In this article, we present the first half of our project on the Iwasawa theory of higher rank Galois deformations over deformations rings of arbitrary dimension. We develop a theory of Coleman maps for a very general class of coefficient rings, devise a dimension reduction procedure for locally restricted Euler systems and finally, put these into use in order to prove a divisibility in a 3-variable main conjecture for nearly ordinary families of Rankin-Selberg convolutions.
We generalize some of the results of Andreatta, Iovita, and Pilloni and the author to Hodge type Shimura varieties having non-empty ordinary locus. For any $p$-adic weight $κ$, we give a geometric definition of the space of overconvergent modular forms of weight $κ$ in terms of sections of a sheaf. We show that our sheaves live in analytic families, interpolating the classical sheaves for integral weights. We define an action of the Hecke algebra, including a completely continuous operator at $p$. In some simple cases, we also build the eigenvariety.
We develop a theory of $p$-adic automorphic forms on unitary groups that allows $p$-adic interpolation in families and holds for all primes $p$ that do not ramify in the reflex field $E$ of the associated unitary Shimura variety. If the ordinary locus is nonempty (a condition only met if $p$ splits completely in $E$), we recover Hida's theory of $p$-adic automorphic forms, which is defined over the ordinary locus. More generally, we work over the $μ$-ordinary locus, which is open and dense. By eliminating the splitting condition on $p$, our framework should allow many results employing Hida's …
We develop a new strategy for studying low weight specializations of $p$-adic families of ordinary modular forms. In the elliptic case, we give a new proof of a result of Ghate--Vatsal which states that a Hida family contains infinitely many classical eigenforms of weight one if and only if it has complex multiplication. Our strategy is designed to explicitly avoid use of the related facts that the Galois representation attached to a classical weight one eigenform has finite image, and that classical weight one eigenforms satisfy the Ramanujan conjecture. We indicate how this strategy might be…
We prove automorphy lifting results for certain essentially conjugate self-dual $p$-adic Galois representations $ρ$ over CM imaginary fields $F$, which satisfy in particular that $p$ splits in $F$, and that the restriction of $ρ$ on any decomposition group above $p$ is reducible with all the Jordan-Hölder factors of dimension at most $2$. We also show some results on Breuil's locally analytic socle conjecture in certain non-trianguline case. The main results are obtained by establishing an $R=\mathbb{T}$-type result over the $\mathrm{GL}_2(\mathbb{Q}_p)$-ordinary families considered by Breuil-…
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…
Several variable p-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations
设F是全实域，G=GSp（4）_{/F}.本文证明了在一个弱假设下，给定一个Hecke本征系λ对G中的一个固定抛物P是（p，P）-平凡的，存在一个包含λ的多元p-adic族Hecke本征系（它们都是（p，P）-近似平凡的）的下划线{λ}.假设λ对于正则系数系统是上同调的。如果F=Q，则变量的个数为3。此外，在这种情况下，我们构造了与下划线{lambda}相关联的伽罗瓦表示的三变量p-adic族rho_{underline{lambda}}。最后，根据几何假设（这将是满意的，如果一个证明，在家庭的伽罗瓦表示来自格罗滕迪克动机），我们表明，ρ_{下划线{lambda}}是几乎普通的双抛物线的P.这段文字是我们的第一个预印本（在“预出版德L 'universite巴黎北部”）的更新版本，并将出现在“科学年鉴德L' E N S”。
Building on the construction of big Heegner points in the quaternionic setting, and their relation to special values of Rankin-Selberg $L$-functions, we obtain anticyclotomic analogues of the results of Emerton-Pollack-Weston on the variation of Iwasawa invariants in Hida families. In particular, combined with the known cases of the anticyclotomic Iwasawa main conjecture in weight $2$, our results yield a proof of the main conjecture for $p$-ordinary newforms of higher weights and trivial nebentypus.
A curve over a field of characteristic $p$ is called ordinary if the $p$-torsion of its Jacobian as large as possible, that is, an $\mathbb{F}_p$ vector space of dimension equal to its genus. In this paper we consider the following question: fix a finite field $\mathbb{F}_q$ and a family $\mathscr{F}$ of curves over $\mathbb{F}_q$. Then, what is the probability that a curve in this family is ordinary? We answer this question when $\mathscr{F}$ is either the Artin-Schreier family in any characteristic or a superelliptic family in characteristic 2.
We extend the Serre-Tate theory of canonical lifts of ordinary abelian varieties to arbitrary unpolarised families of ordinary abelian varieties parameterised by a $p$-adic formal scheme $S$. We show that the canonical lift is the unique lift to $W(S)$ which admits a $δ$-structure in the sense of Joyal, Buium, and Bousfield. We prove analogous statements for families of ordinary $p$-groups and $p$-divisible groups.
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…
Nearly higher Coleman theory and p-adic L-functions for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ and $\mathrm{GSp}(4) \times \mathrm{GL}(2) \times \mathrm{GL}(2)$
The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have…
Pseudonullity for fine Selmer groups over multiple $\mathbb{Z}_p^{d}$-extensions
Morningside Center of Mathematics, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
