P-adic Modular Forms Research Articles

𝑝-adic limit of the Eisenstein series on the exceptional group of type 𝐸_{7,3}

In this paper, we show that the p p -adic limit of a family of Eisenstein series on the exceptional domain where the exceptional group of type E 7 , 3 E_{7,3} acts is an ordinary modular form for a congruence subgroup.

On the Iwasawa main conjecture for generalized Heegner classes in a quaternionic setting

Abstract We prove one divisibility relation of the anticyclotomic Iwasawa Main Conjecture for a higher weight ordinary modular form f and an imaginary quadratic field satisfying a “relaxed” Heegner hypothesis. Let Λ be the anticyclotomic Iwasawa algebra. Following the approach of Howard and Longo–Vigni, we construct the Λ-adic Kolyvagin system of generalized Heegner classes coming from Heegner points on a suitable Shimura curve. As its application, we also prove one divisibility relation in the Iwasawa–Greenberg main conjecture for the p-adic L-function defined by Magrone.

P-adic vertex operator algebras

We postulate axioms for a chiral half of a nonarchimedean 2-dimensional bosonic conformal field theory, that is, a vertex operator algebra in which a p-adic Banach space replaces the traditional Hilbert space. We study some consequences of our axioms leading to the construction of various examples, including p-adic commutative Banach rings and p-adic versions of the Virasoro, Heisenberg, and the Moonshine module vertex operator algebras. Serre p-adic modular forms occur naturally in some of these examples as limits of classical 1-point functions.

A unipotent circle action on 𝑝-adic modular forms

Following a suggestion of Peter Scholze, we construct an action of G m ^ \widehat {\mathbb {G}_m} on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the p p -adic modular curve whose ring of functions is Serre’s space of p p -adic modular functions. This action is a local, p p -adic analog of a global, archimedean action of the circle group S 1 S^1 on the lattice-unstable locus of the modular curve over C \mathbb {C} . To construct the G m ^ \widehat {\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 q ; along the way we also prove a natural generalization of Dwork’s equation τ = log ⁡ q \tau =\log q for extensions of Q p / Z p \mathbb {Q}_p/\mathbb {Z}_p by μ p ∞ \mu _{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 G m ^ \widehat {\mathbb {G}_m} integrates the differential operator θ \theta coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and p p -adic L L -functions.

Asymptotics and Ramanujan's mock theta functions: then and now.

This article is in commemoration of Ramanujan's election as Fellow of The Royal Society 100 years ago, as celebrated at the October 2018 scientific meeting at the Royal Society in London. Ramanujan's last letter to Hardy, written shortly after his election, surrounds his mock theta functions. While these functions have been of great importance and interest in the decades following Ramanujan's death in 1920, it was unclear how exactly they fit into the theory of modular forms-Dyson called this 'a challenge for the future' at another centenary conference in Illinois in 1987, honouring the 100th anniversary of Ramanujan's birth. In the early 2000s, Zwegers finally recognized that Ramanujan had discovered glimpses of special families of non-holomorphic modular forms, which we now know to be Bruinier and Funke's harmonic Maass forms from 2004, the holomorphic parts of which are called mock modular forms. As of a few years ago, a fundamental question from Ramanujan's last letter remained, on a certain asymptotic relationship between mock theta functions and ordinary modular forms. The author, with Ono and Rhoades, revisited Ramanujan's asymptotic claim, and established a connection between mock theta functions and quantum modular forms, which were not defined until 90 years later in 2010 by Zagier. Here, we bring together past and present, and study the relationships between mock modular forms and quantum modular forms, with Ramanujan's mock theta functions as motivation. In particular, we highlight recent work of Bringmann-Rolen, Choi-Lim-Rhoades and Griffin-Ono-Rolen in our discussion. This article is largely expository, but not exclusively: we also establish a new interpretation of Ramanujan's radial asymptotic limits in the subject of topology. This article is part of a discussion meeting issue 'Srinivasa Ramanujan: in celebration of the centenary of his election as FRS'.

Modular forms in the spectral action of Bianchi IX gravitational instantons

We prove a modularity property for the heat kernel and the Seeley-deWitt coefficients of the heat kernel expansion for the Dirac-Laplacian on the Bianchi IX gravitational instantons. We prove, via an isospectrality result for the Dirac operators, that each term in the expansion is a vector-valued modular form, with an associated ordinary (meromorphic) modular form of weight 2. We discuss explicit examples related to well known modular forms. Our results show the existence of arithmetic structures in Euclidean gravity models based on the spectral action functional.

A geometric perspective on p-adic properties of mock modular forms

Bringmann et al. (Trans Am Math Soc 364(5):2393–2410, 2012) showed how to ‘regularize’ 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 a refined form of their results (for good primes p) by employing the geometric theory of harmonic Maass forms developed by Candelori (Math Ann 360(1–2):489–517, 2014) and the theory of overconvergent modular forms due to Katz and Coleman. In particular, our main results imply that the p-adic modular forms in Bringmann et al. (2012) are overconvergent.

Perspectives on mock modular forms

Mock modular forms have played many prominent roles in number theory and other areas of mathematics over the course of the last 15 years. While the term “mock modular form” was not formally defined in the literature until 2007, we now know in hindsight that evidence of this young subject appears much earlier, and that mock modular forms are intimately related to ordinary modular and Maass forms, and Ramanujan's mock theta functions. In this expository article, we offer several different perspectives on mock modular forms – some of which are number theoretic and some of which are not – which together exhibit the strength and scope of their developing theory. They are: combinatorics, q-series and mock theta functions, mathematical physics, number theory, and Moonshine. We also describe some essential results of Bruinier and Funke, and Zwegers, both of which have made tremendous impacts on the development of the theory of mock modular forms. We hope that this article is of interest to both number theorists and enthusiasts – to any reader who is interested in or curious about the history, development, and applications of the subject of mock modular forms, as well as some amount of the mathematical details that go along with them.

P-adic modular forms over unitary Shimura curves and local-global compatibility

p-adic modular forms over unitary Shimura curves and local-global compatibility

Eigenvarieties for Cuspforms Over PEL Type Shimura Varieties With Dense Ordinary Locus

AbstractLet p > 2 be a prime and let X be a compactified PEL Shimura variety of type (A) or (C) such that p is an unramified prime for the PEL datum and such that the ordinary locus is dense in the reduction of X. Using the geometric approach of Andreatta, Iovita, Pilloni, and Stevens, we define the notion of families of overconvergent locally analytic p-adic modular forms of Iwahoric level for X. We show that the systemof eigenvalues of any finite slope cuspidal eigenformof Iwahoric level can be deformed to a family of systems of eigenvalues living over an open subset of the weight space. To prove these results, we actually construct eigenvarieties of the expected dimension that parameterize finite slope systems of eigenvalues appearing in the space of families of cuspidal forms.

On the regulator formulas of Bertolini, Darmon and Rotger

We give a unified, and somewhat simplified, account of the regulator formulas appearing in papers of Bertolini, Darmon and Rotger, describing the syntomic regulator on the first, second and third self-products of modular curves in terms of p-adic modular forms.

Arithmetic properties of Fredholm series forp-adic modular forms

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.

On p-adic modular forms and the Bloch–Okounkov theorem

Bloch–Okounkov studied certain functions on partitions f called shifted symmetric polynomials. They showed that certain q-series arising from these functions (the so-called q-brackets \(\left _q\)) are quasimodular forms. We revisit a family of such functions, denoted \(Q_k\), and study the p-adic properties of their q-brackets. To do this, we define regularized versions \(Q_k^{(p)}\) for primes p. We also use Jacobi forms to show that the \(\left _q\) are quasimodular and find explicit expressions for them in terms of the \(\left _q\).

Explicit congruences for mock modular forms

In recent work of Bringmann, Guerzhoy, and the first author, p-adic modular forms were constructed from mock modular forms. We look at a specific case, starting with the weight −10 mock modular form, and prove explicit congruences.

Special values of anticyclotomic L-functions for modular forms

Abstract In this article, we generalize some works of Bertolini–Darmon and Vatsal on anticyclotomic L-functions attached to modular forms of weight two to higher weight case. We construct a class of anticyclotomic p-adic L-functions for ordinary modular forms and derive the functional equation and the interpolation formula at all critical specializations. Moreover, we prove results on the vanishing of μ-invariant of these p-adic L-functions and the non-vanishing of central L-values with anticyclotomic twists.

On the modularity of certain functions from the Gromov-Witten theory of elliptic orbifolds.

In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov–Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we provide modular completions for several such functions which involve more complicated objects than ordinary modular forms. In particular, we give new closed formulae for special indefinite theta functions of type (1,2) in terms of products of mock modular forms. This formula is also of independent interest.

Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields

This article examines the Fourier expansions of certain non-classical p-adic modular forms of weight one: the eponymous generalised eigenforms of the title, so called because they lie in a generalised eigenspace for the Hecke operators. When this generalised eigenspace contains the theta series attached to a character of a real quadratic field K in which the prime p splits, the coefficients of the attendant generalised eigenform are expressed as p-adic logarithms of algebraic numbers belonging to an idoneous ring class field of K. This suggests an approach to “explicit class field theory” for real quadratic fields which is simpler than the one based on Stark's conjecture or its p-adic variants, and is perhaps closer in spirit to the classical theory of singular moduli.

Mock modular forms and d-distinct partitions

The generating function for partitions into (d+1)-distinct parts (d+1∈N) may be expressed as the q-hypergeometric seriesRd+1(1;q):=∑n⩾0qd+1(n)qn=∑n⩾0q(d+1)(n2)+n(q;q)n, where qd+1(n):=p(n|parts differ by at least d+1). Within combinatorial number theory, this function has long since been of historical importance: a famous identity of Euler and Sylvester implies that the distinct parts function q1/24R1(1;q) is an ordinary modular form, and the celebrated Rogers–Ramanujan identities imply that the 2-distinct parts function q−1/60R2(1;q) is an ordinary modular form. More recently Alder–Andrews, Zagier, and others have studied the general series Rd+1(1;q) for d∈N0. In particular, Zagier has proved for d>1 that these series are never ordinary modular forms; however, other than the cases d=0 (Euler) and d=1 (Rogers–Ramanujan), the precise modular properties of these combinatorial series remain unknown. Here, we prove for integers d⩾1, that the combinatorial q-series Rd+1(1;q), and relevant generalizations, are natural denominators of a new class of mixed mock modular forms. As such, we also obtain many new results as corollaries, including new expressions for Zwegersʼs lauded μ-function, and many of the well-known combinatorial q-series in the subject, confirming the central role that the q-hypergeometric series Rd+1(1;q) play within the theory. Once armed with this realization, we also obtain general theorems on the analytic behavior of Rd+1(1;q) and related series near the unit disk, a priori an impenetrable barrier littered with exponential singularities.

A construction of [formula omitted]-adic modular forms

TextThe classical theory of p-adic (elliptic) modular forms arose in the 1970ʼs in the work of J.-P. Serre [Se1] who took p-adic limits of the q-expansions of these forms. It was soon expanded by N. Katz [Ka1] with a more functorial approach. In the late 1970ʼs, the theory of modular forms associated to Drinfeld modules was born in analogy with elliptic modular forms [Go1,Go2]. The associated expansions at ∞ are quite complicated with no obvious limits at finite primes v. Recently, A. Petrov [Pe1] showed that there is an intermediate expansion at ∞ called the “A-expansion,” and constructed families of cusp forms with such expansions. It is our purpose in this note to show that Petrovʼs results also lead to interesting 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. VideoFor a video summary of this paper, please click here or visit http://youtu.be/xzezUI7-3yc.

Ordinary modular forms and companion points on the eigencurve

We give a new proof of a result due to Breuil and Emerton which relates the splitting behavior at p of the p-adic Galois representation attached to a p-ordinary modular form to the existence of an overconvergent p-adic companion form for f.