Haruzo Hida

Geometric modular forms and elliptic curves

An algebro-geometric tool box elliptic curves geometric modular forms Jacobians and Galois representations modularity problems.

A p‐adic measure attached to the zeta functions associated with two elliptic modular forms II

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Nonvanishing Modulo p of Hecke L -Values

We return to the setting of Sect. 6.4; thus, \(M = \mathbb{Q}[\sqrt{-D}] \subset \overline{\mathbb{Q}}\) with discriminant − D is an imaginary quadratic field in which the fixed prime (p) splits into a product of two primes \(\mathfrak{p}\overline{\mathfrak{p}}\) with \(\mathfrak{p}\neq \overline{\mathfrak{p}}\).

Greenberg's ℒ-invariants of adjoint square Galois representations

Author(s): Hida, Haruzo | Abstract: For a two-dimensional p-adic Galois representation V associated to a p-ordinary Hecke eigen cusp form f of weight k g 1, we identify the L-invariant (of R. Greenberg) of the (three dimensional) adjoint square Ad(V) of V with the derivative of the p-coefficient of the Lambda-adic lift of f. By this result, for a given p-adic analytic family of ordinary Hecke eigenforms, the L-invariant does not vanish for almost all members in the p-adic family (as expected).

Local indecomposability of Tate modules of non-CM abelian varieties with real multiplication

As quoted by Ghate–Vatsal in [GV], Question 1 (and answered there affirmatively to a good extent), it is a fundamental problem posed by R. Greenberg to decide indecomposability of an elliptic modular p-adic Galois representation restricted to the decomposition group at p in the ordinary case (as long as the representation is not of CM type). Jointly with B. Balasubramanyam, they have generalized the result to the cases of Hilbert modular forms for primes p splitting completely in the totally real base field (see [GV1]). There are some good applications of their results (see for example [E] and [G]). We can ask the same question for p-adic Galois representations arising from the Tate module of an abelian variety A with real multiplication defined over a number field k; so, End(A/k) contains the integer ring O of a totally real field of degree dimA. This concerns the p-adic Tate module TpA for a prime p|(p) of O as a module over the decomposition group Dp at a place P over p of k. Suppose that A has good reduction modulo P. If the finite flat group scheme A[p] of p-torsion points is local-local at P, as is well known, the Dp-module A[p](Qp) or TpA⊗Op Fp is often irreducible (as proven in [ALR], IV-38, §A.2.2 for elliptic curves; see also the generalization of Serre’s modulo p modularity conjecture for totally real k; in particular, §3.1 of [BDJ]). In this paper, we limit

Adjoint selmer groups as Iwasawa modules

We fix a primep. In this paper, starting from a given Galois representation ϕ having values inp-adic points of a classical groupG, we study the adjoint action of ϕ on thep-adic Lie algebra of the derived group ofG. We call this new Galois representation the adjoint representation Ad(ϕ) of ϕ. Under a suitablep-ordinarity condition (and ramification conditions outsidep), we define, following Greenberg, the Selmer group Sel(Ad(ϕ))/L for each number fieldL. We scrutinize the behavior of Sel(Ad(ϕ))/E∞ as an Iwasawa module for a fixed ℤp-extensionE∞/E of a number fieldE and deduce an exact control theorem. A key ingredient of the proof is the isomorphism between the Pontryagin dual of the Selmer group and the module of Kähler differentials of the universal nearly ordinary deformation ring of ϕ. WhenG=GL(2), ϕ is a modular Galois representation and the base fieldE is totally real, from a recent result of Fujiwara identifying the deformation ring with an appropriatep-adic Hecke algebra, we conclude some fine results on the structure of the Selmer groups, including torsion-property and an exact limit formula ats=0 of the characteristic power series, after removing the trivial zero.

Elliptic curves and arithmetic invariants

1 Non-triviality of Arithmetic Invariants.- 2 Elliptic Curves and Modular Forms.- 3 Invariants, Shimura Variety and Hecke Algebra.- 4 Review of Scheme Theory.- 5 Geometry of Variety.- 6 Elliptic and Modular Curves over Rings.- 7 Modular Curves as Shimura Variety.- 8 Non-vanishing Modulo p of Hecke L-values.- 9 p-Adic Hecke L-functions and their mu-invariants.- 10 Toric Subschemes in a Split Formal Torus.- 11 Hecke Stable Subvariety is a Shimura Subvariety.- References.- Symbol Index.- Statement Index.- Subject Index.

HECKE FIELDS OF ANALYTIC FAMILIES OF MODULAR FORMS

We make finitenessconjectureson the composite of Hecke fields of classicalmembers of a p-adic analyticfamily of slope 0 elliptic modular forms in the verticalcase (with fixed level varying weight). In the horizontal case (fixed weight varying p-power level), we prove the corresponding statements.

Central critical values of modular Hecke

We give an explicit formula for the central critical value L(1/2, π⊗χ) of the base-change lift π to an imaginary quadratic fieldK of an automorphic representation π as the square of a finite sum of the values of a nearly holomorphic cusp form in π at elliptic curves with complex multiplication byK. As long as the transcendental factor of the value is aCMperiod,χ is basically anyunitary arithmeticHecke character ofK inducing the inverse of the central character of π.

L

Author(s): Hida, H | Abstract: For a quaternion algebra B over a totally real field F unramified at every finite place and ramified most at infinite places of F, we prove that the space of Z[1/E]