17 research outputs found
Weierstrass Points on X 0+(p) and Supersingular J-invariants
We study the arithmetic properties of Weierstrass points on the modular curves X0+(p) for primes p. In particular, we obtain a relationship between the Weierstrass points on X0+(p) and the j-invariants of supersingular elliptic curves in characteristic p
Zeros of some level 2 Eisenstein series
The zeros of classical Eisenstein series satisfy many intriguing properties.
Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc
of the fundamental domain, and recent work by Nozaki explores their interlacing
property. In this paper we extend these distribution properties to a particular
family of Eisenstein series on Gamma(2) because of its elegant connection to a
classical Jacobi elliptic function cn(u) which satisfies a differential
equation. As part of this study we recursively define a sequence of polynomials
from the differential equation mentioned above that allow us to calculate zeros
of these Eisenstein series. We end with a result linking the zeros of these
Eisenstein series to an L-series.Comment: 14 pages, 1 figur
Congruences for the Coefficients of Weakly Holomorphic Modular Forms
Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomenon is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form on any congruence subgroup Γ0 (N). In particular, we give congruences for a wide class of partition functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level
Congruences for the Coefficients of Weakly Holomorphic Modular Forms
Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomenon is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form on any congruence subgroup Γ0 (N). In particular, we give congruences for a wide class of partition functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level
Quantum mock modular forms arising from eta-theta functions
In 2013, Lemke Oliver classified all eta-quotients which are theta functions.
In this paper, we unify the eta-theta functions by constructing mock modular
forms from the eta-theta functions with even characters, such that the shadows
of these mock modular forms are given by the eta-theta functions with odd
characters. In addition, we prove that our mock modular forms are quantum
modular forms. As corollaries, we establish simple finite hypergeometric
expressions which may be used to evaluate Eichler integrals of the odd
eta-theta functions, as well as some curious algebraic identities.Comment: 33 page
Rank Generating Functions as Weakly Holomorphic Modular Forms
Introduction and statement of results. Recent works have illustrated that the Fourier coefficients of harmonic weak Maass forms of weight 1/2 contain a wealth of number-theoretic and combinatorial information. After these works, it is known that many enigmatic q-series (the “mock theta functions” of Ramanujan, and certain rank-generating functions from the theory of partitions, for example) arise naturally as the “holomorphic parts” of such forms. See, for example, Bringmann and Ono [5, 6], Bringmann, Ono, and Rhoades [7], Zwegers [19], Bringmann and Lovejoy [4], Lovejoy and Osburn [12], or see the survey paper [13] for an overview. As another striking example, Bruinier and Ono [9] show that the coefficients of the holomorphic parts of weight 1/2 Maass forms determine the fields of definition of certain Heegner divisors in the Jacobians of modular curves, which in turn determine the vanishing or non-vanishing of derivatives of modular L-function
Zeros of Some Level 2 Eisenstein Series
The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this paper we extend these distribution properties to a particular family of Eisenstein series on Γ(2) because of its elegant connection to a classical Jacobi elliptic function cn(u) which satisfies a differential equation (see formula (1.2)). As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that allow us to calculate zeros of these Eisenstein series. We end with a result linking the zeros of these Eisenstein series to an L-series
Congruences satisfied by eta-quotients
The values of the partition function, and more generally the Fourier
coefficients of many modular forms, are known to satisfy certain congruences.
Results given by Ahlgren and Ono for the partition function and by Treneer for
more general Fourier coefficients state the existence of infinitely many
families of congruences. In this article we give an algorithm for computing
explicit instances of such congruences for eta-quotients. We illustrate our
method with a few examples.Comment: Final version, published in Proceedings of the AM
Constructing Simultaneous Hecke Eigenforms
It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie-Kohnen who considered diagonalization of “bad” Hecke operators on spaces with square free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character
Quantum Mock Modular Forms Arising from Eta–Theta Functions
In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta–theta functions by constructing mock modular forms from the eta–theta functions with even characters, such that the shadows of these mock modular forms are given by the eta–theta functions with odd characters. In addition, we prove that our mock modular forms are quantum modular forms. As corollaries, we establish simple finite hypergeometric expressions which may be used to evaluate Eichler integrals of the odd eta–theta functions, as well as some curious algebraic identities