Computing groups of Hecke characters Pascal Molinand Aurel Page October 7 2022

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Computing groups of Hecke characters

Pascal Molin∗and Aurel Page†

October 7, 2022

Abstract

We describe algorithms to represent and compute groups of Hecke

characters. We make use of an idèlic point of view and obtain the whole

family of such characters, including transcendental ones. We also show

how to isolate the algebraic characters, which are of particular interest

in number theory. This work has been implemented in Pari/GP, and

we illustrate our work with a variety of explicit examples using our

implementation.

1 Introduction

Hecke characters are, from the modern point of view, continuous charac-

ters of idèle class groups, in other words automorphic forms for GL1. They

were introduced by Hecke [13] who proved the functional equation of their

L-function, and are the starting point of many developments that blossom

in modern number theory: automorphic L-functions via Tate’s thesis [39],

`-adic Galois representations via Weil’s notion of algebraic characters [43],

Shimura varieties via CM theory [38], and the Langlands programme via

class ﬁeld theory and the global Weil group [44]. Despite their fundamental

role, Hecke characters have not received a full algorithmic treatment, perhaps

due to the fact that they are considered well-understood compared to auto-

morphic forms on higher rank groups. The existing literature only describes

how to compute with ﬁnite order characters, since they are characters of ray

class groups [7], and algebraic Hecke characters [42]. As part of a collective

eﬀort to enumerate and compute L-functions, automorphic representations

and Galois representations, we believe that the GL1case also deserves close

scrutiny, and this is the goal of the present paper.

We describe algorithms to compute, given a number ﬁeld Fand a mod-

ulus mover F, a basis of the group of Hecke quasi-characters of modulus m

∗Université Paris Cité and Sorbonne Université, CNRS, INRIA, IMJ-PRG, F75013

Paris, France, pascal.molin@imj-prg.fr

†INRIA, Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR5251, F-33400 Talence,

France, aurel.page@inria.fr

arXiv:2210.02716v1 [cs.SC] 6 Oct 2022

(Algorithm 18) and its subgroup of algebraic characters (Algorithm 30), in

a form suitable for evaluation at arbitrary ideals and decomposition into

local characters (Algorithm 17). In particular, we describe a polynomial

time algorithm to compute the maximal CM subﬁeld of F(Algorithm 28).

It is sometimes believed that the adèlic point of view is not suitable for

computational purposes; we claim the contrary, and adopt an adèlic setting

throughout the paper. Our implementation [27] in Pari/GP [31] is avail-

able from version 2.15 of the software. We provide examples that illustrate

the use of our algorithms and showcase some interesting features of Hecke

characters: a presentation of the software interface, small degree examples,

illustrations of automorphic induction from quadratic ﬁelds, examples of CM

abelian varieties with emphasis on the rigorous identiﬁcation of the corre-

sponding Hecke character, illustration of the density of the gamma shifts of

Hecke L-functions in the conjectured space of possible ones (Proposition 44),

examples of provably partially algebraic Hecke characters (Proposition 46)

and of twists of L-functions by Hecke characters.

The only previous work on computation of inﬁnite order Hecke characters

is that of Watkins [42], so we give a short comparison: in Watkins’s paper,

only algebraic characters were considered, and only over a CM ﬁeld, whereas

we treat arbitrary Hecke characters over arbitrary number ﬁelds; the values

of characters were represented exactly by algebraic numbers, whereas we

represent values by approximations since this is forced in the transcendental

case; the emphasis was on individual Hecke characters, which the user had to

construct by hand, whereas our emphasis is on groups of Hecke characters,

which we construct for the user, simply from the modulus.

Our implementation makes it possible to tabulate Hecke characters and

their L-functions systematically by increasing analytic conductor; we think

that this is a valuable project but we leave it for future work.

The paper is organized as follows. In Section 2 we recall the deﬁnitions

and basic properties of Hecke characters and their L-functions. In Section 3

we describe our algorithms to compute groups of Hecke characters and eval-

uate them. In Section 4 we present our algorithms to compute the maximal

CM subﬁeld and groups of algebraic Hecke characters. Finally, Section 5

contains a variety of examples.

Acknowledgements We thank the anonymous reviewers for their careful

reading of our manuscript and their many comments and suggestions. We

also thank Karim Belabas and Bill Allombert for their help in integrating

our code to Pari/GP. The ﬁrst author acknowledges support of ANR FLAIR

ANR-17-CE40-0012. The second author was supported by the grants ANR

CIAO ANR-19-CE48-0008 and ANR CHARM ANR-21-CE94-0003.

2 Hecke characters

We recall the deﬁnition of Hecke characters in the adèlic setting. This ma-

terial is standard and can be found in [18, chap. XIV] or [34].

Let Fbe a number ﬁeld of degree [F:Q] = nand discriminant ∆F.

When K/F is a ﬁnite extension, we denote by NK/F the norm from Kto F;

we also denote N=NF/Qwhen Fis clear from the context. For every

prime ideal pof F, we consider the completion Fpand its ring of integers

Zp. We choose a uniformizer πp∈Zpand denote by vp:F×

pZthe p-adic

valuation. We will always use σto denote an archimedean place of Fand the

corresponding real or complex embedding. For every place v, let nv= [Fv:

Qv], and let |·|vbe the normalized absolute value, i.e. nσ= 1 and |·|σ=|·|

for a real embedding σ,nσ= 2 and |·|σ=|·|2for a complex embedding σ,

and |πp|p= N(p)−1for a prime ideal p. We denote by A×

F=Q0F×

vthe

group of idèles of F. We write FR=F⊗QR∼

=QσFσ∼

=Rr1×Cr2, where r1

(resp. r2) is the number of real embeddings (resp. pairs of non-real complex

embeddings) of F.

Let Udenote the group of complex numbers of absolute value 1. For G

a topological group, G◦will denote the connected component of 1in G.

2.1 Pontryagin duality

We recall some deﬁnitions and properties of locally compact abelian groups

that will be used later. See [28, 29] for general reference.

Let Gbe a locally compact abelian group. A quasi-character of Gis a

continuous morphism

χ:G→C×.

Acharacter of Gis a continuous morphism

χ:G→U.

The group of characters of G, which we denote by b

G, is the Pontrya-

gin dual Homcont(G, U)of G, and is a locally compact abelian group. The

canonical map

G→b

given by g7→ (χ7→ χ(g)) is an isomorphism. Let H⊂Gbe a subgroup. Let

H⊥={χ∈b

G|χ(h)=1for all h∈H}

be the Pontryagin orthogonal of Hin b

G. Then H⊥is a closed subgroup

of b

G, and (H⊥)⊥is the closure of H, where the second orthogonal is taken

in G. If His a closed subgroup of G, then we have canonical isomorphisms

[

G/H ∼

=H⊥and b

G/(H⊥)∼

The group Gis compact if and only if b

Gis discrete.

Pontryagin duality is an exact contravariant functor on the category of

locally compact abelian groups.

Let (x, y)7→ x·ydenote a nondegenerate R-bilinear form on a ﬁnite

dimensional R-vector space V. The pairing V×V→Udeﬁned by (x, y)7→

exp(2iπx ·y)induces an isomorphism V∼

V. We will use this isomorphism

to identify characters on Vwith elements of V.

Let Λbe a full rank lattice in V. The pairing above identiﬁes the dual

lattice Λ∨= Hom(Λ,Z)with the subgroup

Λ⊥={x∈V|x·y∈Zfor all y∈Λ},

which is canonically isomorphic to d

V/Λby the above, and we have b

Λ∼

V/Λ⊥. In particular for V=Rand Λ = Zwe consider the standard bilinear

form and we have d

R/Z=Z⊥=Zand b

Z=R/Z.

The dual V=b

Qof the group of rationals equipped with the discrete

topology, is the compact topological group lim

←nR/nZ, called the solenoid.

2.2 General Hecke characters

A Hecke quasi-character is a quasi-character of CF=A×

F/F ×, and a Hecke

character is a character of CF.

The norm is the Hecke quasi-character

k·k:CF→C×

deﬁned by

x= (xv)v7→ kxk=Y

v|xv|v.

This is a well-deﬁned Hecke quasi-character by the product formula.

Every Hecke quasi-character χis of the form χ=χ0k · ksfor a unique

Hecke character χ0and a unique s∈R. We refer to χ0as the unitary

component of χ. In the algebraic setting, the value w=−2sis the weight

of χ.

We also deﬁne C1

F= ker(k · k:CF→R>0)to be the kernel of the norm,

which is a compact group. We have a canonical embedding

R>0→CF,

by sending t∈R>07→ ((t1/n)σ,1, . . . )∈A×

Fwhere t7→ (t1/n)σdenotes the

diagonal embedding R>0→QσF×

σ, and a canonical decomposition

CF∼

=C1

F×R>0.

As a consequence, it suﬃces to compute the characters of C1

Fto deduce

the full groups of Hecke characters and Hecke quasi-characters

Homcont(CF,C×) = b

CFk·kR=b

Fk·kC.(1)

Every quasi-character χof A×

F(and in particular every Hecke quasi-

character) admits a factorization χ=Qvχv, where χvis a quasi-character

of F×

v. We therefore describe quasi-characters of local ﬁelds.

2.3 Local characters

•Every quasi-character χof C×is of the form

χ(z) = z

|z|k|z|s

C=z

|z|k|z|2s

for a unique pair (k, s)∈Z×C. The quasi-character χis a character

if and only if Re(s)=0, i.e. s=iϕ for some ϕ∈R.

•Every quasi-character χof R×is of the form

χ(x) = sgn(x)k|x|s

for a unique pair (k, s)∈ {0,1} × C. We say that χis unramiﬁed

if k= 0. The quasi-character χis a character if and only if Re(s)=0,

i.e. s=iϕ for some ϕ∈R.

•Let pbe a prime ideal of ZF. Every quasi-character χof F×

pis of the

form

χ(x) = χ0(xπ−vp(x)

pmod pm)χ(p)vp(x)

for a unique m≥0and a unique primitive character χ0of (Zp/pm)×,

and where we write χ(p) = χ(πp)∈C×. Note that in general χ(p)

depends on the choice of uniformizer πp, but χ(p)is well deﬁned up to

the roots of unity of the same order as χ0. We call pmthe conductor

of χand mits conductor exponent. If m= 0 we call χunramiﬁed;

in this case, χ(p)does not depend on the choice of uniformizer, and

the quasi-character χonly depends on χ(p). Regardless of m, the

quasi-character χis a character if and only if χ(p)∈U.

Whenever we write a global idèle character χas a product of local charac-

ters χv, we write its local parameters kσ, ϕσ, and mp, and we let fχ=Qppmp

be the conductor of χ. Note that for a complex place, the pair (kσ, ϕσ)de-

pends on the choice of a complex embedding among the two conjugate ones,

or equivalently on the choice of an isomorphism between the completion of F

and C: we have ϕ¯σ=ϕσand k¯σ=−kσ.

2.4 L-function

Let χbe a Hecke character such that Pσnσϕσ= 0, i.e. that is trivial on

the embedded R>0in (1). Let Nχ=|∆F| · N(fχ). Let

L(χ, s) = Y

p-fχ

(1 −χ(p) N(p)−s)−1

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ComputinggroupsofHeckecharactersPascalMolin*andAurelPageOctober7,2022AbstractWedescribealgorithmstorepresentandcomputegroupsofHeckecharacters.Wemakeuseofanidèlicpointofviewandobtainthewholefamilyofsuchcharacters,includingtranscendentalones.Wealsoshowhowtoisolatethealgebraiccharacters,whichareofpart...

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