Conductors in p-adic families

Given a Weil-Deligne representation of the Weil group of an ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document}-adic number field with coefficients in a domain O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {O}$$\end{document}, we show that its pure specializations have the same conductor. More generally, we prove that the conductors of a collection of pure representations are equal if they lift to Weil-Deligne representations over domains containing O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {O}$$\end{document} and the traces of these lifts are parametrized by a pseudorepresentation over O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {O}$$\end{document}.


Introduction
The aim of this article is to study the variation of automorphic and Galois conductors in p-adic families of automorphic Galois representations, for instance, in Hida families and eigenvarieties. We relate the variation of Galois conductors in families to the purity of p-adic automorphic Galois representations at the finite places not dividing p and for the variation of automorphic conductors, we use local-global compatibility. We establish the constancy of tame conductors at arithmetic points lying along irreducible components of p-adic families.

Motivation
In [9,10], Hida showed that the p-ordinary eigen cusp forms, i.e., the normalized eigen cusp forms whose p-th Fourier coefficients are p-adic units (with respect to fixed embeddings of Q in C and Q p ), can be put in p-adic families. More precisely, he showed that for each positive integer N and an odd prime p with p N and N p ≥ 4, there is a subset of the set of Q p -specializations of the universal p-ordinary Hecke algebra h ord (N ; Z p ), called the set of arithmetic specializations, such that there is a one-to-one correspondence between the arithmetic specializations of h ord (N ; Z p ) and the p-ordinary p-stabilized normalized eigen cusp forms of tame level a divisor of N . It turns out that the tame conductors of the Galois representations attached to the arithmetic specializations remain constant along irreducible components of h ord (N ; Z p ) (see [11,Theorem 4.1]). Following Hida's construction of families of ordinary cusp forms, further examples of families of automorphic Galois representations are constructed, for instance, Hida families of ordinary automorphic representations of definite unitary groups, families of overconvergent forms (see the works of Hida [12], Coleman and Mazur [6], Chenevier [4], Bellaïche and Chenevier [1] et al.). The aim of this article is to understand the variation of automorphic and Galois conductors in these families of automorphic Galois representations. In many cases, the restrictions of padic automorphic Galois representations to decomposition groups at places outside p are known to be pure (i.e. the monodromy filtration and the weight filtration associated to such a local Galois representation are equal up to some shift by an integer, see [18, p. 471], [15, p. 1014] for details). So we focus on the variation of conductors in families of pure representations of local Galois groups.

Results obtained
Let p, be two distinct primes and K be a finite extension of Q . Let O be an integral domain containing Q. In Theorem 3.1, we show that given any Weil-Deligne representation of the Weil group W K of K with coefficients in O, its conductor coincides with the conductors of its pure specializations over Q p . Next, in Theorem 3.2, we show that a collection of pure representations of W K over Q p have the same conductor if they lift to Weil-Deligne representations of W K over domains containing O and the traces of these lifts are parametrized by a pseudorepresentation T : An eigenvariety is an example of a family of Galois representations. The traces of the Galois representations associated to the arithmetic points of an eigenvariety are interpolated by a pseudorepresentation defined over its global sections. By [2,Lemma 7.8.11], this pseudorepresentation lifts to a Galois representation on a finite type module over some integral extension of the normalization of O(U ) for any nonempty admissible open affinoid subset U . But this module is not known to be free. So Theorem 3.2 cannot be used to study the tame conductors of all arithmetic points. To circumvent this problem, we establish Theorem 3.3 which can be used to study the tame conductors of a large class of arithmetic points, for example, if the associated Galois representations are absolutely irreducible and yield pure representations when restricted to decomposition groups at the places outside p. Theorems 3.1, 3.2, and 3.3 apply to p-adic families of automorphic representations, for instance, to ordinary families, overconvergent families, and explain the variation of the tame conductors. We illustrate it using the example of Hida family of ordinary automorphic representations for definite unitary groups (see Theorem 4.1).

Notations
For each field F, fix an algebraic closure F of it and denote the Galois group Gal Denote by q the cardinality of the residue field k of the ring of integers O K of K . Let I K denote the inertia subgroup of G K . Let {G s K } s≥−1 denote the upper numbering filtration on G K by ramification subgroups. The Weil group W K is defined as the subgroup of G K consisting of elements which map to an integral power of the geometric Frobenius element Fr k in G k . Its topology is determined by decreeing that I K with its usual topology is an open subgroup.

Preliminaries
In this section, we provide the definition of Weil-Deligne representations, conductors, etc.
Let (r, N ) be a Weil-Deligne representation on a vector space V with coefficients in a field containing the characteristic roots of all elements of r (W K ). Let r (φ) = r (φ) ss u = ur(φ) ss be the Jordan decomposition of r (φ) as the product of a diagonalizable operator r (φ) ss and a unipotent operator u on V . Following [7, 8.5], definer (σ ) = r (σ )u −v K (σ ) for all σ ∈ W K . Then the pair (r , N ) is also a Weil-Deligne representation on V (by [7, 8.5]). It is called the Frobenius-semisimplification of (r, N ) (cf. [7, 8.6]) and it is denoted by (r, N ) Fr-ss . (r, N ) is Weil-Deligne representation of W K on a vector space V over an algebraically closed field of characteristic zero, then its conductor is defined as

Definition 2.2 If
where V I K ,N =0 denotes the subspace of V on which I K acts trivially and N is zero.

Main results
Denote the fraction field of O by L and the algebraic closure of Q in O by Q cl .
. By Lemma 2.3, the L -dimension of ((r, N ) ⊗ O L ) I K ,N =0 is equal to the Q p -dimension of ( f • (r, N )) I K ,N =0 . Note that G u K is contained in I K for any u ≥ 0 and r (H ) is finite for any subgroup H of I K . Hence dim L ((r, N ) • (r, N )) H by Lemma 2.3. This shows that the conductor of f • (r, N ) is equal to the conductor of (r, N ) ⊗ O L . Now we establish an analogue of the above result for pseudorepresentations of Weil groups. We refer to [17,Section 1] for the definition of pseudorepresentation. They are defined by abstracting the crucial properties of the trace of a group representation.

Theorem 3.2 Let O be an integral domain and res
for any Weil-Deligne representation (r , N ) : W K → GL n (O ) as above. Then Lemma 2.3 shows that the conductors of (r , N ) Let w be a finite place of a number field F not dividing p and assume that O is a Z p -algebra. Suppose T, T 1 , . . . , T n are pseudorepresentations of G F with values in O such that T is equal to T 1 + · · · + T n . By [17,Theorem 1], there exist semisimple representations σ 1 , . . . , σ n of G F over L such that trσ i = T i for all 1 ≤ i ≤ n.

Conductors in families
In this section, using the example of a Hida family of ordinary automorphic representations for definite unitary groups, we show how results of Sect. 3 describes the variation of tame conductors in p-adic families. Let F + denote the maximal totally real subfield of a CM field F. Let n ≥ 2 be an integer such that n[F + : Q] is divisible by 4 if n is even. Let > n be a rational prime such that every prime of F + dividing splits in F. Let K be a finite extension of Q in Q containing the image of every embedding of F in Q . Denote by S the set of places of F + dividing . In the following, R denotes a finite set of finite places of , whose existence follows from [13,Corollaire 5.3]. By [5,Theorem 3.2.3], there exists a unique (up to equivalence) continuous semisimple representation r π : G F → GL n (Q ) (as in [8,Proposition 2.7.2]) associated to WBC(π ).
An ordinary automorphic representation for G is an irreducible constituent π of the G(A ∞,R For a non-Eisenstein maximal ideal m of T ord (in the sense of [8, Section 2.7]), let r m denote the representation of G F + as in [8,Proposition 2.7.4]. Composing the restriction of r m to G F with the projection map GL n (T ord m )×GL 1 (T ord m ) → GL n (T ord m ), we obtain a continuous representation G F → GL n (T ord m ) which we denote by r m by abuse of notation. Since m is non-Eisenstein, by [8, Propositions 2.7.2, 2.7.4], the G F -representations η • r m , r π η are isomorphic for any arithmetic specialization η of T ord m . Theorem 4.1 Let a be a minimal prime of T ord contained in a non-Eisenstein maximal ideal m of T ord . Let w be a finite place of F not dividing . Then there exists a non-negative integer C such that the conductor of WD(r π η | W w ) is equal to C for any arithmetic specialization η of T ord m /a for which the weak base change WBC(π η ) of π η is cuspidal. Moreover, the conductor of WBC(π η ) w is also equal to C for any such specialization η.
Proof If the weak base change of an irreducible constituent π of the G(A ∞,R F + ) × v∈R Iw( v)-representation S λ,{χ v } (Q ) is cuspidal, then from [3, Theorems 1.1, 1.2], the proofs of theorem 5.8 and corollary 5.9 of loc. cit., it follows that the representation r π | G w is pure for any finite place w of F not dividing . Note that the action of the inertia subgroup I w on the representation r m mod a is potentially unipotent by Grothendieck's monodromy theorem (see [16, pp. 515-516]). So Theorem 3.1 gives the first part. Since local Langlands correspondence preserves conductors, the rest follows from [3, Theorem 1.1] on local-global compatibility of cuspidal automorphic representations for GL n .