The classical theory of $p$-adic (elliptic) modular forms arose in the 1970's from the work of
J.-P.\ Serre \cite{se1} who took $p$-adic limits of the $q$-expansions of these forms. It was soon expanded by N.\ Katz \cite{ka1} with a more functorial approach. Since then the theory has grown in a variety of directions. In the late 1970's, the theory of modular forms associated to Drinfeld modules was born in analogy with elliptic modular forms \cite{go1}, \cite{go2}. The associated expansions at $\infty$ are quite complicated and no obvious limits at finite primes ${\mathfrak v}$ were apparent. Recently, however, there has been progress in the $\mathfrak v$-adic theory, \cite{vi1}. Also recently, A.\ Petrov \cite{pe1}, building on previous work of \cite{lo1}, showed that there is an intermediate expansion at $\infty$ called the "$A$-expansion," and he constructed families of cusp forms with such expansions. It is our purpose in this note to show that Petrov's results also lead to interesting ${\mathfrak v}$-adic cusp forms à la Serre. Moreover the existence of these forms allows us to readily conclude a mysterious decomposition of the associated Hecke action.
It is shown that a system of $r$ quadratic forms over a ${\mathfrak p}$-adic field has a non-trivial common zero as soon as the number of variables exceeds $4r$, providing that the residue class field has cardinality at least $(2r)^r$.
We give two specializations of Krasner's mass formula. The first formula yields the number of extensions of a $\mathfrak{p}$-adic field with given, inertia degree, ramification index, discriminant, and ramification polygon. We then refine this formula further to the case where an additional invariant related to the residual polynomials of the segments of the ramification polygon is given.
On $\mathfrak{P}$-adic continued fractions with extraneous denominators: some explicit finiteness results
Dipartimento di Matematica e Fisica, Università degli Studi di Roma Tre（罗马第三大学数学与物理系）；Univ. Grenoble Alpes, CNRS, IF（格勒诺布尔阿尔卑斯大学）；Research Institute CODE, Universität der Bundeswehr München（慕尼黑联邦国防军大学CODE研究所）；Dipartimento di Informatica, Università di Torino（都灵大学计算机科学系）
We interpolate cohomology classes attached to families of Hilbert modular forms. Using this we construct a two variable $p$-adic L-function which interpolates one variable $p$-adic L-functions.
In this paper, we consider the question of correcting mock modular forms in order to obtain $p$-adic modular forms. In certain cases we show that a mock modular form $M^+$ is a $p$-adic modular form. Furthermore, we prove that otherwise the unique correction of $M^+$ is intimately related to the shadow of $M^+$.
We show that Siegel modular forms of level Γ_0(p^m) are p-adic modular forms. Moreover we show that derivatives of such Siegel modular forms are p-adic. Parts of our results are also valid for vector-valued modular forms. In our approach to p-adic Siegel modular forms we follow Serre closely; his proofs however do not generalize to the Siegel case or need some modifications.
Ming-Lun Hsieh constructed three-variable triple product p-adic L-functions attached to triples of primitive Hida families and proved interpolation formulas. We generalize his result in the unbalanced case and construct a three-variable triple product p-adic L-function attached to a primitive Hida family and two more general p-adic families of modular forms.
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 compute the p-adic L-function (in the sense of Pollack-Stevens and Bellaiche) of the "evil twin" Eisenstein series. We use Solomon's Shintani cocycle to construct a p-adic Shintani modular symbol, and show how to specialize this modular symbol to overconvergent modular symbols with the same p-adic L-function as evil/critical-slope Eisenstein series. This gives a new proof of recent results of Bellaiche and Dasgupta.
I will survey some results in the theory of modular representations of a reductive $p$-adic group, in positive characteristic $\ell \neq p$ and $\ell=p$.
We compute the p-adic Abel-Jacobi map of the product of a Hilbert modular surface and a modular curve at a null-homologous (modified) embedding of the modular curve in this product, evaluated on differentials associated to a Hilbert cuspidal form f of weight (2,2) and a cuspidal form of weight 2. We generalize this computation to suitable null-homological cycles in the fibre products of the universal families on the surface and the curve, evaluated at differentials associated to f and g of higher weights. We express the values of the p-adic Abel-Jacobi map at these weights in terms of a p-adic…
We extend previous work of the author using an idea of Buzzard and give an elementary construction of non-ordinary $p$-adic families of Hilbert Modular Eigenforms.
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…
We prove a variant of Oesterlé's conjecture about $p$-adic periods of the modular curve $X_0(p)$, with an additional $Γ(2)$-structure. We use de Shalit's techniques and $p$-adic uniformization of curves whose reduction is semi-stable.
Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of congruences between modular forms modulo $p$ and modulo $p^n$ are given, and the use of a computer for the study of modular forms is illistrated.
In this paper we establish certain identities connecting $p$-adic hypergeometric functions with 4-th twisted Kloosterman sheaf sum. To prove these identities we express certain character sum over finite field in terms of special values of $p$-adic hypergeometric functions. One conjecture of Evans behaves as a bridge to connect $p$-adic hypergeometric functions with Kloosterman sheaf sum. We also connect $p$-adic hypergeometric functions with Fourier coefficients of certain modular forms.
In~\cite{CS04}, Calegari and Stein studied the congruences between classical cusp forms $S_k(Γ_0(p))$ of prime level and made several conjectures about them. In~\cite{AB07} (resp., ~\cite{BP11}) the authors proved one of those conjectures (resp., their generalizations). In this article, we study the analogous conjecture and its generalizations for Drinfeld modular forms.
We show that if an Eisenstein component of the $p$-adic Hecke algebra associated to modular forms is Gorenstein, then it is necessary that the plus-part of a certain ideal class group is trivial. We also show that this condition is sufficient whenever a conjecture of Sharifi holds.
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 prove hypergeometric type identities for a function defined in terms of quotients of the $p$-adic gamma function. We use these identities to prove a supercongruence conjecture of Rodriguez-Villegas between a truncated $_4F_3$ hypergeometric series and the Fourier coefficients of certain weight four modular form.
We introduce a geometric formalism for studying modular forms of half-integral weight and explore some of its basic properties. Geometric Hecke operators are constructed and some basic spaces of $p$-adic forms are introduced. The $p$-adic theory is greatly expanded in subsequent papers, making that part of this paper largely obsolete.
We show how our p-adic method to compute Galois representations occurring in the torsion of Jacobians of algebraic curves can be adapted to modular curves. The main ingredient is the use of "moduli-friendly" Eisenstein series introduced by Makdisi, which allow us to evaluate modular forms at p-adic of modular curves points and dispenses us of the need for equations of modular curves and for q-expansion computations. The resulting algorithm compares very favourably to the complex-analytic method.
We consider the p-adic Galois representation associated to a Hilbert modular form. We show the compatibility with the local Langlands correspondence at a place divising p under a certain assumption. We also prove the monodromy-weight conjecture. The prime-to-p case is established by Carayol.
In this paper, we study the parallel cases of Zagier's and Folsom-Ono's grids of weakly holomorphic (resp. weakly holomorphic and mock modular) forms of weights 3/2 and 1/2, investigating their $p$-adic properties under the action of Hecke operators.
We come back to the construction of p-adic L-functions attached to cusp forms of even weight k in the spirit of G. Stevens, R. Pollack [7] and M. Greenberg [3] with a new unified presentation including the non-ordinary case. This construction is based on Stevens's modular symbols rather than q-developments. We review the proofs in order to obtain an effective algorithm guaranteeing a given p-adic accuracy.
We give a construction of a wide class of modular symbols attached to reductive groups. As an application we construct a p-adic distribution interpolating the special values of the twisted Rankin-Selberg L-function attached to cuspidal automorphic representations of GL(n) and GL(n-1) over number fields. If the representations are ordinary at p, our distribution is bounded and yields analyticity of the associated p-adic L-function.
Bringmann, Guerzhoy and Kane have shown how to correct mock modular forms by a certain linear combination of the Eichler integral of their shadows in order to obtain p-adic modular forms in the sense of Serre. In this paper, we give a new proof of their results (for good primes p) by employing the geometric theory of harmonic Maass forms developed by the first author and the theory of overconvergent modular forms due to Katz and Coleman.
For a totally real number field $F$ and a nonarchimedean prime $\mathfrak{p}$ of $F$ lying above a prime number $p$ we introduce certain sheaf cohomology groups that intertwine the $\mathfrak{p}^{\infty}$-tower of a quaternionic Hilbert modular variety associated to a quaternion algebra $D$ over $F$ that is split at $\mathfrak{p}$ and a $p$-adically admissible representation of $\mbox{PGL}_2(F_{\mathfrak{p}})$. Applied to infinitesimal $p$-adic deformations of the local factor at $\mathfrak{p}$ of a cuspidal automorphic representation $π$ of $D^*(\mathbb{A})$ this yields a natural construction…
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.
We construct a meromorphic function on the eigencurve that interpolates a square root of the ratio of the central values of two quadratic twists of the $L$-function at classical points.
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.
We investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we prove that v-adic modular forms in the sense of Serre, as defined by Goss and Vincent, are preserved under hyperdifferentiation. Moreover, upon multiplication by a Carlitz factorial, hyperdifferentiation preserves v-integrality.
Let $p$ be a rational prime, $v_p$ the normalized $p$-adic valuation on $\mathbb{Z}$, $q>1$ a $p$-power and $A=\mathbb{F}_q[t]$. Let $\wp\in A$ be an irreducible polynomial and $\mathfrak{n}\in A$ a non-zero element which is prime to $\wp$. Let $k\geq 2$ and $r\geq 1$ be integers. We denote by $S_k(Γ_1(\mathfrak{n}\wp^r))$ the space of Drinfeld cuspforms of level $Γ_1(\mathfrak{n}\wp^r)$ and weight $k$ for $A$. Let $n\geq 1$ be an integer and $a\geq 0$ a rational number. Suppose that $\mathfrak{n}\wp$ has a prime factor of degree one and the generalized eigenspace in $S_k(Γ_1(\mathfrak{n}\wp^r…
We examine the Fourier coefficients of modular forms in a canonical basis for the spaces of weakly holomorphic modular forms of weights 4, 6, 8, 10, and 14, and show that these coefficients are often highly divisible by the primes 2, 3, and 5.
We study the behaviour of ordinary parts of the homology modules of modular curves, associated to a decreasing sequence of congruence subgroups $Γ_1(N2^r)$ for $r \geq 2$, and prove a control theorem for these homology modules.
Let $F$ be a totally real number field and $\mathfrak{o}$ the ring of integers of $F$. We study theta functions which are Hilbert modular forms of half-integral weight for the Hilbert modular group $\mathrm{SL}_2(\mathfrak{o})$. We obtain an equivalent condition that there exists a multiplier system of half-integral weight for $\mathrm{SL}_2(\mathfrak{o})$. We determine the condition of $F$ that there exists a theta function which is a Hilbert modular form of half-integral weight for $\mathrm{SL}_2(\mathfrak{o})$. The theta function is defined by a sum on a fractional ideal $\mathfrak{a}$ of $…
Modular and mock modular forms possess many striking $p$-adic properties, as studied by Bringmann, Guerzhoy, Kane, Kent, Ono, and others. Candelori developed a geometric theory of harmonic Maass forms arising from the de Rham cohomology of modular curves. In the setting of over-convergent $p$-adic modular forms, Candelori and Castella showed this leads to $p$-adic analogs of harmonic Maass forms. In this paper we take an analytic approach to construct $p$-adic analogs of harmonic Maass forms of weight $0$ %and $1/2$ with square free level. Although our approaches differ, where the two theories…
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 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…
Let $E$ be a modular elliptic curve over a totally real number field $F$. We prove the weak exceptional zero conjecture which links a (higher) derivative of the $p$-adic $L$-function attached to $E$ to certain $p$-adic periods attached to the corresponding Hilbert modular form at the places above $p$ where $E$ has split multiplicative reduction. Under some mild restrictions on $p$ and the conductor of $E$ we deduce the exceptional zero conjecture in the strong form (i.e.\ where the automorphic $p$-adic periods are replaced by the $\cL$-invariants of $E$ defined in terms of Tate periods) from a…
Let $p$ be a prime $\ge 5$. We establish explicit rates of overconvergence for members of the "Eisenstein family", notably for the $p$-adic modular function $V(E_{(1,0)}^{\ast})/E_{(1,0)}^{\ast}$ ($V$ the $p$-adic Frobenius operator) that plays a pi\-votal role in Coleman's theory of $p$-adic families of modular forms. The proof goes via an in-depth analysis of rates of overconvergence of $p$-adic modular functions of form $V(E_k)/E_k$ where $E_k$ is the classical Eisenstein series of level $1$ and weight $k$ divisible by $p-1$. Under certain conditions, we extend the latter result to a vast g…
Suppose that p > 5 is a rational prime. Starting from a well-known p-adic analytic family of ordinary elliptic cusp forms of level p due to Hida, we construct a certain p-adic analytic family of holomorphic Siegel cusp forms of arbitrary even genus and of level p associated with Hida's p-adic analytic family via the Duke-Imamoglu lifting provided by Ikeda. Moreover, we also give a similar results for the Siegel Eisenstein series of even genus with trivial Nebentypus.
