Variation of Iwasawa invariants in Hida families

By M. Emerton, Robert H. Pollack, T. Weston. 2004

182 citations recorded by the source corpus.

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).

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/525fed479c1d4128dcabd101a381bcaf00a7c776

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Citations in the corpus

On anticyclotomic μ-invariants of modular forms (2006, background): “λ-invariants and congruences In the papers [15, 11] explicit formulae are given for the differences of λ-invariants of cyclotomic Selmer groups of congruent modular forms.”
Rankin--Eisenstein classes and explicit reciprocity laws (2015, background): “Our conventions follow those of [15], for example; but other authors use the term “family” for what we call a “branch” (see below).”
On the two-variable Iwasawa main conjecture (2006, background): “A recent paper [EPW] also proves a similar result on the conjecture 6.”
Variation of Heegner points in Hida families (2006, background): “We recall the basic facts of Hida theory that we need; the reader may refer to [5, 8, 9, 25] for more details.”
Multiplicative reduction and the cyclotomic main conjecture for GL2 (2014, result): “If the analytic or algebraic μ-invariant for some fm (the power of the uniformizer of L dividing L6fm or Ch 6 L ( fm)) — or even for some other ordinary eigenform suitably congruent to f for which the main conjecture holds — were known to be zero, then Theorem A would follow from the main results of”