Matthew Emerton

Variation of Iwasawa invariants in Hida families

Let ρ : GQ → GL2(k) be an absolutely irreducible modular Galois representation over a finite field k of characteristic p. Assume further that ρ is p-ordinary and p-distinguished in the sense that the restriction of ρ to a decomposition group at p is reducible and non-scalar. The Hida family H(ρ) of ρ is the set of all p-ordinary p-stabilized newforms f with mod p Galois representation isomorphic to ρ. (If ρ is unramified at p, then one must also fix an unramified line in ρ and require that the ordinary line of f reduces to this fixed line.) These newforms are a dense set of points in a certain p-adic analytic space of overconvergent eigenforms, consisting of an intersecting system of branches (i.e. irreducible components) T(a) indexed by the minimal primes a of a certain Hecke algebra. To each modular form f ∈ H(ρ) one may associate the Iwasawa invariants μan( f ), λan( f ), μalg( f ) and λalg( f ). The analytic (resp. algebraic) λ-invariants are the number of zeroes of the p-adic L-function (resp. of the characteristic power series of the dual of the Selmer group) of f , while the μ-invariants are the exponents of the powers of p dividing the same objects. In this paper we prove the following results on the behavior of these Iwasawa invariants as f varies over H(ρ).

Supersingular elliptic curves, theta series and weight two modular forms

This paper deals with two subjects and their interaction. The first is the problem of spanning spaces of modular forms by theta series. The second is the commutative algebraic properties of Hecke modules arising in the arithmetic theory of modular forms. Let p be a prime, and let B denote the quaternion algebra over Q that is ramified at p and ∞ and at no other places. If L is a left ideal in a maximal order of B then L is a rank four Z-module equipped in a natural way with a positive definite quadratic form [6, §1]. (We shall say that L is a rank four quadratic space, and remark that the isomorphism class of L as a quadratic space depends only on the left ideal class of L in its maximal order.) Eichler [5] proved that the theta series of L is a weight two modular form on Γ0(p), and that as L ranges over a collection of left ideal class representatives of all left ideals in all maximal orders of B these theta series span the vector space of weight two modular forms on Γ0(p) over Q. In this paper we strengthen this result as follows: if L is as above, then the q-expansion of its theta series Θ(L) has constant term equal to one and all other coefficients equal to even integers. Suppose that f is a modular form whose qexpansion coefficients are even integers, except perhaps for its constant term, which we require merely to be an integer. It follows from Eichler’s theorem that f may be written as a linear combination of Θ(L) (with L ranging over a collection of left ideals of maximal orders of B) with rational coefficients. We show that in fact these coefficients can be taken to be integers. Let T denote the Z-algebra of Hecke operators acting on the space of weight two modular forms on Γ0(p). The proof that we give of our result hinges on analyzing the structure of a certain T-module X . We can say what X is: it is the free Zmodule of divisors supported on the set of singular points of the (reducible, nodal) curve X0(p) in characteristic p. The key properties of X , which imply the above result on theta series, are that the natural map T −→ EndT(X ) is an isomorphism, and that furthermore X is locally free of rank one in a Zariski neighbourhood of the Eisenstein ideal of T. We remark that it is comparatively easy to prove the analogous statements after tensoring with Q, for they then follow from the fact that X is a faithful T-module. Indeed, combining this with the semi-simplicity of the Q-algebra T⊗Z Q, one deduces that X ⊗Z Q is a free T⊗Z Q-module of rank one, and in particular that the map T⊗Z Q −→ EndT⊗ZQ(X ⊗Z Q) is an isomorphism.

Unit

Let f: X -> Y be a separated morphism of schemes of finite type over a finite field of characteristic p, let Lambda be an artinian local Z_p-algebra with finite residue field, let m be the maximal ideal of Lambda, and let L^\bullet be a bounded constructible complex of sheaves of finite free Lambda-modules on the etale site of Y. We show that the ratio of L-functions L(X,L^\bullet)/L(Y,f_! L^\bullet), which is a priori an element of 1+T Lambda[[T]], in fact lies in 1+ m T Lambda [T]. This implies a conjecture of Katz predicting the location of the zeroes and poles of the L-function of a p-adic etale lisse sheaf on the closed unit disk in terms of etale cohomology with compact support.

L

This note summarizes the theory of p-adically completed cohomology. This construction was first introduced in paper [4] (although insufficient attention was given there to the integral aspects of the theory), and then further developed in the papers [2] and [6]. The papers [4] and [2] may give the impression that p-adically completed cohomology is some sort of auxiliary construction that can be used to prove theorems (of either a p-adic or classical nature) about automorphic forms. However, we believe that p-adically completed cohomology is in fact an object of fundamental importance, and that it provides the best approximation that we know of to spaces of p-adic automorphic forms. (In particular, unlike the spaces that go by this name that are sometimes constructed by arithmetico-geometric means in the theory of modular curves, or more generally Shimura varieties, p-adically completed cohomology admits a representation of the p-adic group, and thus allows the introduction of representation-theoretic methods into the study of p-adic properties of automorphic forms.) A systematic exposition of the theory, and of its (largely conjectural, at this point) applications to the p-adic aspects of the Langlands correspondence between automorphic eigenforms and Galois representations, will be given in the paper [3]. These notes provide a summary of some of the basic points of the theory, as well as one of the main conjectures of [3] (Conjecture 6.1 below).

-functions and a conjecture of Katz

Let M be a compact 3-manifold with infinite fundamental group . Given a homomorphism from to a p-adic analytic group G with dense image, we describe the possible mod-p homology growth of covers Mn of M determined by the congruence subgroups Gn. If d D dim.G/ > 3, this growth is always non-trivial, growing at least as fast as Vol.Mn/ 1/=d . Mathematics Subject Classification (2010). 22E40.

Non-abelian Fundamental Groups and Iwasawa Theory: Completed cohomology – a survey

In the paper [BB1], Beilinson and Bernstein used the method of localisation to give a new proof and generalisation of Casselman’s subrepresentation theorem. The key point is to interpret n-homology in geometric terms. The object of this note is to go one step further and describe the Jacquet module functor on Harish-Chandra modules via geometry. Let GR be a real reductive linear algebraic group, and let KR be a maximal compact subgroup of GR. We use lower-case gothic letters to denote the corresponding Lie algebras, and omit the subscript “R” to denote complexifications. Thus (g,K) denotes the Harish-Chandra pair corresponding to GR. Let h be the universal Cartan of g, that is h = b/[b, b] where b is any Borel of g. We equip h with the usual choice of positive roots by declaring the roots of b to be negative. We write ρ ∈ h∗ for half the sum of the positive roots. To any λ ∈ h∗ we associate a character χλ of the centre Z(g) of the universal enveloping algebra U(g) via the Harish-Chandra homomorphism. Under this correspondence, the element ρ ∈ h∗ corresponds to the trivial character χρ. For the rest of this paper, we work with λ ∈ h∗ that is dominant, i.e. α̌(λ) 6∈ {−1,−2, . . .} for any positive coroot α̌. A Harish-Chandra module with infinitesimal character λ is, by definition, a (g,K)module which is finitely generated over U(g) and on which Z(g) acts via the character χλ. We can also view Harish-Chandra modules with infinitesimal character λ as modules over the ring Uλ which is the quotient of U(g) by the two-sided ideal generated by {z−χλ(z) | z ∈ Z(g)}. In light of this, we will sometimes refer to such Harish-Chandra modules simply as (Uλ,K)-modules. Let X be the flag manifold of g, and let Dλ be the sheaf of twisted differential operators with twist λ. By a (Dλ,K)-module we mean a coherent Dλ-module which is K-equivariant. Such a Dλ-module is, by necessity, regular holonomic as K acts on X with finitely many orbits. According to Beilinson-Bernstein [BB2], we have Γ(X,Dλ) = Uλ, and the global sections functor (1.1) Γ : {(Dλ,K)−modules} −→ {(Uλ,K)−modules} is exact and essentially surjective. A section of the functor Γ is given by the localisation functor which takes a (Uλ,K)-module M to the (Dλ,K)-module Dλ ⊗Uλ M. The localisation functor is an equivalence if λ is regular. Let PR be a minimal parabolic subgroup of GR, let nR be the nilpotent radical of pR, and let n̄R be the nilpotent subalgebra of gR opposite to nR. If AR denotes the

Mod-

for all r > 0. (Here Γ denotes the group of units in Zp congruent to one modulo p, Γr denotes the kernel of the reduction of Γ modulo p, and Λ denotes the completed group ring Zp[[Γ]] = lim←− r Zp[Γ/Γr].) Hida proves this through a series of group cohomological calculations combined with his theory of the ordinary part of the p-adic Hecke algebra. In this note we present a simple proof of the same result (Theorem 5.3 below) using only the elementary algebraic topology of the Riemann surfaces Y1(p). As with Hida, we also consider the case of auxiliary Γ1(N)-level structure, for some N prime to p.

p

We show that the mod p cohomology of a smooth projective variety with semistable reduction over K, a finite extension of Qp, embeds into the reduction modulo p of a semistable Galois representation with Hodge-Tate weights in the expected range (at least after semisimplifying, in the case of the cohomological degree > 1). We prove refinements with descent data, and we apply these results to the cohomology of unitary Shimura varieties, deducing vanishing results and applications to the weight part of Serres conjecture.