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Larry Rolen

Publications and source records attributed to Larry Rolen.

At least 19 recordsLinked to original sources
arXiv2026

Some convolution identities for mock modular forms arising from the theory of holomorphic projection

Convolution identities and recursive formulas have long played a role in the theory of holomorphic modular forms and their applications. These have served both as striking formulas and as fundamentally useful tools for applications to combinatorics. Recently, there has been renewed interest in examples arising from non-holomorphic modular forms. These include the famous Hurwitz-Kronecker class number relations dating to 1885, and groundbreaking work of Imamo\u{g}lu, Raum, and Richter from 2014. Imamo\u{g}lu, Raum, and Richter developed a theory of holomorphic projection for products of (vector-valued) harmonic Maass forms and holomorphic modular forms to produce many such formulas. This was related shortly thereafter by Duncan, Griffin, and Ono to replicable-type functions in the sense of Conway and Norton, and such recursions played a key role in their proof of the Umbral Moonshine Conjecture. Here, we develop new results on holomorphic projections of such functions which is more convenient for many natural cases. Imamo\u{g}lu, Raum, and Richter's choice corresponds to the case in which the generalized Pell equation $m^2 - Dn^2 = N$ has a square value for $D$, which has only finitely many solutions for a given value of $N$. Our main results cover the cases of general $D$, in particular those for which the Pell equation has infinitely many solutions. As an application, we resolve a recent conjecture of the second author. More generally, our approach yields 18 convolution identities for mock theta functions, which we boil down to 5 identities that can directly be used to prove the others. In the appendices, we also show how these identities can be proven by more direct $q$-series methods; however, the key utility of the holomorphic projection formulas is that they give a tool to automatically discover and verify such formulas.

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arXiv2026

On $U_p$-congruences for meromorphic modular forms with supersingularity

In this paper, we investigate congruences for meromorphic modular forms $F$ which have a pole at a single point $z$ in the fundamental domain of $\mathrm{SL}_2(\mathbb Z)$. For a prime $p$ with good supersingular reduction at the elliptic curve corresponding to $z$, we show that there exists a cusp form $f$ such that $F|U_p^m \equiv f|U_p^m \pmod{p^{\kappa_m}}$, where $\kappa_m=\alpha m -\beta$ with $\alpha$ only depending on the weight of $F$ and $\beta$ depending on $F$ and $p$ but is independent of $m$. In particular, if the space of cusp forms is trivial, then $F|U_p^m\equiv 0 \pmod{p^{\kappa_m}}$ vanishes $p$-adically to a high order. In order to prove these results, we use the fact that $p$ has supersingular reduction to realize $F$ as an overconvergent modular form and then utilize the theory of overconvergent forms to show the congruences.

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arXiv2025

Strict Log-concavity of $k$-coloured Partitions

In recent years, there has been extensive work on inequalities among partition functions. In particular, Nicolas, and independently DeSalvo--Pak, proved that the partition function $p(n)$ is eventually log-concave. Inspired by this and other results, Chern--Fu--Tang first conjectured log-concavity of $k$-coloured partitions. Three of the authors and Tripp later proved this conjecture by introducing recursive sequences and a strict inequality for fractional partition functions, giving explicit errors. In this paper, we show that the log-concavity is, in fact, strict for $k\geq 2$. We shed further light on this phenomenon by utilizing Hardy--Littlewood--P\'olya's notion of majorizing. We prove that for partitions $\bm{a},\bm{b}$ of $n\in\N$, if $\bm b$ majorizes $\bm a$, then $p_k(\bm{b})>p_k(\bm{a})$. Numerical calculations indicate that our result is sharp.

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arXiv2025

On a Divisor Modular Form and a Theta Lift

In 1975, Zagier introduced the highly influential hyperbolic Poincar\'e series $f_{k,D}$. We connect the divisor modular form of $f_{k,D}$ to a new weak Maass form $\omega_{k+1,D}$. Furthermore, we show that the generating function of $\omega_{k+1,D}$ has the same modularity properties as Kohnen and Zagier's fruitful theta kernel generating the $f_{k,D}$'s. This yields a new theta lift.

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arXiv2025

Summation formulas for Hurwitz class numbers and other mock modular coefficients

We prove a formula for weighted sums of the first $n$ coefficients of mock modular forms of moderate growth and apply it to Hurwitz class numbers and coefficients of negative half integral weight Eisenstein series, which take the form of certain quadratic Dirichlet $L$-values. Our formula is a mock modular version of a Bessel-sum identity proved by Chandrasekharan and Narasimhan for Dirichlet series satisfying a functional equation. Our proof utilizes $L$-functions for mock modular Eisenstein series defined by Shankadhar and Singh.

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arXiv2025

Quantum Modular Forms and Resurgence

In 2010, Zagier described a new phenomenon which he called quantum modularity. This connected various examples coming from disparate fields which exhibit near-modular behavior. In the fifteen years since, Zagier's philosophy has informed new developments in areas such as knot theory, 3-dimensional topology, combinatorics, and physics. More recently, the concept of holomorphic quantum modularity has emerged, pointing to a clearer structure for Zagier's original examples. These new developments suggest connections to perturbative quantum field theory, like the theory of resurgence. In 2024, Fantini and Rella proposed a means of codifying some of these connections under their program of ``modular resurgence." Inspired by their work, we unify all of the examples of quantum modular forms in Zagier's original paper under the umbrella of resurgence. In doing so, we strengthen known quantum modularity results for holomorphic Eichler integrals of half-integer weight modular forms. Our main addition to the literature is a collection of median resummation type results which show that the examples of holomorphic quantum modular forms we consider can be recovered from their asymptotics, or the asymptotics of closely related functions.

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arXiv2025

Flipping operators and locally harmonic Maass forms

In the theory of integral weight harmonic Maass forms of manageable growth, two key differential operators, the Bol operator and the shadow operator, play a fundamental role. Harmonic Maass forms of manageable growth canonically split into two parts, and each operator controls one of these parts. A third operator, called the flipping operator, exchanges the role of these two parts. Maass--Poincar\'e series (of parabolic type) form a convenient basis of negative weight harmonic Maass forms of manageable growth, and flipping has the effect of negating an index. Recently, there has been much interest in locally harmonic Maass forms defined by the first author, Kane, and Kohnen. These are lifts of Poincar\'e series of hyperbolic type, and are intimately related to the Shimura and Shintani lifts. In this note, we prove that a similar property holds for the flipping operator applied to these Poincar\'e series.

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arXiv2024

Zeros of Hook Polynomials and Related Questions

We study the zero set of polynomials built from partition statistics, complementing earlier work in this direction by Boyer, Goh, Parry, and others. In particular, addressing a question of Males with two of the authors, we prove asymptotics for the values of $t$-hook polynomials away from an annulus and isolated zeros of a theta function. We also discuss some open problems and present data on other polynomial families, including those associated to deformations of Rogers-Ramanujan functions.

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arXiv2023

Equidistribution and partition polynomials

Using equidistribution criteria, we establish divisibility by cyclotomic polynomials of several partition polynomials of interest, including $spt$-crank, overpartition pairs, and $t$-core partitions. As corollaries, we obtain new proofs of various Ramanujan-type congruences for associated partition functions. Moreover, using results of Erdös and Turán, we establish the equidistribution of roots of partition polynomials on the unit circle including those for the rank, crank, $spt$, and unimodal sequences. Our results complement earlier work on this topic by Stanley, Boyer-Goh, and others. We explain how our methods may be used to establish similar results for other partition polynomials of interest, and offer many related open questions and examples.

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arXiv2023

The second shifted difference of partitions and its applications

A number of recent papers have estimated ratios of the partition function $p(n-j)/p(n)$, which appears in many applications. Here, we prove an easy-to-use effective bound on these ratios. Using this, we then study second shifted difference of partitions, $f(j,n):= p(n) -2p(n-j) +p(n-2j)$, and give another easy-to-use estimate of $f(j,n)$. As applications of these, we prove a shifted convexity property of $p(n)$, as well as giving new estimates of the $k$-rank partition function $N_k(m,n)$ and non-$k$-ary partitions along with their differences.

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arXiv2023

Period-like polynomials for $L$-series associated with half-integral weight cusp forms

Given the L-series of a half-integral weight cusp form, we construct a cohomology class with coefficients in a finite dimensional vector space in a way that parallels the Eichler cohomology in the integral weight case. We also define a lift of half-integral weight cusp forms to integral weight modular forms that is compatible with the $L$-series of the respective forms.

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arXiv2023

Central $L$-values of newforms and local polynomials

In this paper, we characterize the vanishing of twisted central $L$-values attached to newforms of square-free level in terms of certain polynomials of quadratic forms introduced by Zagier and the action of finitely many Hecke operators thereon. To be more precise, we establish that a twisted central $L$-value attached to a newform vanishes if and only if a certain explicitly computable polynomial is constant. We describe these constants explicitly in two different ways. One of the descriptions involves the generalized Hurwitz class numbers, which were introduced by Pei and Wang in $2003$. We provide some numerical examples and conclude by offering some questions for future work.

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arXiv2023

Oscillating asymptotics and conjectures of Andrews

In 1986, Andrews studied the function $\sigma(q)$ from Ramanujan's ``Lost" Notebook, and made several conjectures on its Fourier coefficients $S(n)$, which count certain partition ranks. In 1988, Andrews-Dyson-Hickerson famously resolved these conjectures, relating the coefficients $S(n)$ to the arithmetic of $\mathbb Q(\sqrt{6})$; this relationship was further expounded upon by Cohen in his work on Maass waveforms, and was more recently extended by Zwegers and by Li and Roehrig. A closer inspection of Andrews' original work on $\sigma(q)$ reveals additional related functions and conjectures, which we study in this paper. In particular, we study the function $v_1(q)$, also from Ramanujan's ``Lost" Notebook, a $q$-hypergeometric series with partition-theoretic Fourier coefficients $V_1(n)$, and prove two of Andrews' conjectures on $V_1(n)$ which are parallel to his original conjectures on $S(n)$. Our methods differ from those used by Andrews-Dyson-Hickerson, and require a blend of novel techniques inspired by Garoufalidis' and Zagier's recent work on asymptotics of Nahm sums, with classical techniques including the Circle Method in Analytic Number Theory; our methods may also be applied to determine the asymptotic behavior of similar $q$-hypergeometric series of interest which are not amenable to classical techniques. We also offer explanations of additional related conjectures of Andrews, ultimately connecting the asymptotics of $V_1(n)$ to the arithmetic of $\mathbb Q(\sqrt{-3})$.

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arXiv2023

Asymptotics of parity biases for partitions into distinct parts via Nahm sums

For a random partition, one of the most basic questions is: what can one expect about the parts which arise? For example, what is the distribution of the parts of random partitions modulo $N$? Since most partitions contain a $1$, and indeed many $1$s arise as parts of a random partition, it is natural to expect a skew towards $1\pmod{N}$. This is indeed the case. For instance, Kim, Kim, and Lovejoy recently established ``parity biases'' showing how often one expects partitions to have more odd than even parts. Here, we generalize their work to give asymptotics for biases $\mod N$ for partitions into distinct parts. The proofs rely on the Circle Method and give independently useful techniques for analyzing the asymptotics of Nahm-type $q$-hypergeometric series.

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arXiv2022

Unimodality of ranks and a proof of Stanton's conjecture

Recently, much attention has been given to various inequalities among partition functions. For example, Nicolas, {and later DeSavlvo--Pak,} proved that $p(n)$ is eventually log-concave, and Ji--Zang showed that the cranks are eventually unimodal. This has led to a flurry of recent activity generalizing such results in different directions. At the same time, Stanton recently made deep conjectures on the positivity of certain polynomials associated to ranks and cranks of partitions, with the ultimate goal of pointing the way to ``deeper'' structure refining ranks and cranks. These have been shown to be robust in recent works, which have identified further infinite families of such conjectures in the case of colored partitions. In this paper, we employ the Circle Method to prove unimodality for ranks. As a corollary, we prove Stanton's original conjecture. This points to future study of the positive, integral coefficients Stanton conjectured to exist, hinting at new combinatorial structure yet to be uncovered.

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arXiv2022

Turán inequalities for the plane partition function

Heim, Neuhauser, and Tröger recently established some inequalities for MacMahon's plane partition function $\mathrm{PL}(n)$ that generalize known results for Euler's partition function $p(n)$. They also conjectured that $\mathrm{PL}(n)$ is log-concave for all $n\geq 12.$ We prove this conjecture. Moreover, for every $d\geq 1$, we prove their speculation that $\mathrm{PL}(n)$ satisfies the degree $d$ Turán inequality for sufficiently large $n$. The case where $d=2$ is the case of log-concavity.

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arXiv2022

Limiting Betti distributions of Hilbert schemes on $n$ points

Hausel and Rodriguez-Villegas recently observed that work of Göttsche, combined with a classical result of Erdős and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes $(\mathbb{C}^2)^{[n]}$ on $n$ points, as $n\rightarrow +\infty,$ is a \textit{Gumbel distribution}. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes $((\mathbb{C}^2)^{[n]})^{T_{α,β}}$ that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdős and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer $A\geq 2.$ Furthermore, if $p_k(A;n)$ denotes the number of partitions of $n$ with exactly $k$ parts that are multiples of $A$, then we obtain the asymptotic $$ p_k(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2π)^k}e^{2π\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}}, $$ a result which is of independent interest.

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