Kevin Buzzard, Toby Gee
We survey the progress (or lack thereof!) that has been made on some questions about the p-adic slopes of modular forms that were raised by the first author in [Buz05], discuss strategies for making further progress, and examine other related questions.
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For ordinary modular forms, there are two constructions of a p-adic L-function attached to the non-unit root of the Hecke polynomial, which are conjectured but not known to coincide. We prove this conjecture for modular forms of CM type, by calculating the the critical-slope L-function arising from Kato's Euler system and comparing this with results of Bellaiche on the critical-slope L-function defined using overconvergent modular symbols.
The slope of a p-adic overconvergent eigenform of weight k is the p-adic valuation of its U_p eigenvalue. We find the slope of all 2-adic finite slope overconvergent eigenforms of tame level 1 and weight 0. As a consequence we prove that any finite slope 2-adic overconvergent eigenform of tame level 1 and weight 0 has coefficients in Q_2. These results provide evidence towards conjectures of the first author that predict the slopes of classical p-adic modular forms under a mild ``Gamma_0(N)''-regular hypothesis on p.
We show that for arithmetic weights with a fixed finite order character, the slopes of $U_p$ (for $p=2$) acting on overconvergent Hilbert modular forms of level $U_0(4)$ are independent of the (algebraic part of the) weight and can be obtained by a simple recipe from the classical slopes in parallel weight $3$.
We show that the slopes of the~$U_5$ operator acting on slopes of 5-adic overconvergent modular forms of weight~$k$ with primitive Dirichlet character~$χ$ of conductor~25 are given by either \[ \left\{{1/4}\cdot\lfloor\frac{8i}{5}\rfloor: i \in \N\right\}\text{or }\left\{{1/4}\cdot\lfloor\frac{8i+4}{5}\rfloor: i \in \N\right\}, \] depending on~$k$ and~$χ$.
Slopes of modular forms and the Ghost conjecture
本文提出一种算法，利用Ghost猜想的结果，从模形式斜率序列的前几项计算完整序列，并指出其对称性对Coleman-Mazur特征曲线有潜在意义。
We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes "at the boundary of weight space".
We give a sufficient condition, namely "Buzzard irregularity", for there to exist a cuspidal eigenform which does not have integral p-adic slope.
Reductions of modular Galois representations of Slope (2,3)
本文基于Bhattacharya-Ghate的工作，计算了p-adic数绝对Galois群的二维结晶表示在斜率(2,3)及任意权重下模p约化的半简化。
In a previous article, we constructed an entire power series over $p$-adic weight space (the 'ghost series') and conjectured, in the $Γ_0(N)$-regular case, that this series encodes the slopes of overconvergent modular forms of any $p$-adic weight. In this paper, we construct 'abstract ghost series' which can be associated to various natural subspaces of overconvergent modular forms. This abstraction allows us to generalize our conjecture to, for example, the case of slopes of overconvergent modular forms with a fixed residual representation that is locally reducible at $p$. Ample numerical evi…
We give a new proof of the slope classicality theorem in classical and higher Coleman theory for modular curves at arbitrary level using the completed cohomology classes attached to overconvergent modular forms. The latter give an embedding of the quotient of overconvergent modular forms by classical modular forms, which is the obstruction space for classicality in either cohomological degree, into a unitary representation of $\mathrm{GL}_2(\mathbb{Q}_p)$. The $U_p$ operator becomes a double-coset, and unitarity yields the slope vanishing.
Let $τ$ be the primitive Dirichlet character of conductor 4, let $χ$ be the primitive even Dirichlet character of conductor 8 and let $k$ be an integer. Then the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $2k+1$ and character $τ$ has slopes in the arithmetic progression ${2,4,...,2n,...}$, and the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $k$ and character $χ\cdot τ^k$ has slopes in the arithmetic progression ${1,2,...,n,...}$. We then show that the characteristic polynomials of the Hecke operators $U_2$ and $T_p$ acting on the space…
We p-adically interpolate the relative de Rham cohomology of the universal elliptic curve over strict neighbourhoods of the ordinary locus of modular curves, together with the Hodge filtration and Gauss-Manin connection. Sections of these sheaves provide the so called nearly overconvergent modular forms. This extends previous work of Andreatta, Iovita and Pilloni where we p-adically interpolate powers of the Hodge bundle and in that case the sections coincide with Coleman overconvergent modular forms. We also show that, under suitable assumptions, one can p-adically interpolate the Gauss-Manin…
We give an explicit description of the matrix associated to the $U_p$ operator acting on spaces of overconvergent Hilbert modular forms over totally real fields. Using this, we compute slopes for weights in the centre and near the boundary of weight space for certain real quadratic fields. \added[id=h]{Near the boundary of weight space we see that the slopes do not appear to be given by finite unions of arithmetic progressions but instead can be produced by a simple recipe from which we make a conjecture on the structure of slopes. We also prove a lower bound on the Newton polygon of the $U_p$…
Let χ be the primitive Dirichlet character of conductor 49 defined by χ(3)=ζ, for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U_7 operator acting on the space of overconvergent modular forms on X_1(49) with weight k and character either χ^{7k-6} or χ^{8-7k}, depending on the embedding of Q(ζ) into C_7. By applying results of Coleman, and of Cohen-Oesterlé, we are then able to conclude the slopes of U_7 acting on all classical Hecke newforms of the same weight and character.
The ghost conjecture, formulated by this article's authors, predicts the list of p-adic valuations of the non-zero p-th eigenvalues ("slopes") for overconvergent p-adic modular eigenforms in terms of the Newton polygon of an easy-to-describe power series (the "ghost series"). The prediction is restricted to eigenforms whose Galois representation modulo p is reducible on a decomposition group at p. It has been discovered, however, that the conjecture is not formulated correctly. Here we explain the issue and propose a salvage.
We study $p$-adic families of eigenforms for which the $p$-th Hecke eigenvalue $a_p$ has constant $p$-adic valuation ("constant slope families"). We prove two separate upper bounds for the size of such families. The first is in terms of the logarithmic derivative of $a_p$ while the second depends only on the slope of the family. We also investigate the numerical relationship between our results and the former Gouvêa--Mazur conjecture.
We formulate a conjecture on slopes of overconvergent p-adic cuspforms of any p-adic weight in the Gamma_0(N)-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes "at the boundary of weight space".
We consider the Galois representation associated with a finite slope $p$-adic family of modular forms. We prove that the Lie algebra of its image contains a congruence Lie subalgebra of a non-trivial level. We describe the largest such level in terms of the congruences of the family with $p$-adic CM forms.
Slopes of modular forms and geometry of eigencurves
本文在更强的普遍性条件下证明了模形式斜率与特征曲线几何的局部类比，并推导了多个相关猜想的结论。
Let f be a modular form of weight k and Nebentypus $ψ$. By generalizing a construction of Dabrowski and Delbourgo, we construct a p-adic L-function interpolating the special values of the L-function $L(s,\mathrm{Sym}^2(f)\otimes ξ)$, where $ξ$ is a Dirichlet character. When s=k-1 and $ξ=ψ^{-1}$, this p-adic L-function vanishes due to the presence of a so-called trivial zero. We give a formula for the derivative at s=k-1 of this p-adic L-function when the form f is Steinberg at p. If the weight of f is even, the conductor is even and squarefree, and the Nebentypus is trivial this formula implie…
The goal of this paper is to construct the p-adic analytic family of overconvergent half-integral weight modular forms using Hecke-equivariant overconvergent Shintani lifting. The classical Shintani map is the Hecke-equivariant map from the space of cusp forms of integral weight to the space of cusp forms of half-integral weight. Glenn Stevens proved in [St1] that there is Lamda-adic lifting of this map to the Hida family of ordinary cusp forms of integral weight and consequently constructed Lamda-adic modular eigen form of half-integral weight. The natural thing to do is generalizing his resu…
We establish formulae for the Iwasawa invariants of Mazur--Tate elements of cuspidal eigenforms, generalizing known results in weight 2. Our first theorem deals with forms of "medium" weight, and our second deals with forms of small slope . We give examples illustrating the strange behavior which can occur in the high weight, high slope case.
In this preprint we prove that any finite slope modular form fits into a p-adic family of modular forms which is indexed by the weight. Here, the term p-adic family means that p-adic congruences between weights entail certain p-adic congruences between the corresponding modular forms. We also show that the dimension of the slope subspace of the space of modular forms does not depend on the weight. Both statements are predicted by the Mazur-Gouvea Conjecture, which has been proven by Coleman using methods from rigid analytic geometry. In contrast our proof is based on a comparison of (topologic…
In this article we will show how to compute $U_p$ acting on spaces of overconvergent $p$-adic modular forms when $X_0(p)$ has genus 1. We first give a construction of Banach bases for spaces of overconvergent $p$-adic modular forms, and then give an algorithm to approximate both the characteristic power series of the $U_p$ operator and eigenvectors of finite slope for $U_p$, and present some explicit examples. We will also relate this to the conjectures of Clay on the slopes of overconvergent modular forms.
Slope distributions of $\mathscr{L}$-invariants for modular forms
本文利用幽灵级数，在局部可约且强一般性条件下，确定了模形式的 $\mathscr{L}$-不变量的 $p$-adic 斜率分布，证明了其整数性及等分布性质，证实了 Bergdall--Pollack 猜想。
Let $F$ be a function field over $\mathbb{F}_q$, $A$ its ring of regular functions outside a place $\infty$ and $\mathfrak{p}$ a prime ideal of $A$. First, we develop Hida theory for Drinfeld modular forms of rank $r$ which are of slope zero for a suitably defined Hecke operator $\mathrm{U}_{\mathfrak{p}}$. Second, we show the existence in the finite slope case of families of Drinfeld modular forms varying continuously with respect to the weight. Finally, we show a classicity result: an overconvergent Drinfeld modular form of sufficiently small slope with respect to the weight is a classical D…
We study Kato and Perrin-Riou's critical slope p-adic L-function attached to an ordinary modular form, using the methods of our earlier work with Lei. We show that it may be decomposed as a sum of two bounded measures multiplied by explicit distributions depending only on the local properties of the modular form at p. We use this decomposition to prove results on the zeros of the p-adic L-function, and we show that our results match the behaviour observed in examples calculated by Pollack and Stevens.
Twisted Triple Product $p$-adic $L$-function for Finite Slope Families of Hilbert Modular Forms
本文构造了Hilbert模簇上相对de Rham上同调类的过收敛族，证明其Gauss-Manin联络的$p$-adic迭代收敛性，并据此构造了Hilbert模形式有限斜率族的$p$-adic扭曲三重积$L$-函数。
$p$-adic Asai and twisted triple product $L$-functions for finite slope families
本文利用Hilbert模曲面的近过度收敛高阶Coleman理论，定义了有限斜率Hilbert模形式族的二元$p$-进Asai及扭曲三重积$L$-函数，推广了已有构造并拓展了经典$L$-值的插值范围。
The main result of this paper is an instance of the conjecture made by Gouvea and Mazur (Math. Res. Lett., 1995) which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k should be spanned by the finite slope Hecke eigenforms. Using methods adapted from the work of Buzzard and Calegari I show that for p=2, k = 0, N = 1, this holds for all r in (5/12, 7/12). The proof relies on a certain factorisation of the U_p operator which is known in this case but I conjecture also holds for p = 3 and 5; this conjecture also implies exact for…
Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct p-adic L-functions for non-critical slope rational modular forms, the theory has been extended to construct p-adic L-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope…
We define a pro-$p$ Abelian sheaf on a modular curve of a fixed level $N \geq 5$ divisible by a prime number $p \neq 2$. Every $p$-adic representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ associated to an eigenform is obtained as a quotient of its étale cohomology. For any compact $\mathbb{Z}_p[[1 + N \mathbb{Z}_p]]$-algebra $Λ_1$ satisfying certain suitable conditions, we construct a representation of $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ over $Λ_1$ associated to a $Λ_1$-adic cuspidal eigenform of finite slope as a scalar extension of a quotient of the étale cohomology.
We consider $p$-adic families of Siegel eigenforms of genus $2$ and finite slope, defined as local pieces of an eigenvariety and equipped with a suitable integral structure. Under some assumptions on the residual image, we show that the image of the Galois representation associated with a family is big, in the sense that a Lie algebra attached to it contains a congruence subalgebra of non-zero level. We call Galois level of the family the largest such level. We show that it is trivial when the residual representation has full image. When the residual representation is a symmetric cube, the zer…
Let $p$ be a prime number. We study the slopes of $U_p$-eigenvalues on the subspace of modular forms that can be transferred to a definite quaternion algebra. We give a sharp lower bound of the corresponding Newton polygon. The computation happens over a definite quaternion algebra by Jacquet-Langlands correspondence; it generalizes a prior work of Daniel Jacobs who treated the case of $p=3$ with a particular level. In case when the modular forms have a finite character of conductor highly divisible by $p$, we improve the lower bound to show that the slopes of $U_p$-eigenvalues grow roughly li…
Let $F$ be a totally real field and $p$ be an odd prime which splits completely in $F$. We prove that the eigenvariety associated to a definite quaternion algebra over $F$ satisfies the following property: over a boundary annulus of the weight space, the eigenvariety is a disjoint union of countably infinitely many connected components which are finite over the weight space; on each fixed connected component, the ratios between the $U_\mathfrak{p}$-slopes of points and the $p$-adic valuations of the $\mathfrak{p}$-parameters are bounded by explicit numbers, for all primes $\mathfrak{p}$ of $F$…
We prove a gluing lemma for sections of line bundles on a rigid analytic variety. We apply the lemma, in conjunction with a result of Buzzard's, to give a proof of (a generalization) of Coleman's theorem which states that overconvergent modular forms of small slope are classical. The proof is "geometric" in nature, and is suitable for generalization to other PEL Shimura varieties.
On the variation of slopes over moduli stacks of Galois representations
本文研究了模形式p进斜率随伽罗瓦表示模p变化的关系，并证明了幽灵猜想在直和下具有稳定性。
Let F be a totally real field and p a rational prime unramified in F. We prove a partial classicality theorem for overconvergent Hilbert modular forms: when the slope is small compared to certain but not all weights, an overconvergent form is partially classical. We use the method of analytic continuation.
Following McShane, we employ the stable norm on the homology of the modular torus to investigate the Markov ordering on the set of relatively prime integer pairs $(q,p)$ with $q\ge p\ge0$. Our main theorem is a characterization of slopes along which the Markov ordering is monotone with respect to $q$, confirming conjectures of Lee-Li-Rabideau-Schiffler that refine conjectures of Aigner. The main tool is an explicit computation of the slopes at the corners of the stable norm ball for the modular torus.
Given two newforms $f$ and $g$ of respective weights $k$ and $l$ with $k<l$, Hida constructed a $p$-adic $L$-function interpolating the values of the Rankin convolution of $f$ and $g$ in the critical strip $l \leq s \leq k$. However, this construction works only if $f$ is an ordinary form. Using a method developed by Panchishkin to construct $p$-adic $L$-function associated with modular forms, we generalize this construction to the case where the slope of $f$ is small.
Continuing the study of the Iwasawa theory of symmetric powers of CM modular forms at supersingular primes begun by the first author and Antonio Lei, we prove a Main Conjecture equating the "admissible" $p$-adic $L$-functions to characteristic ideals of "finite-slope" Selmer modules constructed by the second author. As a key ingredient, we improve Rubin's result on the Main Conjecture of Iwasawa theory for imaginary quadratic fields to an equality at inert primes.
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.
Functional equation of the $p$-adic $L$-function of Bianchi modular forms
本文研究类数为 1 的虚二次域上 Bianchi 模形式的 $p$-进 $L$-函数，证明小斜率 $p$-稳定化情形的函数方程，并借助 $p$-进族推广至 $\Sigma$-光滑基变换形式。
The ghost conjecture is first provided by Bergdall and Pollack in [BP-1,BP-2] to study the Up-slopes of spaces of modular forms, which, so far, has already brought plenty of important results. The local version of this conjecture under genericity condition has been solved by Liu-Truong-Xiao-Zhao in [LTXZ-1, LTXZ-2]. In the current paper, we prove a necessary and sufficient condition for a sequence of local ghost series to satisfy that their product has the same Newton polygon to the ghost series build from the direct sum of their associated modules. That answers a common question asked in both…
The theory of overconvergent modular symbols, developed by Rob Pollack and Glenn Stevens, gives a beautiful and effective construction of the $p$-adic $L$-function of a modular form. In this paper, we give an analogue of their results for Bianchi modular forms, that is, modular forms over imaginary quadratic fields. In particular, we prove control theorems that say that the canonical specialisation map from overconvergent to classical Bianchi modular symbols is an isomorphism on small slope eigenspaces of suitable Hecke operators. We also give an explicit link between the classical modular sym…
We construct $p$-adic $L$-functions associated with $p$-refined cohomological cuspidal Hilbert modular forms over any totally real field under a mild hypothesis. Our construction is canonical, varies naturally in $p$-adic families, and does not require any small slope or non-criticality assumptions on the $p$-refinement. The main new ingredients are an adelic definition of a canonical map from overconvergent cohomology to a space of locally analytic distributions on the relevant Galois group and a smoothness theorem for certain eigenvarieties at critically refined points.
We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the U_p operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.
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…
$p$-adic rigidity of eigenforms of infinite slope
本文提出了允许斜率为无穷的 Hecke 特征形 $p$-进族概念，并证明了无穷斜率族仅存在特定扭曲或复乘形式，其证明结合了局部 Galois 表示形变与全局模形式猜想。
Critical Lambda-adic modular forms and bi-ordinary complexes
本文构造了Λ-adic临界斜率过收敛模形式的平坦Λ-模，实现Hida型插值，并插值双普通复形，证明相关Hecke代数的R=T定理，且产生度移位Hecke作用及Stark单位作用。
On the Brauer class of Modular Endomorphism Algebras
本文研究非CM形式所附带的动机的端射影代数的Brauer类，并揭示其分枝性与归一化斜率的关系。
Gouvea and Mazur made a precise conjecture about slopes of modular forms. Weaker versions of this conjecture were established by Coleman and Wan. In this note, we exhibit explicit examples contradicting the full conjecture as it currently stands.
A symmetric $p$-adic symbol for triples of modular forms
本文基于 Garrett-Rankin 三重积 p-adic L-函数引入新的 p-adic 三重符号，证明其在置换三个模形式时满足对称性，并扩展 Lauder 算法以支持普通投影，提供计算示例及高效计算高权重 Poincaré 配对的方法。
The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions L_p^\sharp(f,T) and L_p^\flat(f,T) for a weight two modular form \sum a_n q^n and a good prime p. This generalizes work of Pollack who worked in the supersingular case and also assumed a_p=0. The Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: We bound the rank and estimate the growth of the Tate-Shafarevich group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.
In this work we construct an eigencurve for p-adic modular forms attached to an indefinite quaternion algebra over Q. Our theory includes the definition, both as rules on test objects and sections of line bundle, of p-adic modular forms, convergent and overconvergent, of any p-adic weight. We prove that our modular forms can be put in analytic families over the weight space and we introduce the Hecke operators U and T_l, that can also be put in families. We show that the U-operator acts compactly on the space of overconvergent modular forms. We finally construct the eigencurve, a rigid analyti…
The strong Ramanujan bound for Drinfeld modular forms
本研究针对Drinfeld模形式，证明了1级时𝔭-Hecke算子迹次数的强拉马努金界，改进了原有界并分类了其尖锐性，还得到了与Atkin算子最小斜率及Drinfeld模Frobenius多项式模p分布相关的结果。
Let $F$ be a totally real field in which $p$ is unramified. We prove that, if a cuspidal overconvergent Hilbert cuspidal form has small slopes under $U_p$-operators, then it is classical. Our method follows the original cohomological approach of Coleman. The key ingredient of the proof is giving an explicit description of the Goren-Oort stratification of the special fiber of the Hilbert modular variety. A byproduct of the proof is to show that, at least when $p$ is inert, of the rigid cohomology of the ordinary locus has the same image as the classical forms in the Grothendieck group of Hecke …
Localized Gouvêa-Mazur conjecture
本文在剩余 Galois 表示不可约且限制在 Gal(Q̄_p/Q_p) 上可约且非常一般的条件下，证明了局部化 Gouvêa-Mazur 猜想。
On $p$-adic Asai $L$-functions of Bianchi modular forms at non-ordinary primes and their decomposition into bounded $p$-adic $L$-functions
本文研究了Bianchi模形式在非普通素数处的p-进AsaiL函数的构造及其分解为有界p-进L函数的方法。
Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $Γ$ be the Galois group of the cyclotomic extension of $K$. Fontaine's theory gives a classification of $p$-adic representations of $\mathrm{Gal}\left(\overline{K}/K\right)$ in terms of $(φ,Γ)$-modules. A useful aspect of this classification is Berger's dictionary which expresses invariants coming from $p$-adic Hodge theory in terms of these $\left(φ,Γ\right)$-modules. In this paper, we use the theory of locally analytic vectors to generalize this dictionary to the setting where $Γ$ is the Galois group of a Lubin-Tate extension of $K$. …
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