HKU Scholars Hubhttp://hub.hku.hkThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Fri, 23 Oct 2020 08:35:27 GMT2020-10-23T08:35:27Z50561- Sums of class numbers and mixed mock modular formshttp://hdl.handle.net/10722/252829Title: Sums of class numbers and mixed mock modular forms
Authors: Kane, BR
Abstract: In this talk, we discuss Hurwitz class numbers of integral binary quadratic forms of negative discriminant. The generating function for these is a certain modular object now known as a mock modular form and sums of these turn out to be coefficients of what are now known as mixed mock modular forms. Using the theory of mixed mock modular forms, we obtain explicit formulas for sums of class numbers, proving a conjecture of Brown, Calkin, Flowers, James, Smith, and Stout. This is based on joint work with Kathrin Bringmann.
Description: Invited Talk
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10722/2528292017-01-01T00:00:00Z
- Locally harmonic Maass forms and the kernel of the Shintani lifthttp://hdl.handle.net/10722/199118Title: Locally harmonic Maass forms and the kernel of the Shintani lift
Authors: Bringmann, K; Kohnen, W; Kane, BR
Abstract: In this paper we define a new type of modular object and construct explicit examples of such functions. Our functions are closely related to cusp forms constructed by Zagier which played an important role in the construction by Kohnen and Zagier of a kernel function for the Shimura and Shintani lifts between half-integral and integral weight cusp forms. Although our functions share many properties in common with harmonic weak Maass forms, they also have some properties which strikingly contrast those exhibited by harmonic weak Maass forms. As a first application of the new theory developed in this paper, one obtains a new perspective on the fact that the even periods of Zagier's cusp forms are rational as an easy corollary.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10722/1991182015-01-01T00:00:00Z
- The aliquot constanthttp://hdl.handle.net/10722/192200Title: The aliquot constant
Authors: Bosma, W; Kane, B
Abstract: The average value of log s(n)/n taken over the first N even integers is shown to converge to a constant λ when N tends to infinity; moreover, the value of this constant is approximated and proved to be less than 0. Here, s(n) sums the divisors of n less than n. Thus, the geometric mean of s(n)/n, the growth factor of the function s, in the long run tends to be less than 1. This could be interpreted as probabilistic evidence that aliquot sequences tend to remain bounded.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10722/1922002012-01-01T00:00:00Z
- Duality and differential operators for harmonic Maass formshttp://hdl.handle.net/10722/192202Title: Duality and differential operators for harmonic Maass forms
Authors: Bringmann, K; Kane, B; Rhoades, RC
Abstract: Due to the graded ring nature of classical modular forms, there are many interesting relations between the coefficients of different modular forms. We discuss additional relations arising from Duality, Borcherds products, theta lifts. Using the explicit description of a lift for weakly holomorphic forms, we realize the differential operator ${D}^{k-1} := {( \frac{1} {2\pi \mathrm{i}} \frac{\partial } {\partial z})}^{k-1}$ acting on a harmonic Maass form for integers k > 2 in terms of ${\xi }_{2-k} := 2\mathrm{i}{y}^{2-k}\overline{ \frac{\partial } {\partial \overline{z}}}$ acting on a different form. Using this interpretation, we compute the image of D k − 1. We also answer a question arising in recent work on the p-adic properties of mock modular forms. Additionally, since such lifts are defined up to a weakly holomorphic form, we demonstrate how to construct a canonical lift from holomorphic modular forms to harmonic Maass forms.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10722/1922022013-01-01T00:00:00Z
- The triangular theorem of eight and representation by quadratic polynomialshttp://hdl.handle.net/10722/192201Title: The triangular theorem of eight and representation by quadratic polynomials
Authors: Bosma, W; Kane, B
Abstract: We investigate here the representability of integers as sums of triangular numbers, where the n-th triangular number is given by Tn = n(n+1)/2. In particular, we show that f(x1, x2, ...,xk) = b1Tx1+· · ·+bkTxk, for fixed positive integers b1, b2,. . ., bk, represents every nonnegative integer if and only if it represents 1, 2, 4, 5, and 8. Moreover, if 'cross-terms' are allowed in f, we show that no finite set of positive integers can play an analogous role, in turn showing that there is no overarching finiteness theorem which generalizes the statement from positive definite quadratic forms to totally positive quadratic polynomials. © 2012 American Mathematical Society.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10722/1922012013-01-01T00:00:00Z
- Multiplicative q-hypergeometric series arising from real quadratic fieldshttp://hdl.handle.net/10722/192195Title: Multiplicative q-hypergeometric series arising from real quadratic fields
Authors: Bringmann, K; Kane, B
Abstract: Andrews, Dyson, and Hickerson showed that 2 $ q$-hypergeometric series, going back to Ramanujan, are related to real quadratic fields, which explains interesting properties of their Fourier coefficients. There is also an interesting relation of such series to automorphic forms. Here we construct more such examples arising from interesting combinatorial statistics.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/10722/1921952011-01-01T00:00:00Z
- On cycle integrals of weakly holomorphic modular formshttp://hdl.handle.net/10722/208221Title: On cycle integrals of weakly holomorphic modular forms
Authors: Bringmann, K; Guerzhoy, P; Kane, BR
Abstract: In this paper, we investigate cycle integrals of weakly holomorphic modular forms. We show that these integrals coincide with the cycle integrals of classical cusp forms. We use these results to define a Shintani lift from integral weight weakly holomorphic modular forms to half-integral weight holomorphic modular forms.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10722/2082212015-01-01T00:00:00Z
- A problem of Petersson about weight 0 meromorphic modular formshttp://hdl.handle.net/10722/227319Title: A problem of Petersson about weight 0 meromorphic modular forms
Authors: Bringmann, K; Kane, BR
Abstract: In this paper, we provide an explicit construction of weight 0 meromorphic modular forms. Following work of Petersson, we build these via Poincaré series. There are two main aspects of our investigation which differ from his approach. Firstly, the naive definition of the Poincaré series diverges and one must analytically continue via Hecke's trick. Hecke's trick is further complicated in our situation by the fact that the Fourier expansion does not converge everywhere due to singularities in the upper half-plane so it cannot solely be used to analytically continue the functions. To explain the second difference, we recall that Petersson constructed linear combinations from a family of meromorphic functions which are modular if a certain principal parts condition is satisfied. In contrast to this, we construct linear combinations from a family of non-meromorphic modular forms, known as polar harmonic Maass forms, which are meromorphic whenever the principal parts condition is satisfied.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10722/2273192016-01-01T00:00:00Z
- Half-integral weight p-adic coupling of weakly holomorphic and holomorphic modular formshttp://hdl.handle.net/10722/220626Title: Half-integral weight p-adic coupling of weakly holomorphic and holomorphic modular forms
Authors: Bringmann, K; Guerzhoy, P; Kane, BR
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10722/2206262015-01-01T00:00:00Z
- Equidistribution of Heegner points and ternary quadratic formshttp://hdl.handle.net/10722/192196Title: Equidistribution of Heegner points and ternary quadratic forms
Authors: Jetchev, D; Kane, B
Abstract: We prove new equidistribution results for Galois orbits of Heegner points with respect to single reduction maps at inert primes. The arguments are based on two different techniques: primitive representations of integers by quadratic forms and distribution relations for Heegner points. Our results generalize an equidistribution result with respect to a single reduction map established by Cornut and Vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow. Moreover, for fixed fundamental discriminant and variable conductor, we deduce an effective surjectivity theorem for the reduction map from Heegner points to supersingular points at a fixed inert prime. Our results are applicable to the setting considered by Kolyvagin in the construction of the Heegner points Euler system.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/10722/1921962011-01-01T00:00:00Z
- Analogues of the Ramanujan-Mordell Theoremhttp://hdl.handle.net/10722/231991Title: Analogues of the Ramanujan-Mordell Theorem
Authors: Cooper, S; Kane, BR; Ye, D
Abstract: The Ramanujan-Mordell Theorem for sums of an even number of squares is extended to other quadratic forms and quadratic polynomials.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10722/2319912017-01-01T00:00:00Z
- Modular local polynomialshttp://hdl.handle.net/10722/208220Title: Modular local polynomials
Authors: Bringmann, K; Kane, BR
Abstract: In this paper, we consider modular local polynomials. These functions satisfy modularity while they are locally defined as polynomials outside of an exceptional set. We prove an inequality for the dimension of the space of such forms when the exceptional set is given by certain natural geodesics related to binary quadratic forms of (positive) discriminant D. We furthermore show that the dimension is the largest possible if and only if D is an even square. Following this, we describe how to use the methods developped in this paper to establish an algorithm which explicitly determines the space of modular local polynomials for each D.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/10722/2082202011-01-01T00:00:00Z
- Sign changes of Fourier coefficients of cusp forms and representations of integers by quadratic polynomialshttp://hdl.handle.net/10722/251930Title: Sign changes of Fourier coefficients of cusp forms and representations of integers by quadratic polynomials
Authors: Kane, BR
Abstract: In this talk, we discuss applications of sign changes of Fourier coefficients of cusp forms to the theory of quadratic polynomials. There are two applications discussed; firstly, an algorithm for finding an elliptic curve with a given endomorphism ring is shown to halt, and secondly possible future directions are discussed.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10722/2519302017-01-01T00:00:00Z
- Meromorphic modular forms and polar harmonic Maass formshttp://hdl.handle.net/10722/240110Title: Meromorphic modular forms and polar harmonic Maass forms
Authors: Kane, BR
Abstract: In this talk, we discuss applications of a new modular object known as polar harmonic Maass forms on some old questions of Petersson about meromorphic modular forms. These applications include an explicit version of the Riemann-Roch Theorem and Fourier coefficients for meromorphic modular forms.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10722/2401102015-01-01T00:00:00Z
- The Bruinier--Funke pairing and the orthogonal complement of unary theta functionshttp://hdl.handle.net/10722/242356Title: The Bruinier--Funke pairing and the orthogonal complement of unary theta functions
Authors: Kane, BR; Man, SH
Abstract: We describe an algorithm for computing the inner product between a holomorphic modular form and a unary theta function, in order to determine whether the form is orthogonal to unary theta functions without needing a basis of the entire space of modular forms and without needing to use linear algebra to decompose this space completely.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10722/2423562016-01-01T00:00:00Z
- Inequalities for differences of Dyson's rank for all odd modulihttp://hdl.handle.net/10722/192192Title: Inequalities for differences of Dyson's rank for all odd moduli
Authors: Bringmann, K; Kane, B
Abstract: Kathrin Bringmann (Mathematisches Institut, Universität Köln, Weyertal 86-90, D-50931 Köln, Germany)
Ben Kane (Wiskunde Afdeling, Radboud Universiteit, Postbus 9010, 6500 GL, Nijmegen, Netherlands)
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10722/1921922010-01-01T00:00:00Z
- Faber polynomials and poincaré serieshttp://hdl.handle.net/10722/192197Title: Faber polynomials and poincaré series; Faber polynomials and poincare series
Authors: Kane, B
Abstract: In this paper we consider weakly holomorphic modular forms (i.e., those meromorphic modular forms for which poles only possibly occur at the cusps) of weight 2−k∈2\Z for the full modular group \SL2(\Z). The space has a distinguished set of generators f2−k,m. Such weakly holomorphic modular forms have been classified in terms of finitely many Eisenstein series, the unique weight 12 newform Δ, and certain Faber polynomials in the modular invariant j(z), the Hauptmodul for \SL2(\Z). We employ the theory of harmonic weak Maass forms and (non-holomorphic) Maass–Poincaré series in order to obtain the asymptotic growth of the coefficients of these Faber polynomials. Along the way, we obtain an asymptotic formula for the partial derivatives of the Maass–Poincaré series with respect to y as well as extending an asymptotic for the growth of the ℓth repeated integral of the Gauss error function at x to include ℓ∈\R and a wider range of x.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/10722/1921972011-01-01T00:00:00Z
- Regularized Petersson inner products for meromorphic modular formshttp://hdl.handle.net/10722/232365Title: Regularized Petersson inner products for meromorphic modular forms
Authors: Kane, BR
Abstract: We investigate the history of inner products within the theory of modular forms. We first give the history of the applications of Petersson's original definition for the inner product of $S_{2k}$ and then recall Zagier's extension to a non-degenerate (but not necessarily positive-definite) inner product on all holomorphic modular forms. We then recall the history of the so-called ``regularization'' of the inner product to extend it to weakly holomorphic modular forms originally by Petersson and then later independently rediscovered by Harvey--Moore and Borcherds, as well as its applications to theta lifts by Borcherds, Bruinier--Funke, and many more recent authors. This has been recently extended to a well-defined inner product on all weakly holomorphic modular forms by Bringmann, Diamantis, and Ehlen. Finally, we consider inner products on meromorphic modular forms which have poles in the upper half-plane. Petersson also defined a regularization in this case by cutting out small neighborhoods around each pole occurring in the fundamental domain; Bringmann, von Pippich, and the author have recently constructed an extension of this regularization, which, when combined with the regularization of Bringmann, Diamantis, and Ehlen, yields an inner product that is well-defined and finite on all meromorphic modular forms.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10722/2323652016-01-01T00:00:00Z
- New identities involving sums of the tails related to real quadratic fieldshttp://hdl.handle.net/10722/192193Title: New identities involving sums of the tails related to real quadratic fields
Authors: Bringmann, K; Kane, B
Abstract: In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of $\mathbb {Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ . Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields at negative integers.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10722/1921932010-01-01T00:00:00Z
- Universal sums of polygonal numbershttp://hdl.handle.net/10722/253704Title: Universal sums of polygonal numbers
Authors: Kane, BR
Abstract: In this talk, we will consider certain “finiteness theorems”. In a celebrated result of Conway and Schneeberger, it was shown that every positive-definite integral quadratic form is universal if and only if it represents every integer up to 15 (the proof was later simplified and generalized by Bhargava). Applying this result to arbitrary repeated sums of squares (i.e., diagonal quadratic forms), we consider generalizations where the quadratic form is replaced with sums of m-gonal numbers (the m = 4 case is sums of squares). One finds that for each m there exists a finiteness result of the above type, and the main result in this talk is a bound on the growth of the constant up to which one must check for universality. This is joint work with Jingbo Liu.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10722/2537042018-01-01T00:00:00Z
- On signs of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ringhttp://hdl.handle.net/10722/253292Title: On signs of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ring
Authors: Kane, Ben; Fung, King Cheon
Abstract: In this talk, we will consider an application coming from the alternation of signs of Fourier coefficients of (half-integral weight) cusp forms. In particular, we consider certain cusp forms coming from the difference of two ternary theta functions associated to the norm map applied to trace zero elemeents within a maximal order of the definite quaternion algebra ramified precisely at p and ∞. It was conjectured by Chevyrev and Galbraith that one such theta function could not “dominate” the other, i.e., its Fourier coefficients essentially could not always be larger. We prove this conjecture by recognizing that the coefficients of the difference change sign infinitely often due to recent work of others on sign changes. The conjecture of Chevyreve and Galbraith was originally made because it implies that a certain algorithm they had developed would halt with the correct answer, and hence as a corollary we conclude that their algorithm indeed halts. This is joint work with my Masters student King Cheong Fung, who is investigating further related directions.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10722/2532922016-01-01T00:00:00Z
- Secord-order cusp forms and mixed mock modular formshttp://hdl.handle.net/10722/192203Title: Secord-order cusp forms and mixed mock modular forms
Authors: Bringmann, K; Kane, B
Abstract: In this paper, we consider the space of second order cusp forms. We determine that this space is precisely the same as a certain subspace of mixed mock modular forms. Based upon Poincaré series of Diamantis and O'Sullivan (Trans. Am. Math. Soc. 360:5629-5666, 2008) which span the space of second order cusp forms, we construct Poincaré series which span a natural (more general) subspace of mixed mock modular forms. © 2012 Springer Science+Business Media, LLC.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10722/1922032013-01-01T00:00:00Z
- Polar harmonic Maass formshttp://hdl.handle.net/10722/250152Title: Polar harmonic Maass forms
Authors: Kane, BR
http://hdl.handle.net/10722/250152
- Regularized inner products and meromorphic modular formshttp://hdl.handle.net/10722/253736Title: Regularized inner products and meromorphic modular forms
Authors: Kane, BR
Abstract: In this talk, we consider a regularization of Petersson's inner product which is well-defined (and finite) between two meromorphic modular forms and agrees with Petersson's inner product whenever the latter exists. We take the inner product between a special family of meromorphic modular forms and show a connection with the automorphic Green's function and certain functions which are called polar harmonic Maass forms. We then discuss some applications of these polar harmonic Maass forms, including formulas and asymptotics for Fourier coefficients of meromorphic modular forms, construction of a basis of meromorphic modular forms of non-positive weight, and an algorithm to compute the divisor of a given meromorphic modular form given only its Fourier expansion. Most of the talk is joint work with Kathrin Bringmann and Anna von Pippich, while the first two applications are joint work with Kathrin Bringmann and the application to divisors of modular forms is joint work with Kathrin Bringmann, Steffen Loebrich, Ken Ono, and Larry Rolen.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10722/2537362017-01-01T00:00:00Z
- On almost universal mixed sums of squares and triangular numbershttp://hdl.handle.net/10722/192194Title: On almost universal mixed sums of squares and triangular numbers
Authors: Kane, B; Sun, ZW
Abstract: In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than $ 2719$ can be represented by the famous Ramanujan form $ x^2+y^2+10z^2$; equivalently the form $ 2x^2+5y^2+4T_z$ represents all integers greater than 1359, where $ T_z$ denotes the triangular number $ z(z+1)/2$. Given positive integers $ a,b,c$ we employ modular forms and the theory of quadratic forms to determine completely when the general form $ ax^2+by^2+cT_z$ represents sufficiently large integers and to establish similar results for the forms $ ax^2+bT_y+cT_z$ and $ aT_x+bT_y+cT_z$. Here are some consequences of our main theorems: (i) All sufficiently large odd numbers have the form $ 2ax^2+y^2+z^2$ if and only if all prime divisors of $ a$ are congruent to 1 modulo 4. (ii) The form $ ax^2+y^2+T_z$ is almost universal (i.e., it represents sufficiently large integers) if and only if each odd prime divisor of $ a$ is congruent to 1 or 3 modulo 8. (iii) $ ax^2+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4. (iv) When $ v_2(a)\not=3$, the form $ aT_x+T_y+T_z$ is almost universal if and only if all odd prime divisors of $ a$ are congruent to 1 modulo 4 and $ v_2(a)\not=5,7,\ldots$, where $ v_2(a)$ is the $ 2$-adic order of $ a$.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10722/1921942010-01-01T00:00:00Z
- Representing sets with sums of triangular numbershttp://hdl.handle.net/10722/192198Title: Representing sets with sums of triangular numbers
Authors: Kane, B
Abstract: We investigate here sums of triangular numbers f (x) := ∑ i b iT xi where T n is the nth triangular number. We show that for a set of positive integers S, there is a finite subset S 0 such that f represents S if and only if f represents S 0. However, computationally determining S 0 is ineffective for many choices of S. We give an explicit and efficient algorithm to determine the set S 0 under certain generalized Riemann hypotheses, and implement the algorithm to determine S 0 when S is the set of all odd integers. © The Author 2009. Published by Oxford University Press. All rights reserved.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10722/1921982009-01-01T00:00:00Z
- Representations of integers by ternary quadratic formshttp://hdl.handle.net/10722/192191Title: Representations of integers by ternary quadratic forms
Authors: Kane, B
Abstract: We investigate the representation of integers by quadratic forms whose theta series lie in Kohnen's plus space , where p is a prime. Conditional upon certain GRH hypotheses, we show effectively that every sufficiently large discriminant with bounded divisibility by p is represented by the form, up to local conditions. We give an algorithm for explicitly calculating the bounds. For small p, we then use a computer to find the full list of all discriminants not represented by the form. Finally, conditional upon GRH for L-functions of weight 2 newforms, we give an algorithm for computing the implied constant of the Ramanujan–Petersson conjecture for weight 3/2 cusp forms of level 4N in Kohnen's plus space with N odd and squarefree.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10722/1921912010-01-01T00:00:00Z
- On the Andrews-Zagier asymptotics for partitions without sequenceshttp://hdl.handle.net/10722/238657Title: On the Andrews-Zagier asymptotics for partitions without sequences
Authors: Bringmann, K; Kane, BR; Parry, D; Rhoades, R
Abstract: In this paper, we establish asymptotics of radial limits for certain functions of Wright. These functions appear in bootstrap percolation and the generating function for partitions without sequences of k consecutive part sizes. We specifically establish asymptotics numerically obtained by Zagier in the case k=3.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10722/2386572017-01-01T00:00:00Z
- Inequalities for full rank differences of 2-marked Durfee symbolshttp://hdl.handle.net/10722/192199Title: Inequalities for full rank differences of 2-marked Durfee symbols
Authors: Bringmann, K; Kane, B
Abstract: In this paper, we obtain infinitely many non-trivial identities and inequalities between full rank differences for 2-marked Durfee symbols, a generalization of partitions introduced by Andrews. A certain strict inequality, which almost always holds, shows that identities for Dysonʼs rank, similar to those proven by Atkin and Swinnerton-Dyer, are quite rare. By showing an analogous strict inequality, we show that such non-trivial identities are also rare for the full rank, but on the other hand we obtain an infinite family of non-trivial identities, in contrast with the partition theoretic case.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10722/1921992012-01-01T00:00:00Z
- On simultaneous s-cores/t-coreshttp://hdl.handle.net/10722/192190Title: On simultaneous s-cores/t-cores
Authors: Aukerman, D; Kane, B; Sze, L
Abstract: In this paper, the authors investigate the question of when a partition of n∈N is an s-core and also a t-core when s and t are not relatively prime. A characterization of all such s/t-cores is given, as well as a generating function dependent upon the polynomial generating functions for s/t-cores when s and t are relatively prime. Furthermore, characterizations and generating functions are given for s/t-cores which are self-conjugate and also for (e,r)/(e′,r)-cores.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10722/1921902009-01-01T00:00:00Z
- An algebraic approach to the Siegel-Weil average for binary quadratic formshttp://hdl.handle.net/10722/268818Title: An algebraic approach to the Siegel-Weil average for binary quadratic forms
Authors: Kane, BR
Abstract: In this talk, we will consider the celebrated results of Siegel and Weil about the number of representations by the genus of a given positive-definite integral quadratic form. By restricting to the (very special) case of binary quadratic forms representing integers and investigating the question via the associated algebraic theory of quadratic fields and Gauss's composition law, we obtain a new proof that these are coefficients of certain Eisenstein series and obtain nice explicit formulas for their evaluations. This is based on joint work with Pavel Guerzhoy.
Description: Invited Talk - Venue: Harish-Chandra Research Institute - Second program in the series `International conference on class groups of number fields and related topics'
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10722/2688182018-01-01T00:00:00Z
- Polar harmonic Maass forms and their applicationshttp://hdl.handle.net/10722/229217Title: Polar harmonic Maass forms and their applications
Authors: Bringmann, K; Kane, BR
Abstract: In this survey, we present recent results of the authors about non-meromorphic modular objects known as polar harmonic Maass forms. These include the computation of Fourier coefficients of meromorphic modular forms and relations between inner products of meromorphic modular forms and higher Green's functions evaluated at CM-points.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10722/2292172016-01-01T00:00:00Z
- Central L-values of elliptic curves, Tunnell's Theorem, and locally harmonic Maass formshttp://hdl.handle.net/10722/267952Title: Central L-values of elliptic curves, Tunnell's Theorem, and locally harmonic Maass forms
Authors: Kane, BR
Abstract: In this talk, we will discuss a way to study the vanishing of central L-values of elliptic curves via locally harmonic Maass forms. In particular, we will discuss some combinatorial applications, such as an alternative formula for Tunnell's resolution of the congruent number problem (assuming BSD) and for representations of primes as the sum of two cubes. This talk is based on joint work with Stephan Ehlen, Pavel Guerzhoy, and Larry Rolen.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10722/2679522018-01-01T00:00:00Z
- Regularized inner products of meromorphic modular forms and higher Green's functionshttp://hdl.handle.net/10722/252092Title: Regularized inner products of meromorphic modular forms and higher Green's functions
Authors: Bringmann, K; Kane, BR; von Pippich, A
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10722/2520922019-01-01T00:00:00Z
- On sign changes of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ringhttp://hdl.handle.net/10722/231990Title: On sign changes of cusp forms and the halting of an algorithm to construct a supersingular elliptic curve with a given endomorphism ring
Authors: FUNG, KC; Kane, B
Abstract: Chevyrev and Galbraith recently devised an algorithm which inputs a maximal order of the quaternion algebra ramified at one prime and infinity and constructs a supersingular elliptic curve whose endomorphism ring is precisely this maximal order. They proved that their algorithm is correct whenever it halts, but did not show that it always terminates. They did however prove that the algorithm halts under a reasonable assumption which they conjectured to be true. It is the purpose of this paper to verify their conjecture and in turn prove that their algorithm always halts. More precisely, Chevyrev and Galbraith investigated the theta series associated with the norm maps from primitive elements of two maximal orders. They conjectured that if one of these theta series 'dominated' the other in the sense that the nth (Fourier) coefficient of one was always larger than or equal to the nth coefficient of the other, then the maximal orders are actually isomorphic. We prove that this is the case. © 2017 American Mathematical Society.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10722/2319902018-01-01T00:00:00Z
- Central L-values of elliptic curves and local polynomialshttp://hdl.handle.net/10722/279178Title: Central L-values of elliptic curves and local polynomials
Authors: Ehlen, S; Guerzhoy, P; Kane, B; Rolen, L
Abstract: Here we study the recently introduced notion of a locally harmonic Maass form and its applications to the theory of L-functions. In particular, we find finite formulas for certain twisted central L-values of a family of elliptic curves in terms of finite sums over canonical binary quadratic forms. This yields vastly simpler formulas related to work of Birch and Swinnerton-Dyer for such L-values, and extends beyond their framework to special non-CM elliptic curves.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10722/2791782020-01-01T00:00:00Z
- An extension of Rohrlich's Theorem to the j-functionhttp://hdl.handle.net/10722/279451Title: An extension of Rohrlich's Theorem to the j-function
Authors: Bringmann, K; Kane, B
Abstract: We start by recalling the following theorem of Rohrlich [17]. To state it, let denote half of the size of the stabilizer of in and for a meromorphic function let be the order of vanishing of at. Moreover, define, where, and set, where. Rohrlich's theorem may be stated in terms of the Petersson inner product, denoted by. © The Author(s) 2020.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10722/2794512020-01-01T00:00:00Z
- An algebraic approach to the Siegel-Weil average for binary quadratic formshttp://hdl.handle.net/10722/268076Title: An algebraic approach to the Siegel-Weil average for binary quadratic forms
Authors: Kane, BR
Abstract: In this talk, we will consider the celebrated results of Siegel and Weil about the number of representations by the genus of a given positivedefinite integral quadratic form. By restricting to the (very special) case of binary quadratic forms representing integers and investigating the question via the associated algebraic theory of quadratic fields and Gauss’s composition law, we obtain a new proof that these are coefficients of certain Eisenstein series and obtain nice explicit formulas for their evaluations. This is based on joint work with Pavel Guerzhoy.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10722/2680762019-01-01T00:00:00Z
- An algebraic and analytic approach to spinor exceptional behavior in translated latticeshttp://hdl.handle.net/10722/259723Title: An algebraic and analytic approach to spinor exceptional behavior in translated lattices
Authors: Haensch, A; Kane, BR
Abstract: In this announcement we discuss the representation problem for translations of positive-definite lattices via a discussion of representation by inhomogeneous quadratic polynomials. In particular, we give a survey of the extent to which algebraic and analytic methods are useful in determining how the behavior of the spinor genus contributes to failure of the local-global principle.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10722/2597232016-01-01T00:00:00Z
- Explicit congruences for mock modular formshttp://hdl.handle.net/10722/223853Title: Explicit congruences for mock modular forms
Authors: Kane, BR; Waldherr, M
Abstract: 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 a weight -10 mock modular form, and prove explicit congruences.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10722/2238532016-01-01T00:00:00Z
- Number theoretic generalization of the Monster denominator formulahttp://hdl.handle.net/10722/247471Title: Number theoretic generalization of the Monster denominator formula
Authors: Bringmann, K; Kane, BR; Löbrich, S; Rolen, L; Ono, K
Abstract: The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z) - j( au)$ in terms of the Hecke system of $SL2(Z)$-modular functions $j_n(z)$. This formula can be reformulated entirely number theoretically. Namely, it is equivalent to the description of the generating function for the $j_n(z)$ as a weight 2 modular form in with a pole at $z$. Although these results rely on the fact that $X_0(1)$ has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the $X_0(N)$ modular curves. In this survey of recent work, we discuss this generalization, and we offer an introduction to the theory of polar harmonic Maass forms. We conclude with applications to formulas of Ramanujan and Green's functions.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10722/2474712017-01-01T00:00:00Z
- Ramanujan and coefficients of meromorphic modular formshttp://hdl.handle.net/10722/223854Title: Ramanujan and coefficients of meromorphic modular forms
Authors: Bringmann, K; Kane, BR
Abstract: The study of Fourier coefficients of meromorphic modular forms dates back to Ramanujan, who, together with Hardy, studied the reciprocal of the weight 6 Eisenstein series. Ramanujan conjectured a number of further identities for other meromorphic modular forms and quasi-modular forms which were subsequently established by Berndt, Bialek, and Yee. In this paper, we place these identities into the context of a larger family by making use of Poincaré series introduced by Petersson and a new family of Poincaré series which we construct here and which are of independent interest. In addition we establish a number of new explicit identities. In particular, we give the first examples of Fourier expansions for meromorphic modular form with third-order poles and quasi-meromorphic modular forms with second-order poles.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10722/2238542017-01-01T00:00:00Z
- Regularized inner products and weakly holomorphic Hecke eigenformshttp://hdl.handle.net/10722/249999Title: Regularized inner products and weakly holomorphic Hecke eigenforms
Authors: Bringmann, K; Kane, BR
Abstract: An interpretation of Guerzhoy's integral weight weakly holomorphic Hecke eigenforms via regularized inner products is given in this paper. In particular, Guerzhoy's eigenforms are eigenvectors in the quotient space of weakly holomorphic modular forms modulo a certain subspace, and it is shown in this paper that the subspace which is factored out is precisely the subspace that is orthogonal to all weakly holomorphic modular forms under an appropriate (regularize) inner product.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10722/2499992017-01-01T00:00:00Z
- Universal sums of m -gonal numbershttp://hdl.handle.net/10722/266440Title: Universal sums of m -gonal numbers
Authors: Kane, B; Liu, J
Abstract: In this paper we study universal quadratic polynomials which arise as sums of polygonal numbers. Specifically, we determine an asymptotic upper bound (as a function of m) on the size of the (minimal) subset of the natural numbers such that if a sum of m-gonal numbers represents every element of that sent, then it represents all natural numbers.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10722/2664402019-01-01T00:00:00Z
- The mathematical key to unlocking the mysteries of cryptographyhttp://hdl.handle.net/10722/271528Title: The mathematical key to unlocking the mysteries of cryptography
Authors: Kane, BR
Abstract: We discuss the basic mathematics behind cryptography and the role that number theory has played in its development. This is a talk aimed at undergraduates and should also be suitable for interested advanced high school students.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10722/2715282018-01-01T00:00:00Z
- Interesting identities involving weighted representations of integers as sums of arbitrarily many squareshttp://hdl.handle.net/10722/277448Title: Interesting identities involving weighted representations of integers as sums of arbitrarily many squares
Authors: Jang, MJ; Kane, B; Kohnen, W; Man, SH
Abstract: We consider the number of ways to write an integer as a sum of squares, a problem with a long history going back at least to Fermat. The previous studies in this area generally fix the number of squares which may occur and then either use algebraic techniques or connect these to coefficients of certain complex analytic functions with many symmetries known as modular forms, from which one may use techniques in complex and real analysis to study these numbers. In this paper, we consider sums with arbitrarily many squares, but give a certain natural weighting to each representation. Although there are a very large number of such representations of each integer, we see that the weighting induces massive cancellation, and we furthermore prove that these weighted sums are again coefficients of modular forms, giving precise formulas for them in terms of sums of divisors of the integer being represented.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10722/2774482019-01-01T00:00:00Z
- Differential operators on polar harmonic Maass forms and elliptic dualityhttp://hdl.handle.net/10722/269429Title: Differential operators on polar harmonic Maass forms and elliptic duality
Authors: Bringmann, K; Jenkins, P; Kane, BR
Abstract: In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincaré series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10722/2694292019-01-01T00:00:00Z
- Sign changes of Fourier coefficients of cusp forms and representations of integers by quadratic polynomialshttp://hdl.handle.net/10722/285137Title: Sign changes of Fourier coefficients of cusp forms and representations of integers by quadratic polynomials
Authors: Kane, BR
Abstract: In this talk, we discuss an application of recent results involving sign changes of Fourier coefficients of cusp forms. In particular, we discuss their applications to the study of representations of integers by totally positive ternary quadratic polynomials. Part of this is work in progress, while another is based on joint work with K.C. Fung, with a specific application of the latter part in showing the halting of an algorithm to compute a supersingular elliptic with a given enfomorphism ring.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10722/2851372017-01-01T00:00:00Z
- Regular ternary polygonal formshttp://hdl.handle.net/10722/282229Title: Regular ternary polygonal forms
Authors: HE, Z; Kane, B
Abstract: Inspired by Dickson's classification of regular ternary quadratic forms, we prove that there are no primitive regular m-gonal forms when m is sufficiently large. In order to do so, we construct sequences of primes that are inert in a certain quadratic field and show that they satisfy a certain inequality bounding the next prime by the product of the previous primes, a question of independent interest.
Wed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10722/2822292020-01-01T00:00:00Z
- On divisors of modular formshttp://hdl.handle.net/10722/251433Title: On divisors of modular forms
Authors: Bringmann, K; Kane, BR; Löbrich, S; Ono, K; Rolen, L
Abstract: The denominator formula for the Monster Lie algebra is the product expansion for the modular function $J(z)−J(τ)$ given in terms of the Hecke system of $SL_2(Z)$-modular functions $j_n(τ)$. It is prominent in Zagier’s seminal paper on traces of singular moduli, and in the Duncan-Frenkel work on Moonshine. The formula is equivalent to the description of the generating function for the $j_n(z)$ as a weight 2 modular form with a pole at z. Although these results rely on the fact that $X_0(1)$ has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the $X_0(N)$ modular curves. We use these functions to study divisors of modular forms.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10722/2514332018-01-01T00:00:00Z