Iudica, Francesco Maria · The International Islamic University Malaysia Repository (The International Islamic University Malaysia) · 2021
The aim of this thesis is to study p-adic modular forms, which are, roughly \nspeaking, p-adic limits of classical modular forms. In particular, we give a \nself-contained and detailed overview of Serre’s p-adic modular forms, the objects introduced in the foundational paper "Formes modulaires et fonctions zêta p-adiques", [18]. Additionally, a brief introduction to the work of Haruzo Hida on ∧-adic forms will be given
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Simone Maletto · Spectrum Research Repository (Concordia University) · 2018
The aim of this thesis will be introducing an analogue of the classical modular \nforms that can work in the p-adic environment. \nTo do so, we will first try to make sense of a modulo-p concept of modular \nforms. As the classical object is defined over the complex number there is not an \nimmediate way to make this reduction. In order to do so, we have to utilize the \nq-expansion principle to obtain an “integral" object (we use the quote-on-quote \nto remind that the q -expansion of a modular from lives in the localization of ℤ \nat a prime). So the first idea will be to work with those object, to do so we will \nfollow [8]. \nOnce speaking of modular forms modulo p , and modulo pn makes sense, \nwe will start talking about the p-adic theory as described in [6]. This first \nconstruction will be quite easy, but it will have important consequences on the \nnotion of weight of a modular form. \nWhile the approach described above is quite natural and efficient in order \nto have something to work with (we will end up with q -expansion of modular \nforms automatically), to retrive the geometrical nature of those object will be \nmuch harder if we proceed on this path. Therefore we look at the theory of \nmodular forms as section of the sheaf of invariant differentials on the modular \ncurve, following [3]. \nIn the end we will end up with two different definitions, one which gives us \nobjects that are easier to grasp (and to compute), the other which has a more \nclear geometric nature (which is the reason why we study modular forms in \nfirst place). The last section of this thesis show the relation between those two, \nproving that we can recover one object in the first form by object defined in the \nsecond way and vice-versa.
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Ananyo Kazi · Spectrum Research Repository (Concordia University) · 2019
We study the approach of N.M. Katz to define p-adic modular forms, first as sections of tensor powers of the sheaf of invariant differentials on the formal ordinary locus and then as functions on the Katz modula satisfying certain transformation properties under the action of a Galois group, depending on the weight. Next we study the Igusa tower and the big Igusa tower and describe an action of the multiplicative formal group on the Igusa tower which restricts to an action on the Katz tower. We show, that one gets back the [theta]-operator of Serre by differentiating this action.
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Kathrin Bringmann, Pavel Guerzhoy, Ben Kane · Transactions of the American Mathematical Society · 2012
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 $M^+$.
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Siegfried Böcherer · SARAJEVO JOURNAL OF MATHEMATICS · 2024
There is a sophisticated theory of nearly holomorphic Siegel modular forms by Shimura. Using previous results by Nagaoka and myself on Rankin-Cohen operators and theta-operators we will present a proof that quasimodular forms (defined as constant terms or as holomorphic part of a nearly holomorphic Siegel modular form) are always p-adic. * This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.
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Rodolfo Venerucci · Archivio Istituzionale della Ricerca (Universita Degli Studi Di Milano) · 2013
The theme of this Thesis is Iwasawa theory of Hida p-adic analytic families of modular forms. Our main goal is to describe special values of Hida?s p-adic L-functions in the context of a p-adic Birch and Swinnerton-Dyer conjecture for the weight variable.
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染川睦郎, Mutsuro Somekawa · Tokyo Tech Research Repository (Tokyo Institute of Technology) · 2006
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Lawren Smithline · 2000
Let p be prime, N be a positive integer prime to p, and k be an integer. Let Pk(t) be the characteristic series for Atkin’s U operator as an endomorphism of p-adic overconvergent modular forms of tame level N and weight k. Motivated by conjectures of Gouvêa and Mazur, we strengthen a congruence in [W] between coefficients of Pk and Pk ′ for k ′ p-adically close to k. For p − 1 | 12, N = 1, k = 0, we compute a matrix for U whose entries are coefficients in the power series of a rational function of two variables. We apply this computation to show for p = 3 a parabola below the Newton polygon N0 of P0, which coincides with N0 infinitely often. As a consequence, we find a polygonal curve above N0. This tightest bound on N0 yields the strongest congruences between coefficients of P0 and Pk for k of large 3-adic valuation. 1 Overview and background Let p be a prime number, N be a positive integer relatively prime to p, and k be an integer. Let B be a p-adic ring between Zp and Øp, the ring of integers in Cp. Denote by Mk(N,B) the p-adic overconvergent modular forms of tame level N and weight k and by Sk(N,B) the subspace of overconvergent cusp forms. For every weight k, Atkin’s U operator is an endomorphism of Mk(N,B) stabilizing Sk(N,B). Denote by U (k) the restriction of U to Mk(N,B) and by U (k) the restriction to Sk(N,B). These are compact operators, so the characteristic series Pk(t) = det(1 − tU (k)), Qk(t) = det(1 − tU (k)) exist. Let am(Pk) be the coefficient of t m in Pk(t). As a function on a suitably defined space of weights k, am(Pk) is a rigid analytic function of k. Wan[W], and Buzzard[B] construct ˆ N(m), which grows as O(m 2) and depends on p and N and not on k such that vp(am(Pk))> ˆ N(m). Gouvêa and Mazur[GM] note, in an earlier work, the existence of ˆ N(m) and show, for prime p ≥ 5, integer l and positive integer n, vp(am(Pk) − am(P k+lp n (p−1))) ≥ n + 1. (1) 1 Following a remark in [Ka], the result in Equation (1) extends to p = 2,3. In section 2, we show vp(am(Pk) − am(P k+lp n (p−1))) ≥ ˆ N(m − 2) + n + 1. (2) In section 3, for each p = 2,3,5,7,13, N = 1, we construct a matrix M for U (0) with respect to an explicit basis. We show, for Mij the entries of M, i=1 j=1 Mijx i y j is the power series expansion of a rational function of two variables. In section 4, we show for p = 3, m
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Lorenzo La Porta · 2017
The first definition that one usually encounters of modular forms (automorphic functions of the complex Poincare half-plane) might not appear self-explanatory at first, and the understanding
of such definition is made much easier by the geometric interpretation of the modular forms as k-differential forms on the Riemann surfaces associated to each congruence subgroup of SL2(Z), that are one first instance of modular curves. This interpretation of the modular forms can be used the free oneself from the limitation of considering modular
forms over the complex numbers, and give a definition in a much more abstract context.
Similarly, one can proceed from the usual definition over the complex numbers and consider some congruences of the q-expansions of modular forms with integral or p-integral coefficients, and through elementary techniques this leads to the definition of the p-adic modular forms. These, exactly as the classical ones, admit a geometric description as global
sections of a sheaf on a modular curve. This sheaf’s existence is linked to some p-adic representations, and our goal is trace out its construction in the second Chapter, while keeping in mind the analogy with the classical situation, and the elementary definition of p-adic modular forms that is detailed as well in the first Chapter.
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Nicholas M. Katz · Lecture notes in mathematics · 2006
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Matthew Greenberg, M. Seveso · Annales mathématiques du Québec · 2016
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Series on number theory and its applications · 2021
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Yubo Jin · arXiv (Cornell University) · 2022
The purpose of this paper is to study the special values of the standard $L$-functions for quaternionic modular forms using the doubling method. We obtain an integral representation for the $L$-function twisted by a character and construct the $p$-adic measure interpolating certain special $L$-values.
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Siegfried Boecherer, Shōyū Nagaoka · arXiv (Cornell University) · 2013
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.
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Sasaki, Shu · OpenGrey (Institut de l'Information Scientifique et Technique) · 2008
The underlying motivation of the thesis is to generalise the techniques of Buzzard-Taylor and Buzzard to a totally real field F. Their novel approach to modularity of Galois representations via knowing conditions under which an overconvergent Up-eigenform is indeed a classical modular forms, was instrumental in proving new cases of Arlin conjecture for Q and has also been important in the modern theory of p-adic modular forms. My thesis is an exploration of their ideas in the Hilbert case. Analytic continuation of overconvergent Hilbert eigenfor~s depends fundamentally on the rigid geometry of Hilbert modular varieties. The model Deligne-Pappas considered, is highly singular at fibres over ramified primes and in order to avoid technical problems arising from geometry, I construct, in the first chapter, a model over the integers of a finite extension of Qp which desingularises the Deligne-Pappas model, using ideas from Pappas-Rapoport on local models. In the second chapter, I prove an analogue. in the Hilbert case where p splits completely in F, of Coleman's theorem- an overconvergent Up-eigenform of small slope is classical. The methodology is to generalise the earlier work of Buzzard on analytic continuation of overconvergent Up-eigenforms (non-ordinary loci) and then apply Kassaei's idea (ordinary loci) to glue overconvergent forms in the absence of companion forms.
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Kengo Fukunaga · arXiv (Cornell University) · 2019
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.
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Sean Howe · arXiv (Cornell University) · 2020
Following a suggestion of Peter Scholze, we construct an action of $\hat{\mathbb{G}}_m$ on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the $p$-adic modular curve whose ring of functions is Serre's space of $p$-adic modular functions. This action is a local, $p$-adic analog of a global, archimedean action of the circle group $S^1$ on the lattice-unstable locus of the modular curve over $\mathbb{C}$. To construct the $\hat{\mathbb{G}}_m$-action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates $q$; along the way we also prove a natural generalization of Dwork's equation $τ=\log q$ for extensions of $\mathbb{Q}_p/\mathbb{Z}_p$ by $μ_{p^\infty}$ valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of $\hat{\mathbb{G}}_m$ integrates the differential operator $θ$ coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and $p$-adic $L$-functions.
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Mihir Deo · Research in Number Theory · 2025
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Takashi Ichikawa · Journal für die reine und angewandte Mathematik (Crelles Journal) · 2012
Abstract. We study vector-valued p -adic Siegel modular forms based on the arithmetic theory of vector-valued Siegel modular forms over rings. We construct Fourier expansions and Hecke operators for these modular forms, and especially, using the description of these p -adic forms on the Igusa tower, we give a natural definition of p -adic operators.
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Adriana Sofer · 1994
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