In this paper we develop a new method to study Iwasawa theory and Eisenstein families for unitary groups $\mathrm{U}(r,s)$ of general signature over a totally real field $F$. As a consequence we prove that for a motive corresponding to a regular algebraic cuspidal automorphic representation $π$ on $\mathrm{U}(r,s)_{/F}$ which is ordinary at $p$, twisted by a Hecke character, if its Selmer group has rank $0$, then the corresponding central $L$-value is nonzero. This generalizes a result of Skinner-Urban in their ICM 2006 report in the special case when $F=\mathbb{Q}$ and the motive is conjugate self-dual. Along the way we also obtain $p$-adic functional equations for the corresponding $p$-adic $L$-functions and $p$-adic families of Klingen Eisenstein series. Our method does not involve computing Fourier-Jacobi coefficients (as opposed to previous work which only work in low rank cases, e.g. $\mathrm{U}(1,1)$, $\mathrm{U}(2,0)$ and $\mathrm{U}(1,0)$) whose automorphic interpretation is unclear in general.
A result due to R. Greenberg gives a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic $\mathbb Z_p$-extensions at good ordinary primes $p$. We extend Greenberg's result to more general $p$-adic Galois representations, including a large subclass of those attached to $p$-ordinary modular forms of level $Γ_0(N)$ with $p\nmid N$.
This paper contains a complete proof of Fukaya's and Kato's epsilon$-isomorphism conjecture in [23] for invertible Λ-modules (the case of V = V_0(r) where V_0 is unramified of dimension 1). Our results rely heavily on Kato's unpublished proof of (commutative) epsilon-isomorphisms for one dimensional representations of G_{Q_p} in [27], but apart from fixing some sign-ambiguities in (loc.\ cit.) we use the theory of (ϕ,Γ)-modules instead of syntomic cohomology. Also, for the convenience of the reader we give a slight modification or rather reformulation of it in the language of [23] and extend i…
We prove a `Weierstrass Preparation Theorem' and develop an explicit descent formalism in the context of Whitehead groups of non-commutative Iwasawa algebras. We use these results to describe the precise connection between the main conjecture of non-commutative Iwasawa theory (in the sense of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is both a converse to a theorem of Fukaya and Kato and also provides an important means of deriving explicit consequences of the main conjecture.
This is a contribution to the ICM 2002. We explain the relation between the (equivariant) Bloch-Kato conjecture for special values of L-functions and the Main Conjecture of (non-abelian) Iwasawa theory. On the way we will discuss briefly the case of Dirichlet characters in the abelian case. We will also discuss how "twisting" in the non-abelian case would allow to reduce the general conjecture to the case of number fields. This is one the main motivations for a non-abelian Main Conjecture.
We construct an Euler system of $p$-adic zeta elements over the eigencurve which interpolates Kato's zeta elements over all classical points. Applying a big regulator map gives rise to a purely algebraic construction of a two-variable $p$-adic $L$-function over the eigencurve. As a first application of these ideas, we prove the equality of the $p$-adic $L$-functions associated with a critical-slope refinement of a modular form by the works of Bellaïche/Pollack-Stevens and Kato/Perrin-Riou.
The aim of this article is to study Bloch-Kato's exponential map and Perrin-Riou's "big" exponential map purely in terms of (ϕ,Γ)-modules over the Robba ring. We first generalize the definition of Bloch-Kato's exponential map for all the (ϕ,Γ)-modules without using Fontaine's rings B_{cris}, B_{dR} of p-adic periods and then we generalize the construction of Perrin-Riou's "big" exponential map for all the de Rham (ϕ,Γ)-modules and prove that this map interpolates our Bloch-Kato's exponential map and the dual exponential map. Finally, we prove a theorem concerning to the determinant of our "big…
We study the adjoint Bloch--Kato Selmer groups attached to a classical point in the cuspidal eigenvariety associated with $\mathrm{GSp}_{2g}$. Our strategy is based on the study of families of Galois representations on the eigenvariety, which is inspired by the book of J. Bellaiche and G. Chenevier.
Kato and Fukaya conjectured existence of certain elements in certain ($p$-adic) Iwasawa algebras which are related to Deligne-Langland's ($l$-adic) local epsilon constants. We prove the existence of these elements in the $l \neq p$ case.
Let $ρ$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic, polarizable automorphic representation of $GL_n$. Assuming only that $ρ$ satisfies an irreducibility condition, we prove the vanishing of the adjoint Bloch--Kato Selmer group attached to $ρ$. This generalizes previous work of the author and James Newton.
Let $p>2$ be a prime. Under mild assumptions, we prove the Iwasawa main conjecture of Kato, for modular forms with general weight and conductor prime to $p$.
We prove the Bloch--Kato conjecture for certain critical values of degree 8 $L$-functions associated to cusp forms on $\mathrm{GSp}_4 \times \mathrm{GL}_2$. We also construct a $p$-adic Eichler--Shimura isomorphism in Hida families for $\mathrm{GSp}_4$, relating $H^2$ of automorphic vector bundles with $\mathbf{D}_{\mathrm{cris}}$ of a subquotient of étale cohomology.
The aim of this paper is to prove inequalities towards instances of the Bloch-Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the $L$-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.
This is essentially the author's thesis submited to The University of Chicago (May 1997). I prove the validity of Tamagawa number conjecture of Bloch-Kato for certain Hecke characters. I study the exponential map and local Tamagawa number for all odd primes (both ordinary and supersingular), using Kato's explicit reciprocity law for one dimensional Lubin-Tate formal group. I also study p-part of Shafarevich-Tate group for motives associated to Hecke characters, using Rubin's Main Conjecture in Iwasawa theory. Interestingly a congruence property between p-adic periods of elliptic curve and weig…
Noncommutative Iwasawa theory has created a lot of interest in Whitehead groups of Iwasawa algebras of compact p-adic Lie groups with a quotient isomorphic to the additive group of p-adic integers. In this paper we compute Whitehead groups of Iwasawa algebra of a pro-p compact p-adic Lie group of dimension one. We also give results on Whitehead groups of the localisation of such Iwasawa algebras at the canonical Ore set defined by Coates, Fukaya, Kato, Sujatha and Venjakob.
In 2005 Coates, Fukaya, Kato, Sujatha, and Venjakob formulated a noncommutative Iwasawa main conjecture for l-adic Lie extensions of number fields. To provide evidence for this main conjecture we formulate and prove an analogous statement for l-adic Lie extensions of separated schemes of finite type over a finite field of characteristic prime to l.
In this paper, we generalise the construction of the Bloch-Kato exponential map to complete discrete valuation fields of mixed characteristic (0,p) whose residue fields have a finite p-basis. As an application we prove an explicit reciprocity law, extending a result of Cherbonnier and Colmez in the classical case. This result relies on the calculation of the Galois cohomology of a p-adic representation V in terms of its (phi,G)-module.
We study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic Zp-extension of Q under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.
We discuss abelian equivariant Iwasawa theory for elliptic curves over $\mathbb{Q}$ at good supersingular primes and non-anomalous good ordinary primes. Using Kobayashi's method, we construct equivariant Coleman maps, which send the Beilinson-Kato element to the equivariant $p$-adic $L$-functions. Then we propose equivariant main conjectures and, under certain assumptions, prove one divisibility via Euler system machinery. As an application, we prove a case of a conjecture of Mazur-Tate.
In this article, we study the Iwasawa theory for Hilbert modular forms over the anticyclotomic extension of a CM field. We prove a one sided divisibility result toward the Iwasawa main conjecture. The proof relies on the first and second reciprocity law relating theta elements to Heegner points Euler system. As a by-product we also prove certain Bloch-Kato type result in the rank $0$ case and a parity conjecture
We show an Iwasawa functional equation for a two dimensional $p$-adic representation of the absolute Galois group of $\mathbf{Q}_p$. This allows us to complete Nakamura's proof of Kato's local $ε$-conjecture in dimension $2$.
In this article I generalise previous computations (by K. Kato, T. Hara and myself) of K_1 (only up to p-power torsion) of p-adic group rings of finite non-abelian p-groups in terms of p-adic group rings of abelian subquotients of the group. Such computation have applications in non-commutative Iwasawa theory due to a strategy proposed by D. Burns, K. Kato (and a modification by T. Hara) for deducing non-commutative main conjectures from commutative main conjectures and certain congruences between special L-values. However, I do not say anything about Iwasawa theory in this article. The detail…
In this paper, we are going to establish a simultaneous generalization of the relative Iwasawa theory proposed by Kedlaya-Pottharst and the relative $p$-adic Hodge theory after Kedlaya-Liu. We call this Hodge-Iwasawa theory in the sense that one could apply the theory to study noncommutative Iwasawa cohomology and noncommutative Iwasawa theories in families and meanwhile one could apply the theory to study the deformation theory of étale local systems or families of representations of fundamental groups or the equivariant constructible $p$-adic sheaves, with more sophisticated point of view co…
We discuss the formalism of Iwasawa theory descent in the setting of the localized K_1-groups of Fukaya and Kato. We then prove interpolation formulas for the `leading terms' of the global Zeta isomorphisms that are associated to certain Tate motives and of the p-adic L-functions that are associated to certain critical motives.
Let f be a newform of weight 2k-2 and level 1. In this paper we provide evidence for the Bloch-Kato conjecture for modular forms. We demonstrate an implication that under suitable hypothesis if a prime divides the algebraic part of L(k,f), then the prime divides the order of the Selmer group associated to f. We demonstrate this by establishing a congruence between the Saito-Kurokawa lift of f and a cuspidal Siegel eigenform that is not a Saito-Kurokawa lift. We then examine what this congruence says in terms of Galois representations to produce a non-trivial p-torsion element in the Selmer gro…
In this article we construct a $p$-adic three dimensional Eigenvariety for the group $U(2,1)(E)$, where $E$ is a quadratic imaginary field and $p$ is inert in $E$. The Eigenvariety parametrizes Hecke eigensystems on the space of overconvergent, locally analytic, cuspidal Picard modular forms of finite slope. The method generalized the one developed in Andreatta-Iovita-Pilloni by interpolating the coherent automorphic sheaves when the ordinary locus is empty. As an application of this construction, we reprove a particular case of the Bloch-Kato conjecture for some Galois characters of $E$, exte…
Following Ribet's seminal 1976 paper there have been many results employing congruences between stable cuspforms and lifted forms to construct non-split extensions of Galois representations. We show how this strategy can be extended to construct elements in the Bloch-Kato Selmer groups of +/--Asai (or tensor induction) representations associated to Bianchi modular forms. We prove, in particular, how the Galois representation associated to a suitable low weight Siegel modular form produces elements in the Selmer group for exactly the Asai representation (+ or -) that is critical in the sense of…
We formulate and prove an analogue of the non-commutative Iwasawa Main Conjecture for $\ell$-adic representations of the Galois group of a function field of characteristic $p$. We also prove a functional equation for the resulting non-commutative $L$-functions. As corollaries, we obtain non-commutative generalisations of the main conjecture for Picard-$1$-motives of Greither and Popescu and a main conjecture for abelian varieties over function fields in precise analogy to the $\operatorname{GL}_2$ main conjecture of Coates, Fukaya, Kato, Sujatha and Venjakob.
For a crystalline p-adic representation of the absolute Galois group of Qp, we define a family of Coleman maps (linear maps from the Iwasawa cohomology of the representation to the Iwasawa algebra), using the theory of Wach modules. Let f = sum(a_n q^n) be a normalized new modular eigenform and p an odd prime at which f is either good ordinary or supersingular. By applying our theory to the p-adic representation associated to f, we define two Coleman maps with values in the Iwasawa algebra of Zp^* (after extending scalars to some extension of Qp). Applying these maps to the Kato zeta elements …
We use $p$-adic families of automorphic forms for an unitary group in three variables, containing some non-tempered forms constructed by Rogawski, to prove some cases of the Bloch-Kato conjectures.
For an elliptic curve $E$ over $\mathbb{Q}$, putting $K=\mathbb{Q}(E[p])$ which is the $p$-th division field of $E$ for an odd prime $p$, we study the ideal class group $\mathrm{Cl}_K$ of $K$ as a $\mathrm{Gal}(K/\mathbb{Q})$-module. More precisely, for any $j$ with $1\leqslant j \leqslant p-2$, we give a condition that $\mathrm{Cl}_K\otimes \mathbb{F}_p$ has the symmetric power $\mathrm{Sym}^j E[p]$ of $E[p]$ as its quotient $\mathrm{Gal}(K/\mathbb{Q})$-module, in terms of Bloch-Kato's Tate-Shafarevich group of $\mathrm{Sym}^j V_p E$. Here $V_p E$ denotes the rational $p$-adic Tate module of …
For the $p$-cyclotomic tower of $\mathbb{Q}_p$ Fontaine established a description of local Iwasawa cohomology with coefficients in a local Galois representation $V$ in terms of the $ψ$-operator acting on the attached etale $(φ,Γ)$-module $D(V)$. In this article we generalize Fontaine's result to the case of arbitratry Lubin-Tate towers $L_\infty$ over finite extensions $L$ of $\mathbb{Q}_p$ by using the Kisin-Ren/Fontaine equivalence of categories between Galois representations and $(φ_L,Γ_L)$-module and extending parts of [Herr L.: Sur la cohomologie galoisienne des corps $p$-adiques. Bull. S…
We study the Eisenstein ideal for modular forms of even weight $k>2$ and prime level $N$. We pay special attention to the phenomenon of $\mathit{extra \ reducibility}$: the Eisenstein ideal is strictly larger than the ideal cutting out reducible Galois representations. We prove a modularity theorem for these extra reducible representations. As consequences, we relate the derivative of a Mazur-Tate $L$-function to the rank of the Hecke algebra, generalizing a theorem of Merel, and give a new proof of a special case of an equivariant main conjecture of Kato. In the second half of the paper, we r…
For certain algebraic Hecke characters chi of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL_2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-value L(0,chi). We further prove that its index is bounded from above by the order of the Selmer group of the p-adic Galois character associated to chi^{-1}. This uses the work of R. Taylor et al. on a…
In this paper we concentrate on the relations between the structure of small Galois groups, arithmetic of fields, Bloch-Kato conjecture, and Galois groups of maximal pro-$p$-quotients of absolute Galois groups.
In the present paper, we study the $p$-adic $L$-functions and the (strict) Selmer groups over $\mathbb{Q}_{\infty}$, the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, of the $p$-adic weight one cusp forms $f$, obtained via the $p$-stabilization of weight one Eisenstein series, under the assumption that a certain Eisenstein component of the $p$-ordinary universal cuspidal Hecke algebra is Gorenstein. As an application, we compute the Iwasawa invariants of ordinary modular forms of weight $k\geq 2$ with the same residual Galois representations as the one of $f$, which in our setting, is r…
A difference formula of $p$-adic height pairings via the Bloch-Kato logarithm map
Joint Graduate School of Mathematics for Innovation, Kyushu University（九州大学数学创新联合研究生院）；Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）；Graduate School for Mathematics, Kyushu University（九州大学数学研究院）
In this article, we discuss Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms. After the overview on the situation of the ordinary case of Hida family, we will introduce a Coleman map for Coleman family for the non-ordinary case (Coleman family) which was obtained as a joint work with Filippo Nuccio [NO16] and we give some results on Iwasawa Theory for Coleman families as applications of . First, we give a construction of a two-variable $p$-adic $L$-function for a Coleman family thanks to the ingredients given in [NO16] (Theorem 5.1). Combining this result and Coleman…
In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that if the central critical value of the Rankin-Selberg $L$-function does not vanish, then the Bloch-Kato Selmer group with coefficients in a favorable field of the corresponding motive vanishes. We also show that if the class in the Bloch-Kato Selmer group constructed from certain diagonal cycle does not vanish, which is conjecturally equivalent to the nonvanis…
The purpose of this article is to give formulas for Bloch-Kato's exponential map and its dual for an absolutely crystalline p-adic representation V, in terms of the (phi,Gamma)-module associated to that representation. As a corollary of these computations, we can give a very simple (and slightly improved) description of Perrin-Riou's exponential map (which interpolates Bloch-Kato's exponentials for V(k)). This new description directly implies Perrin-Riou's reciprocity formula.
Let $k > 3$ be an integer and $p$ a prime with $p > 2k-2$. Let $f$ be a newform of weight $2k-2$ and level 1 so that $f$ is ordinary at $p$ and $\barρ_{f}$ is irreducible. Under some additional hypotheses we prove that $ord_{p}(L_{alg}(k,f)) \leq ord_{p}(# S)$ where $S$ is the Pontryagin dual of the Selmer group associated to $ρ_{f} \otimes ε^{1-k}$ with $ε$ the $p$-adic cyclotomic character. We accomplish this by first constructing a congruence between the Saito-Kurokawa lift of $f$ and a non-CAP Siegel cusp form. Once this congruence is established, we use Galois representations to obtain th…
In this article, we construct zeta morphisms for the universal deformations of odd absolutely irreducible two dimensional mod p Galois representations satisfying some mild assumptions, and prove that our zeta morphisms interpolate Kato's zeta morphisms for Galois representations associated to Hecke eigen cusp newforms. The existence of such morphisms was predicted by Kato's generalized Iwasawa main conjecture. Based on Kato's original construction, we construct our zeta morphisms using many deep results in the theory of p-adic (local and global) Langlands correspondence for GL_{2/Q}. As an app…
We provide a simple and efficient numerical criterion to verify the Iwasawa main conjecture and the indivisibility of derived Kato's Euler systems for modular forms of weight two at any good prime under mild assumptions. In the ordinary case, the criterion works for all members of a Hida family once and for all. The key ingredient is the explicit computation of the integral image of the derived Kato's Euler systems under the dual exponential map. We provide explicit new examples at the end. This work does not appeal to the Eisenstein congruence method at all.
In this paper, we prove the Iwasawa main conjecture of totally real fields for certain specific non-commutative $p$-adic Lie extensions, using the integral logarithms introduced by Oliver and Taylor. Our result gives certain generalization of Kazuya Kato's proof of the main conjecture for Galois extensions of Heisenberg type.
