π, α) is noncritical if and only if it is Galois-noncritical.
The import of these definitions is that whenever (π, α) is weakly noncritical, the )-isotypic subspace of
(Hcq (Y1 (np), Dκ,w ) ⊗L E) [mπ,α ]
×
$
is a one-dimensional E-vector space for each ) ∈ π"
0 (F∞ ), and we may choose a generator δπ,α , well-defined
×
$
up to multiplication by an element of E . When (π, α) is noncritical, iκ,w (δπ,α ) generates the )-isotypic subspace of (Hcq (Y1 (np), Lκ,w (L)) ⊗L E) [mπ,α ]. It’s also worth remarking that noncriticality belies a vanishing
theorem for overconvergent cohomology, since the target of iκ,w vanishes in all degrees outside the interval
d − r2 ≤ i ≤ d after localization at mπ,α .
On the other hand, we construct a canonical L-linear map
Perκ,w : Hcq (Y1 (np), Dκ,w ) → D(ΓF ) ⊗Qp L,
the period map in weight (κ, w). This map is the heart of the matter, and is the p-adic analogue of the
restriction map
C (GL2 (AF ))
f
→ C (A×
)
# F
x
#→ f
1
$
.
*
$
Extending scalars from L to E, we now set µ(π, α) = $ Perκ,w (δπ,α
). Having done this, the interpolation
and growth properties of Lp (π, α) follow from fairly routine calculations; as expected, the Uv -operators play
$
a key role. It also transpires that Perκ,w is S -equivariant, so in fact the Amice transform of Perκ,w (δπ,α
)
$
is precisely Lp (π, α).
We must next recall some basic ideas on eigenvarieties in order to discuss the variation and canonicity
results meaningfully. Suppose now that F is totally real. We interpolate the cohomological weights (κ, w)
into a rigid analytic weight space W , equidimensional of dimension d + 1; let λ denote a typical point of
W . For any admissible affinoid open Ω ⊂ W , we construct a Frechet O(Ω)-module DΩ , together with
∼
canonical specialization maps DΩ ⊗O(Ω) O(Ω)/mλ → Dκ(λ),w(λ) for every cohomological weight λ ∈ Ω. For
some sufficiently small affinoid Ω containing (κπ , wπ ) and some sufficiently large fixed rational number h, the
theory of slope decompositions yields a Hecke-stable direct summand
Hcd (Y1 (np), DΩ )h ⊂ Hcd (Y1 (np), DΩ )
which is module-finite over O(Ω). Let TΩ,h denote the commutative subalgebra of
EndO(Ω) (Hc∗ (Y1 (np), DΩ )h )
generated by T(n) ⊗Q O(Ω); the eigenvariety X is constructed by gluing the affinoid local pieces XΩ,h =
SpTΩ,h together with their structure maps w : XΩ,h → Ω and φ : T(n) → O(XΩ,h ), but in this discussion we
4
only work locally. Let xπ,α ∈ XΩ,h be the point corresponding to (π, α), with mπ,α ⊂ TΩ,h be the associated
maximal ideal. Using the spectral sequences introduced in (Han12), we show that the map
Hcd (Y1 (np), DΩ )h ⊗O(Ω) k(λ) → Hcd (Y1 (np), Dκ,w )h
becomes an isomorphism after localizing at mπ,α ; here λ = (κ, w) is the point in Ω corresponding to the
weight of π. We are then able to adapt a beautiful proof of Chenevier to show that XΩ,h is reduced and
smooth at xπ,α and that w is etale at xπ,α . Let U be the minimal Zariski-open and -closed subspace of XΩ,h
containing xπ,α ; shrinking Ω if necessary, U is reduced, smooth, and disconnected from (the Zariski closure
of) its complement in XΩ,h , with O(U ) module-finite and etale over O(Ω), and with the O(U ) ⊗ S -module
M = Hcd (Y1 (np), DΩ )h ⊗TΩ,h O(U ) free of rank one. Note that M is a Hecke-stable direct summand of
Hcd (Y1 (np), DΩ ). The affinoid U contains a Zariski-dense set of points U nc accumulating at xπ,α 2 such that
for any x ∈ U nc :
i. w(x) ∈ Ω corresponds to a cohomological weight (κ(x), w(x)), and
ii. there is a canonical isomorphism
M (x) = M ⊗O(U) k(x) , Hcd (Y1 (np), Dκ(x),w(x) )[mπ(x),α(x) ]
of T(n) ⊗ S -modules, for a unique noncritically refined eigenform (π(x), α(x)) of cohomological weight
(κ(x), w(x)).
We then define a big period map
such that the diagram
,
PerΩ : Hcd (Y1 (np), DΩ ) → D(ΓF )⊗O(Ω)
Hcd (Y1 (np), DΩ )
⊗k(λ)
! H d (Y1 (np), Dλ )
c
Perλ
PerΩ
"
,
D(ΓF )⊗O(Ω)
"
⊗k(λ)
! D(ΓF ) ⊗ k(λ)
commutes for every λ ∈ Ω. We may regard PerΩ as an element of
,
HomO(Ω) (Hcd (Y1 (np), DΩ ), O(Ω)⊗D(Γ
F )).
Since M is a direct summand of Hcd (Y1 (np), DΩ ), the restriction of PerΩ to M yields an element of
.
,
,
HomO(Ω) M , O(Ω)⊗D(Γ
= HomO(Ω) (M , O(Ω)) ⊗O(Ω) O(Ω)⊗D(Γ
F)
F)
,
HomO(Ω) (M , O(Ω)) ⊗D(Γ
F ).
The module HomO(Ω) (M , O(Ω)) is free of rank one over O(U ) ⊗ S ; fixing an isomorphism between these
,
,
two modules, PerΩ defines an element of S ⊗ O(U )⊗D(Γ
F ), and we define Lp ∈ O(G × U ) , O(U )⊗D(Γ
F)
,
,
as the image of this element under the natural contraction S ⊗ O(U )⊗D(ΓF ) → O(U )⊗D(ΓF ) induced by
the action of S on D(ΓF ). The variation result follows in a straightforward manner, and canonicity follows
from a simple observation: U nc contains a further Zariski-dense subset U enc of extremely non-critical points,
and for any x ∈ U enc , the L-function Lp (π(x), α(x)) is determined up to a unit solely by its interpolation
and growth properties.
2 A subset S of a rigid analytic space X accumulates at a point x ∈ X if, for any connected affinoid open subset Y ⊂ X
containing x, Y ∩ S is Zariski-dense in Y .
5
Iwasawa theory
Fix a place v|p of F and an algebraic closure Fv /Fv together with an isomorphism Fv , Qp . Let Fv,0
be the maximal unramified subfield of Fv , and let Fv* be the maximal subfield of Fv (ζp∞ ) unramified over
Fv . Set Γv = Gal(Fv (ζp∞ )/Fv ). Let B†rig be the topological Qp -algebra defined in (Ber02); this ring is
equipped with a continuous action of GQp and a commuting operator ϕ. The Robba ring is the ring Rv =
(B†rig )Gal(Qp /Fv (ζp∞ )) , with its natural actions of ϕ and Γv . There is an isomorphism
/
0
Rv , f (πv ) | f (T ) ∈ Fv* [[T, T −1 ]] with f convergent on r ≤ |T | < 1 for some r = rf < 1 .
Here πv is a certain indeterminate arising from the theory of the field of norms; ϕ and Γv act on the coefficients
of f through the absolute Frobenius and the natural map Γv → Gal(Fv* /Fv ), respectively, but the actions on
πv are noncanonical in general. The topological ring Rv is naturally an LF-space, i.e. a strict inductive limit
of Frechet spaces: setting
/
0
Rvr,s = f (T ) ∈ Fv* [[T, T −1]] | f convergent on r ≤ |T | ≤ s
with its natural Banach structure, Rv = ∪r<1 ∩s<1 Rvr,s . In particular, for any affinoid algebra A, the
, Qp A is well-defined, and the ϕ- and Γv -actions extend naturally.
completed tensor product Rv,A = Rv ⊗
Definition. A (ϕ, Γv )-module over A is a finite projective Rv,A -module D equipped with commuting
Rv -semilinear actions of ϕ and Γv such that the matrix of ϕ lies in GLn (Rv,A ) for some basis of D, and
such that the map Γv → End(D) is continuous.
We write Mod(ϕ, Γv )/A for the category of (ϕ, Γv )-modules over A. On the other hand, let Rep(GFv )/A
denote the category of pairs (V, ρ) where V is a finite projective A-module and ρ : GFv → EndA (V ) is a
continuous group homomorphism. The significance of (ϕ, Γv )-modules arises from the following result (due
to many people).
Theorem. There is an exact, fully faithful functor
Rep(GFv )/A
V
→ Mod(ϕ, Γv )/A
#
→
D(V )
whose formation commutes with base change in A. When A is Artinian and V is free over A, D(V ) is free
over Rv,A and is realized explicitly as
)
+Gal(Qp /Fv (ζp∞ ))
D(V ) = B†rig ⊗Qp V
.
This discussion “sheafifies” over general rigid analytic spaces in an unsurprising manner.
Now choose a sufficiently large finite set of places S containing all places dividing p; write S = S p ∪ {v|p}.
An oriented family of Galois representations V is a triple V = (X, V, {Dv,+ }v|p ) where:
• X is a separated and irreducible rigid analytic space,
• V is a coherent locally free OX -module with a continuous homomorphism ρ : GF,S → EndOX (V ),
• Dv,+ is choice, for each v|p, of a coherent Rv,X -saturated (ϕ, Γv )-submodule of Dv = D(V |GFv ).
Given an oriented family, we define a Selmer complex RΓ(F, V) ∈ Dbfg (X) as the mapping cone

•
Cone Ccts
(GF,S , V ) ⊕
3
v|p
•
Cϕ,Γ
(Dv,+ ) ⊕
v
3
•
Ccts
(GFv , V )unr
v∈S p
P
v iv ◦resv −av
−→
3
v|p
•
Cϕ,Γ
(Dv ) ⊕
v
3
v∈S p
•
Ccts
(GFv , V ) [−1].
•
•
Here Galois cohomology is computed via the usual bar resolution, resv : Ccts
(GF,S , V ) → Ccts
(GFv , V ) is the
∼
•
natural restriction map, iv is the identity if v ! p and is the natural quasiisomorphism iv : Ccts
(GFv , V ) →
6

•
•
•
•
Cϕ,Γ
(Dv ) if v|p, and av is the natural morphism Cϕ,Γ
(Dv,+ ) → Cϕ,Γ
(Dv ) (resp. Ccts
(GFv , V )unr →
v
v
v
•
Ccts (GFv , V )). The fact that the claimed maps exist at all and that RΓ(F, V) has coherent cohomology follows
from the work of many people, especially that of Nekovar, Liu and Pottharst. Set H̃ i (V) = H i (RΓ(F, V)).
We apply this formalism to associate a Selmer group with (π, α) when π is regular. We let Vπ,α be the
oriented family with:
• X = G ×SpQp SpE.
• V = ρπ (χp ) ⊗Qp OG with the natural diagonal action, where GF acts on OG through the universal
character χG : GF → ΓF #→ O(G )× . (Here χp is the cyclotomic character).
• The refinement α determines a rank one (ϕ, Γv )-submodule Dv,αv ⊂ D(ρπ |GFv ) by the requirement
fv
that Dcrys (Dv,α ) = Dcrys (ρπ |GFv )ϕv =αv . We then set
Dv,+
=
Dv,αv ⊗Rv,Qp D(OG (χG χp )|GFv )
⊂
D(V |GFv ).
Set Sp (π, α) = H̃ 2 (Vπ,α ).
Conjecture 1.5 (“Main Conjecture”). The support of Sp (π, α) as an OX -module is strictly of dimension < dimG . If (π, α) is regular, noncritical and Galois-noncritical, and Lp (π, α) has no exceptional zero,
then Lp (π, α) generates the characteristic ideal of Sp (π, α).
This is an “interpolation” of the Bloch-Kato conjecture. More precisely, let χx denote the specialization
of χG at x ∈ G ; then for any x with χx as in Theorem 1.1.ii, we have Lp (π, α)(χx ) ≈ L(0, ι−1 (ρπ ⊗ χx χp )),
while on the other hand
∗
Sp (π, α) ⊗ k(x) ∼
= Hf1 (F, ρ∗π ⊗ χ−1
x ) .
Are there lots of noncritical refinements?
We need to discuss the rarity of critical and noncritical refinements, and why one might believe Conjecture
1.4.
Definition. An algebra homomorphism φ : T(n) → A is Eisenstein if there is a pair of A-valued Hecke
characters η1 , η2 such that φ(Tv ) = η1 ((v ) + η2 ((v ) for almost all places v.
Conjecture 1.6. Suppose F is totally real, and let x ∈ X be any point in the eigenvariety. If
Hci (Y1 (np), Dλ(x) )mx 2= 0 for some i 2= d, the residue homomorphism φ(x) : T(n) → k(x) is Eisenstein.
Theorem 1.7. Suppose F is totally real, and that Conjecture 1.6 is true. Then any refined regular form
(π, α) which is Galois-noncritical is noncritical.
Conjecture 1.8. Suppose π corresponds to a non-CM elliptic curve. Then every refinement of π is
noncritical.
References
[AV75] Yvette Amice and Jacques Vélu, Distributions p-adiques associées aux séries de Hecke, Journées
Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, Bordeaux, 1974), Soc. Math. France, Paris,
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