Proceedings of the American Mathematical Society, 2008
In a recent paper, Folsom and Ono constructed a canonical sequence of weight 1/2 mock theta funct... more In a recent paper, Folsom and Ono constructed a canonical sequence of weight 1/2 mock theta functions and a canonical sequence of weight 3/2 weakly holomorphic modular forms, both using Poincaré series. They show a remarkable symmetry in the coefficients of these functions and conjecture that all the coefficients are integers. We prove that this conjecture is true by giving an explicit construction for the weight 1/2 mock theta functions, using some results found by Guerzhoy.
It is well known that G. H. Hardy travelled in a taxicab numbered 1729 to an English nursing home... more It is well known that G. H. Hardy travelled in a taxicab numbered 1729 to an English nursing home to visit his bedridden colleague S. Ramanujan. Hardy was disappointed that his cab had such a mundane number, but to his surprise when he mentioned this to Ramanujan, the brilliant Indian mathematician found 1729 to be quite interesting, for it is the smallest integer that has two distinct representations as a sum of two cubes: 1729 = 1 3 + 12 3 = 9 3 + 10 3. J. H. Silverman used this famous anecdote to motivate the study of elliptic curves in a recent article [8]. Recently I learned that other permutations of the digits 1, 2, 7, and 9 are significant to the Ramanujan story. Two permutations involve Bruce Berndt, the diligent editor of Ramanujan's notebooks. Bruce has devoted most of his professional career to undertaking the daunting task of proving many of Ramanujan's identities (written in notebooks without proofs), but to my surprise his fascination with Ramanujan has profoundly impacted his life outside mathematics. Sonja, Bruce's youngest daughter, was born in 1972. Is this a coincidence, or could it be an example of "Ramanujan family planning?" With more sleuthing I discovered that Bruce's home is in Urbana, Illinois 61802-7219. Could there be any truth to the rumor that Bruce paid the postmaster a mere $12.79 for this vanity zipcode? In a more serious direction, consider the number 2719, which came to my attention in joint work with K. Soundararajan [5]. We begin with the following footnote from Ramanujan's 1916 paper on quadratic forms [6, p. 14]: ". .. the even numbers which are not of the form x 2 + y 2 + 10z 2 are the numbers 4 λ (16µ + 6), while the odd numbers that are not of that form, viz.,
We identify a parameterized family of K3 surfaces with generic Picard number 19, and we employ el... more We identify a parameterized family of K3 surfaces with generic Picard number 19, and we employ elementary methods to determine their local zeta functions. In addition, we explicitly determine those surfaces which are modular.
In this survey article we discuss the problem of determining the number of representations of an ... more In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study reveals several interesting results. If n ≥ 0 is a non-negative integer, then the n th triangular number is Tn = n(n+1) 2 . Let k be a positive integer. We denote by δ k (n) the number of representations of n as a sum of k triangular numbers. Here we use the theory of modular forms to calculate δ k (n). The case where k = 24 is particularly interesting. It turns out that if n ≥ 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 2 12 (2 12 -1)δ 24 (n -3). Furthermore the formula for δ 24 (n) involves the Ramanujan τ (n)-function. As a consequence, we get elementary congruences for τ (n). In a similar vein, when p is a prime we demonstrate δ 24 (p k -3) as a Dirichlet convolution of σ 11 (n) and τ (n). It is also of interest to know that this study produces formulas for the number of lattice points inside k-dimensional spheres.
The theory of congruences for the partition function p(n) depends heavily on the properties of ha... more The theory of congruences for the partition function p(n) depends heavily on the properties of halfintegral weight Hecke operators. The subject has been complicated by the absence of closed formulas for the Hecke images P (z) | T (2), where P (z) is the relevant modular generating function. We obtain such formulas using Euler's Pentagonal Number Theorem and the denominator formula for the Monster Lie algebra. As a corollary, we obtain congruences for certain powers of Ramanujan's Delta-function.
Although much is known about the partition function, little is known about its parity. For the po... more Although much is known about the partition function, little is known about its parity. For the polynomials D(x) := (Dx 2 + 1)/24, where D ≡ 23 (mod 24), we show that there are infinitely many m (resp. n) for which p(D(m)) is even (resp. p(D(n)) is odd) if there is at least one such m (resp. n). We bound the first m and n (if any) in terms of the class number h(−D). For prime D we show that there are indeed infinitely many even values. To this end we construct new modular generating functions using generalized Borcherds products, and we employ Galois representations and locally nilpotent Hecke algebras.
We prove that the coefficients of certain weight -1/2 harmonic Maass forms are "traces" of singul... more We prove that the coefficients of certain weight -1/2 harmonic Maass forms are "traces" of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight -2 harmonic weak Maass forms to spaces of weight -1/2 vectorvalued harmonic weak Maass forms on Mp 2 (Z), a result which is of independent interest. We then prove a general theorem which guarantees (with bounded denominator) when such Maass singular moduli are algebraic. As an example of these results, we derive a formula for the partition function p(n) as a finite sum of algebraic numbers which lie in the usual discriminant -24n + 1 ring class field. We indicate how these results extend to general weights. In particular, we illustrate how one can compute theta lifts for general weights by making use of the Kudla-Millson kernel and Maass differential operators.
Let R(w; q) be Dyson's generating function for partition ranks. For roots of unity ζ = 1, it is k... more Let R(w; q) be Dyson's generating function for partition ranks. For roots of unity ζ = 1, it is known that R(ζ ; q) and R(ζ ; 1/q) are given by harmonic Maass forms, Eichler integrals, and modular units. We show that modular forms arise from G(w; q), the generating function for ranks of partitions into distinct parts, in a similar way. If D(w; q) := (1 + w)G(w; q) + (1 − w)G(−w; q), then for roots of unity ζ = ±1 we show that q 1 12 • D(ζ ; q)D(ζ −1 ; q) is a weight 1 modular form. Although G(ζ ; 1/q) is not well defined, we show that it gives rise to natural sequences of q-series whose limits involve infinite products (and modular forms when ζ = 1). Our results follow from work of Fine on basic hypergeometric series.
MATHEMATICS AND ITS APPLICATIONS-DORDRECHT-, Mar 31, 1999
The D-quadratic twist of F, denoted FD, is given by n= l and for Re (s)» 0 its L-function is give... more The D-quadratic twist of F, denoted FD, is given by n= l and for Re (s)» 0 its L-function is given by L (FD, s)=£^= i xD (n) a (n) n-*. These L-functions have analytic continuations to C and satisfy well known functional equations. If A (F, s)=(27r)-T (s) Af'/2L (F, s), then where e=±1 is the so-called sign of the functional equation, and the quadratic twists satisfy
Gauss's hypergeometric function gives a modular parameterization of period integrals of elliptic ... more Gauss's hypergeometric function gives a modular parameterization of period integrals of elliptic curves in Legendre normal form We study a modular function which "measures" the variation of periods for the isomorphic curves E(λ) and E λ λ-1 , and we show that it padically "interpolates" the cusp form for the "congruent number" curve E(2), the case where these pairs collapse to a single curve.
In a recent paper, Kaneko and Zagier studied a sequence of modular forms F k (z) which are soluti... more In a recent paper, Kaneko and Zagier studied a sequence of modular forms F k (z) which are solutions of a certain second order differential equation. They studied the polynomials e F k (j) = Y τ ∈H/Γ−{i,ω} (j − j(τ)) ord τ (F k) , where ω = e 2πi/3 and H/Γ is the usual fundamental domain of the action of SL 2 (Z) on the upper half of the complex plane. If p ≥ 5 is prime, they proved that e F p−1 (j) (mod p) is the nontrivial factor of the locus of supersingular j-invariants in characteristic p. Here we consider the irreducibility of these polynomials, and consider their Galois groups.
Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has ... more Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms. These non-holomorphic modular forms play central roles in many subjects: arithmetic geometry, combinatorics, modular forms, and mathematical physics. Here we outline the general facets of the theory, and we give several applications to number theory: partitions and q-series, modular forms, singular moduli, Borcherds products, extensions of theorems of Kohnen-Zagier and Waldspurger on modular L-functions, and the work of Bruinier and Yang on Gross-Zagier formulae. What is surprising is that this story has an unlikely beginning: the pursuit of the solution to a great mathematical mystery.
In this paper we list all the weight 2 newforms f (τ) that are products and quotients of the Dede... more In this paper we list all the weight 2 newforms f (τ) that are products and quotients of the Dedekind eta-function η(τ) := q 1/24 ∞ n=1 (1 − q n), where q := e 2πiτ. There are twelve such f (τ), and we give a model for the strong Weil curve E whose Hasse-Weil L−function is the Mellin transform for each of them. Five of the f (τ) have complex multiplication, and we give elementary formulae for their Fourier coefficients which are sums of Hecke Grössencharacter values. These formulae follow easily from well known q−series infinite product identities.
Proceedings of the Japan Academy, Series A, Mathematical Sciences
Here we are interested in the arithmetic nature of sums of certain values of Hecke Grössencharakt... more Here we are interested in the arithmetic nature of sums of certain values of Hecke Grössencharakters, sums we call Shimura sums. In particular, we exhibit all primes p as the square root of a Shimura sum associated with a weight k = 3 Hecke Grössencharakter of K = Q(√ −2). The formula shows that primes p come from ideals in O K with norm 4p 2 − n 2. It will be shown that the non-existence of identities of this type imply certain cases of the Atkin Conjecture as well as Lehmer's Conjecture on the non-vanishing of Fourier coefficients.
Congruences for Fourier coefficients of integer weight modular forms have been the focal point of... more Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall exhibit congruences for Fourier coefficients of a slightly different type. Let f (z) = P ∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coefficients. If is prime, then we shall be interested in congruences of the form a(N) ≡ 0 mod
Euler proved the following recurrence for p(n), the number of partitions of an integer n : (1) p(... more Euler proved the following recurrence for p(n), the number of partitions of an integer n : (1) p(n) + ∞ X k=1 (−1) k (p(n − ω(k)) + p(n − ω(−k))) = 0 for ω(k) = 3k 2 +k 2. Using the Jacobi Triple Product identity we show analogues of Euler's recurrence formula for common restricted partition functions. Moreover following Kolberg, these recurrences allow us to determine that these partition functions are both even and odd infinitely often. Using the theory of modular forms, these recurrences may be viewed as infinite product identities involving Dedekind's η-function. Specifically, if the generating function for an arithmetical function is a modular form, then one often obtains analogous recurrence formulas; in particular here we get recurrence relations involving the number of t-core partitions, the number of representations of sums of squares, certain divisor functions, the number of points in finite fields on certain elliptic curves with complex multiplication, the Ramanujan τ −function and some appropriate analogs. In some cases recurrences hold for almost all n, and in others these recurrences hold for all n where the equality is replaced by a congruence mod m for any fixed integer m. These new recurrences are consequences of some of the theory of modular forms as developed by Deligne, Ribet, Serre, and Swinnerton-Dyer. p(n) ∼ 1 4n √ 3 e π √ 2n 3
Proceedings of the American Mathematical Society, 2008
In this paper we prove a p-adic supercongruence conjecture of van Hamme by placing it in the cont... more In this paper we prove a p-adic supercongruence conjecture of van Hamme by placing it in the context of the Beukers-like supercongruences of Rodriguez-Villegas. This conjecture is a p-adic analog of a formula of Ramanujan.
Proceedings of the American Mathematical Society, 2008
In a recent paper, Folsom and Ono constructed a canonical sequence of weight 1/2 mock theta funct... more In a recent paper, Folsom and Ono constructed a canonical sequence of weight 1/2 mock theta functions and a canonical sequence of weight 3/2 weakly holomorphic modular forms, both using Poincaré series. They show a remarkable symmetry in the coefficients of these functions and conjecture that all the coefficients are integers. We prove that this conjecture is true by giving an explicit construction for the weight 1/2 mock theta functions, using some results found by Guerzhoy.
It is well known that G. H. Hardy travelled in a taxicab numbered 1729 to an English nursing home... more It is well known that G. H. Hardy travelled in a taxicab numbered 1729 to an English nursing home to visit his bedridden colleague S. Ramanujan. Hardy was disappointed that his cab had such a mundane number, but to his surprise when he mentioned this to Ramanujan, the brilliant Indian mathematician found 1729 to be quite interesting, for it is the smallest integer that has two distinct representations as a sum of two cubes: 1729 = 1 3 + 12 3 = 9 3 + 10 3. J. H. Silverman used this famous anecdote to motivate the study of elliptic curves in a recent article [8]. Recently I learned that other permutations of the digits 1, 2, 7, and 9 are significant to the Ramanujan story. Two permutations involve Bruce Berndt, the diligent editor of Ramanujan's notebooks. Bruce has devoted most of his professional career to undertaking the daunting task of proving many of Ramanujan's identities (written in notebooks without proofs), but to my surprise his fascination with Ramanujan has profoundly impacted his life outside mathematics. Sonja, Bruce's youngest daughter, was born in 1972. Is this a coincidence, or could it be an example of "Ramanujan family planning?" With more sleuthing I discovered that Bruce's home is in Urbana, Illinois 61802-7219. Could there be any truth to the rumor that Bruce paid the postmaster a mere $12.79 for this vanity zipcode? In a more serious direction, consider the number 2719, which came to my attention in joint work with K. Soundararajan [5]. We begin with the following footnote from Ramanujan's 1916 paper on quadratic forms [6, p. 14]: ". .. the even numbers which are not of the form x 2 + y 2 + 10z 2 are the numbers 4 λ (16µ + 6), while the odd numbers that are not of that form, viz.,
We identify a parameterized family of K3 surfaces with generic Picard number 19, and we employ el... more We identify a parameterized family of K3 surfaces with generic Picard number 19, and we employ elementary methods to determine their local zeta functions. In addition, we explicitly determine those surfaces which are modular.
In this survey article we discuss the problem of determining the number of representations of an ... more In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study reveals several interesting results. If n ≥ 0 is a non-negative integer, then the n th triangular number is Tn = n(n+1) 2 . Let k be a positive integer. We denote by δ k (n) the number of representations of n as a sum of k triangular numbers. Here we use the theory of modular forms to calculate δ k (n). The case where k = 24 is particularly interesting. It turns out that if n ≥ 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 2 12 (2 12 -1)δ 24 (n -3). Furthermore the formula for δ 24 (n) involves the Ramanujan τ (n)-function. As a consequence, we get elementary congruences for τ (n). In a similar vein, when p is a prime we demonstrate δ 24 (p k -3) as a Dirichlet convolution of σ 11 (n) and τ (n). It is also of interest to know that this study produces formulas for the number of lattice points inside k-dimensional spheres.
The theory of congruences for the partition function p(n) depends heavily on the properties of ha... more The theory of congruences for the partition function p(n) depends heavily on the properties of halfintegral weight Hecke operators. The subject has been complicated by the absence of closed formulas for the Hecke images P (z) | T (2), where P (z) is the relevant modular generating function. We obtain such formulas using Euler's Pentagonal Number Theorem and the denominator formula for the Monster Lie algebra. As a corollary, we obtain congruences for certain powers of Ramanujan's Delta-function.
Although much is known about the partition function, little is known about its parity. For the po... more Although much is known about the partition function, little is known about its parity. For the polynomials D(x) := (Dx 2 + 1)/24, where D ≡ 23 (mod 24), we show that there are infinitely many m (resp. n) for which p(D(m)) is even (resp. p(D(n)) is odd) if there is at least one such m (resp. n). We bound the first m and n (if any) in terms of the class number h(−D). For prime D we show that there are indeed infinitely many even values. To this end we construct new modular generating functions using generalized Borcherds products, and we employ Galois representations and locally nilpotent Hecke algebras.
We prove that the coefficients of certain weight -1/2 harmonic Maass forms are "traces" of singul... more We prove that the coefficients of certain weight -1/2 harmonic Maass forms are "traces" of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight -2 harmonic weak Maass forms to spaces of weight -1/2 vectorvalued harmonic weak Maass forms on Mp 2 (Z), a result which is of independent interest. We then prove a general theorem which guarantees (with bounded denominator) when such Maass singular moduli are algebraic. As an example of these results, we derive a formula for the partition function p(n) as a finite sum of algebraic numbers which lie in the usual discriminant -24n + 1 ring class field. We indicate how these results extend to general weights. In particular, we illustrate how one can compute theta lifts for general weights by making use of the Kudla-Millson kernel and Maass differential operators.
Let R(w; q) be Dyson's generating function for partition ranks. For roots of unity ζ = 1, it is k... more Let R(w; q) be Dyson's generating function for partition ranks. For roots of unity ζ = 1, it is known that R(ζ ; q) and R(ζ ; 1/q) are given by harmonic Maass forms, Eichler integrals, and modular units. We show that modular forms arise from G(w; q), the generating function for ranks of partitions into distinct parts, in a similar way. If D(w; q) := (1 + w)G(w; q) + (1 − w)G(−w; q), then for roots of unity ζ = ±1 we show that q 1 12 • D(ζ ; q)D(ζ −1 ; q) is a weight 1 modular form. Although G(ζ ; 1/q) is not well defined, we show that it gives rise to natural sequences of q-series whose limits involve infinite products (and modular forms when ζ = 1). Our results follow from work of Fine on basic hypergeometric series.
MATHEMATICS AND ITS APPLICATIONS-DORDRECHT-, Mar 31, 1999
The D-quadratic twist of F, denoted FD, is given by n= l and for Re (s)» 0 its L-function is give... more The D-quadratic twist of F, denoted FD, is given by n= l and for Re (s)» 0 its L-function is given by L (FD, s)=£^= i xD (n) a (n) n-*. These L-functions have analytic continuations to C and satisfy well known functional equations. If A (F, s)=(27r)-T (s) Af'/2L (F, s), then where e=±1 is the so-called sign of the functional equation, and the quadratic twists satisfy
Gauss's hypergeometric function gives a modular parameterization of period integrals of elliptic ... more Gauss's hypergeometric function gives a modular parameterization of period integrals of elliptic curves in Legendre normal form We study a modular function which "measures" the variation of periods for the isomorphic curves E(λ) and E λ λ-1 , and we show that it padically "interpolates" the cusp form for the "congruent number" curve E(2), the case where these pairs collapse to a single curve.
In a recent paper, Kaneko and Zagier studied a sequence of modular forms F k (z) which are soluti... more In a recent paper, Kaneko and Zagier studied a sequence of modular forms F k (z) which are solutions of a certain second order differential equation. They studied the polynomials e F k (j) = Y τ ∈H/Γ−{i,ω} (j − j(τ)) ord τ (F k) , where ω = e 2πi/3 and H/Γ is the usual fundamental domain of the action of SL 2 (Z) on the upper half of the complex plane. If p ≥ 5 is prime, they proved that e F p−1 (j) (mod p) is the nontrivial factor of the locus of supersingular j-invariants in characteristic p. Here we consider the irreducibility of these polynomials, and consider their Galois groups.
Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has ... more Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms. These non-holomorphic modular forms play central roles in many subjects: arithmetic geometry, combinatorics, modular forms, and mathematical physics. Here we outline the general facets of the theory, and we give several applications to number theory: partitions and q-series, modular forms, singular moduli, Borcherds products, extensions of theorems of Kohnen-Zagier and Waldspurger on modular L-functions, and the work of Bruinier and Yang on Gross-Zagier formulae. What is surprising is that this story has an unlikely beginning: the pursuit of the solution to a great mathematical mystery.
In this paper we list all the weight 2 newforms f (τ) that are products and quotients of the Dede... more In this paper we list all the weight 2 newforms f (τ) that are products and quotients of the Dedekind eta-function η(τ) := q 1/24 ∞ n=1 (1 − q n), where q := e 2πiτ. There are twelve such f (τ), and we give a model for the strong Weil curve E whose Hasse-Weil L−function is the Mellin transform for each of them. Five of the f (τ) have complex multiplication, and we give elementary formulae for their Fourier coefficients which are sums of Hecke Grössencharacter values. These formulae follow easily from well known q−series infinite product identities.
Proceedings of the Japan Academy, Series A, Mathematical Sciences
Here we are interested in the arithmetic nature of sums of certain values of Hecke Grössencharakt... more Here we are interested in the arithmetic nature of sums of certain values of Hecke Grössencharakters, sums we call Shimura sums. In particular, we exhibit all primes p as the square root of a Shimura sum associated with a weight k = 3 Hecke Grössencharakter of K = Q(√ −2). The formula shows that primes p come from ideals in O K with norm 4p 2 − n 2. It will be shown that the non-existence of identities of this type imply certain cases of the Atkin Conjecture as well as Lehmer's Conjecture on the non-vanishing of Fourier coefficients.
Congruences for Fourier coefficients of integer weight modular forms have been the focal point of... more Congruences for Fourier coefficients of integer weight modular forms have been the focal point of a number of investigations. In this note we shall exhibit congruences for Fourier coefficients of a slightly different type. Let f (z) = P ∞ n=0 a(n)q n be a holomorphic half integer weight modular form with integer coefficients. If is prime, then we shall be interested in congruences of the form a(N) ≡ 0 mod
Euler proved the following recurrence for p(n), the number of partitions of an integer n : (1) p(... more Euler proved the following recurrence for p(n), the number of partitions of an integer n : (1) p(n) + ∞ X k=1 (−1) k (p(n − ω(k)) + p(n − ω(−k))) = 0 for ω(k) = 3k 2 +k 2. Using the Jacobi Triple Product identity we show analogues of Euler's recurrence formula for common restricted partition functions. Moreover following Kolberg, these recurrences allow us to determine that these partition functions are both even and odd infinitely often. Using the theory of modular forms, these recurrences may be viewed as infinite product identities involving Dedekind's η-function. Specifically, if the generating function for an arithmetical function is a modular form, then one often obtains analogous recurrence formulas; in particular here we get recurrence relations involving the number of t-core partitions, the number of representations of sums of squares, certain divisor functions, the number of points in finite fields on certain elliptic curves with complex multiplication, the Ramanujan τ −function and some appropriate analogs. In some cases recurrences hold for almost all n, and in others these recurrences hold for all n where the equality is replaced by a congruence mod m for any fixed integer m. These new recurrences are consequences of some of the theory of modular forms as developed by Deligne, Ribet, Serre, and Swinnerton-Dyer. p(n) ∼ 1 4n √ 3 e π √ 2n 3
Proceedings of the American Mathematical Society, 2008
In this paper we prove a p-adic supercongruence conjecture of van Hamme by placing it in the cont... more In this paper we prove a p-adic supercongruence conjecture of van Hamme by placing it in the context of the Beukers-like supercongruences of Rodriguez-Villegas. This conjecture is a p-adic analog of a formula of Ramanujan.
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