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... 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.
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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... 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
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The Eichler-Selberg trace formulas express the traces of Hecke operators on a spaces of cusp forms in terms of weighted sums of Hurwitz-Kronecker class numbers. For cusp forms on $\text {\rm SL}_2(\mathbb{Z}),$ Zagier proved these... more
The Eichler-Selberg trace formulas express the traces of Hecke operators on a spaces of cusp forms in terms of weighted sums of Hurwitz-Kronecker class numbers. For cusp forms on $\text {\rm SL}_2(\mathbb{Z}),$ Zagier proved these formulas by cleverly making use of the weight 3/2 nonholomorphic Eisenstein series he discovered in the 1970s. The holomorphic part of this form, its so-called {\it mock modular form}, is the generating function for these class numbers. In this expository note we revisit Zagier's method, and we show how to obtain such formulas for congruence subgroups, working out the details for $\Gamma_0(2)$ and $\Gamma_0(4).$ The trace formulas fall out naturally from the computation of the Rankin-Cohen brackets of Zagier's mock modular form with specific theta functions.
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We explicitly construct the Dirichlet series L_Tam(s):=∑_m=1^∞P_Tam(m)/m^s, where P_Tam(m) is the proportion of elliptic curves E/ℚ in short Weierstrass form with Tamagawa product m. Although there are no E/ℚ with everywhere good... more
We explicitly construct the Dirichlet series L_Tam(s):=∑_m=1^∞P_Tam(m)/m^s, where P_Tam(m) is the proportion of elliptic curves E/ℚ in short Weierstrass form with Tamagawa product m. Although there are no E/ℚ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is P_Tam(1)=0.5053…. As a corollary, we find that L_Tam(-1)=1.8193… is the average Tamagawa product for elliptic curves over ℚ. We give an application of these results to canonical and Weil heights.
In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers ℕ as limiting values of q-series as q→ζ a root of unity... more
In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers ℕ as limiting values of q-series as q→ζ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of ℕ by analogous structures in the integer partitions 𝒫. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of Dirichlet series arising from Dirichlet convolutions. Here the authors prove that Wang's extension has a partition-theoretic analogue as well, yielding new q-series density formulas for any subset of ℕ. To do so, we outline a theory of q-series density calculations from first principles, based on a statistic we call the "q-density" of a given subset. This theory in turn yields infinite families of further formula...
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We compute the Moore-Witten regularized u-plane integral on CP^2, and we confirm their conjecture that it is the generating function for the SO(3)-Donaldson invariants of CP^2. We prove this conjecture using the theory of mock theta... more
We compute the Moore-Witten regularized u-plane integral on CP^2, and we confirm their conjecture that it is the generating function for the SO(3)-Donaldson invariants of CP^2. We prove this conjecture using the theory of mock theta functions and harmonic Maass forms. We also derive further such generating functions for the SO(3)-Donaldson invariants with 2N_f massless monopoles using the geometry of certain rational elliptic surfaces (N_f=0,2,3,4). We show that the partition function for N_f=4 is nearly modular. When combined with one of Ramanujan's mock theta functions, we obtain a weight 1/2 modular form. This fact is central to the proof of the conjecture.
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For a finite group G, let K(G) denote the field generated over Q by its character values. For n>24, G. R. Robinson and J. G. Thompson proved that K(A_n)=Q ({√(p^*) : p≤ n an odd prime with p≠ n-2}), where p^*:=(-1)^p-1/2p. Confirming a... more
For a finite group G, let K(G) denote the field generated over Q by its character values. For n>24, G. R. Robinson and J. G. Thompson proved that K(A_n)=Q ({√(p^*) : p≤ n an odd prime with p≠ n-2}), where p^*:=(-1)^p-1/2p. Confirming a speculation of Thompson, we show that arbitrary suitable multiquadratic fields are similarly generated by the values of A_n-characters restricted to elements whose orders are only divisible by ramified primes. To be more precise, we say that a π-number is a positive integer whose prime factors belong to a set of odd primes π:= {p_1, p_2,..., p_t}. Let K_π(A_n) be the field generated by the values of A_n-characters for even permutations whose orders are π-numbers. If t≥ 2, then we determine a constant N_π with the property that for all n> N_π, we have K_π(A_n)=Q(√(p_1^*), √(p_2^*),..., √(p_t^*)).
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In a recent important paper, Hoffstein and Hulse generalized the notion of Rankin-Selberg convolution L-functions by defining shifted convolution L-functions. We investigate symmetrized versions of their functions. Under certain mild... more
In a recent important paper, Hoffstein and Hulse generalized the notion of Rankin-Selberg convolution L-functions by defining shifted convolution L-functions. We investigate symmetrized versions of their functions. Under certain mild conditions, we prove that the generating functions of certain special values are linear combinations of weakly holomorphic quasimodular forms and "mixed mock modular" forms.
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We revisit the mathematics that Ramanujan developed in connection with the famous "taxi-cab" number 1729. A study of his writings reveals that he had been studying Euler's diophantine equation a^3+b^3=c^3+d^3. It turns out... more
We revisit the mathematics that Ramanujan developed in connection with the famous "taxi-cab" number 1729. A study of his writings reveals that he had been studying Euler's diophantine equation a^3+b^3=c^3+d^3. It turns out that Ramanujan's work anticipated deep structures and phenomena which have become fundamental objects in arithmetic geometry and number theory. We find that he discovered a K3 surface with Picard number 18, one which can be used to obtain infinitely many cubic twists over Q with rank ≥ 2.
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We give algorithms for computing the singular moduli of suitable nonholomorphic modular functions F(z). By combining the theory of isogeny volcanoes with a beautiful observation of Masser concerning the nonholomorphic Eisenstein series... more
We give algorithms for computing the singular moduli of suitable nonholomorphic modular functions F(z). By combining the theory of isogeny volcanoes with a beautiful observation of Masser concerning the nonholomorphic Eisenstein series E_2*(z), we obtain CRT-based algorithms that compute the class polynomials H_D(F;x), whose roots are the discriminant D singular moduli for F(z). By applying these results to a specific weak Maass form F_p(z), we obtain a CRT-based algorithm for computing partition class polynomials, a sequence of polynomials whose traces give the partition numbers p(n). Under the GRH, the expected running time of this algorithm is O(n^5/2+o(1)). Key to these results is a fast CRT-based algorithm for computing the classical modular polynomial Phi_m(X,Y) that we obtain by extending the isogeny volcano approach previously developed for prime values of m.
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For positive integers 1≤ i≤ k, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews-Gordon q-series ∏_1≤ n≡ 0,± i2k+11/1-q^n. This study is motivated by their appearance in conformal... more
For positive integers 1≤ i≤ k, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews-Gordon q-series ∏_1≤ n≡ 0,± i2k+11/1-q^n. This study is motivated by their appearance in conformal field theory, where these series are essentially the irreducible characters of (2,2k+1) Virasoro minimal models. We determine the vanishing of such Wronskians, a result whose proof reveals many partition identities. For example, if P_b(a;n) denotes the number of partitions of n into parts which are not congruent to 0, ± a b, then for every positive integer n we have P_27(12; n)=P_27(6;n-1) + P_27(3;n-2). We also show that these quotients classify supersingular elliptic curves in characteristic p. More precisely, if 2k+1=p, where p≥ 5 is prime, and the quotient is non-zero, then it is essentially the locus of characteristic p supersingular j-invariants in characteristic p.
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Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves E/Q. We show... more
Mock modular forms, which give the theoretical framework for Ramanujan's enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves E/Q. We show that mock modular forms which arise from Weierstrass ζ-functions encode the central L-values and L-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by Hövel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values for the quadratic twists of E. We employ results of Bruinier and the third author, which builds on seminal work of Gross, Kohnen, Shimura, Waldspurger, and Zagier. We also obtain p-adic formulas for the corresponding weight 2 newform using the action of the Hecke algebra on the Weierstrass mock modular form.
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For integers k≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2-k. The operator ξ_2-k (resp. D^k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage... more
For integers k≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2-k. The operator ξ_2-k (resp. D^k-1) defines a map to the space of weight k cusp forms (resp. weakly holomorphic modular forms). We leverage these operators to study coefficients of harmonic weak Maass forms. Although generic harmonic weak Maass forms are expected to have transcendental coefficients, we show that those forms which are "dual" under ξ_2-k to newforms with vanishing Hecke eigenvalues (such as CM forms) have algebraic coefficients. Using regularized inner products, we also characterize the image of D^k-1.
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We celebrate the 100th anniversary of Srinivasa Ramanujan's election as a Fellow of the Royal Society, which was largely based on his work with G. H. Hardy on the asymptotic properties of the partition function. After recalling this... more
We celebrate the 100th anniversary of Srinivasa Ramanujan's election as a Fellow of the Royal Society, which was largely based on his work with G. H. Hardy on the asymptotic properties of the partition function. After recalling this revolutionary work, marking the birth of the "circle method", we present a contemporary example of its legacy in topology. We deduce the equidistribution of Hodge numbers for Hilbert schemes of suitable smooth projective surfaces.
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Answering a question posed by Conway and Norton in their seminal 1979 paper on moonshine, we prove the existence of a graded infinite-dimensional module for the sporadic simple group of O'Nan, for which the McKay--Thompson series are... more
Answering a question posed by Conway and Norton in their seminal 1979 paper on moonshine, we prove the existence of a graded infinite-dimensional module for the sporadic simple group of O'Nan, for which the McKay--Thompson series are weight 3/2 modular forms. The coefficients of these series may be expressed in terms of class numbers, traces of singular moduli, and central critical values of quadratic twists of weight 2 modular L-functions. As a consequence, for primes p dividing the order of the O'Nan group we obtain congruences between O'Nan group character values and class numbers, p-parts of Selmer groups, and Tate--Shafarevich groups of certain elliptic curves. This work represents the first example of moonshine involving arithmetic invariants of this type.
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Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such... more
Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as "generating functions" for central values and derivatives of quadratic twists of weight 2 modular L-functions. To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weak Maass forms. The connection with periods, Fourier coefficients, derivatives of L-functions, and points in the Jacobian of modular curves is obtained by analyzing the properties of these differentials using works of Scholl, Waldschmidt, and Gross and Zagier.
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For each prime ℓ, let |·|_ℓ be an extension to of the usual ℓ-adic absolute value on . Suppose g(z) = ∑_n=0^∞ c(n)q^n ∈ M_k+(N) is an eigenform whose Fourier coefficients are algebraic integers. Under a mild condition, for all but... more
For each prime ℓ, let |·|_ℓ be an extension to of the usual ℓ-adic absolute value on . Suppose g(z) = ∑_n=0^∞ c(n)q^n ∈ M_k+(N) is an eigenform whose Fourier coefficients are algebraic integers. Under a mild condition, for all but finitely many primes ℓ there are infinitely many square-free integers m for which |c(m)|_ℓ = 1. Consequently we obtain indivisibility results for "algebraic parts" of central critical values of modular L-functions and class numbers of imaginary quadratic fields. These results partially answer a conjecture of Kolyvagin regarding Tate-Shafarevich groups of modular elliptic curves. Similar results were obtained earlier by Jochnowitz by a completely different method. Our method uses standard facts about Galois representations attached to modular forms, and pleasantly uncovers surprising Kronecker-style congruences for L-function values. For example if Δ(z) is Ramanujan's cusp form and g(z)=∑_n=1^∞c(n)q^n is the cusp form for which L(Δ_D,6)=()πD^6...
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Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we... more
Ramanujan (and others) proved that the partition function satisfies a number of striking congruences modulo powers of 5, 7 and 11. A number of further congruences were shown by the works of Atkin, O'Brien, and Newman. In this paper we prove that there are infinitely many such congruences for every prime modulus exceeding 3. In addition, we provide a simple criterion guaranteeing the truth of Newman's conjecture for any prime modulus exceeding 3 (recall that Newman's conjecture asserts that the partition function hits every residue class modulo a given integer M infinitely often).
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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... 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 vector-valued 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.
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Using predictions in mirror symmetry, Căldăraru, He, and Huang recently formulated a "Moonshine Conjecture at Landau-Ginzburg points" for Klein's modular j-function at j=0 and j=1728. The conjecture asserts that the... more
Using predictions in mirror symmetry, Căldăraru, He, and Huang recently formulated a "Moonshine Conjecture at Landau-Ginzburg points" for Klein's modular j-function at j=0 and j=1728. The conjecture asserts that the j-function, when specialized at specific flat coordinates on the moduli spaces of versal deformations of the corresponding CM elliptic curves, yields simple rational functions. We prove this conjecture, and show that these rational functions arise from classical _2F_1-hypergeometric inversion formulae for the j-function.
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In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer α is a value of τ(n). For any given odd α, Murty, Murty, and Shorey proved that τ(n)≠α for... more
In the spirit of Lehmer's speculation that Ramanujan's tau-function never vanishes, it is natural to ask whether any given integer α is a value of τ(n). For any given odd α, Murty, Murty, and Shorey proved that τ(n)≠α for sufficiently large n. Several recent papers consider the case of odd α. In this note, we determine examples of even integers that are not tau-values. Namely, for the indicated primes ℓ, we prove that τ(n)∉{± 2· 691} ∪{2ℓ : 3≤ℓ≤ 97 with ℓ≠ 43, 79} ∪{-2ℓ : 3≤ℓ≤ 97 with ℓ≠ 5, 17, 41, 47, 59, 89}∪{-2ℓ^2 : 3≤ℓ≤ 23}. The method of proof applies mutatis mutandis to newforms with residually reducible mod 2 Galois representation and is easily adapted to generic newforms with integer coefficients.
In this note we obtain effective lower bounds for the canonical heights of non-torsion points on E(Q) by making use of suitable elliptic curve ideal class pairings Ψ_E,-D: E(Q)× E_-D(Q)CL(-D). In terms of the class number H(-D) and... more
In this note we obtain effective lower bounds for the canonical heights of non-torsion points on E(Q) by making use of suitable elliptic curve ideal class pairings Ψ_E,-D: E(Q)× E_-D(Q)CL(-D). In terms of the class number H(-D) and T_E(-D), a logarithmic function in D, we prove h(P)> |E_tor(Q)|^2/( H(-D)+ |E_tor(Q)|)^2· T_E(-D).
Ideal class pairings map the rational points of rank r≥ 1 elliptic curves E/ to the ideal class groups (-D) of certain imaginary quadratic fields. These pairings imply that h(-D) ≥1/2(c(E)-ε)(log D)^r/2 for sufficiently large... more
Ideal class pairings map the rational points of rank r≥ 1 elliptic curves E/ to the ideal class groups (-D) of certain imaginary quadratic fields. These pairings imply that h(-D) ≥1/2(c(E)-ε)(log D)^r/2 for sufficiently large discriminants -D in certain families, where c(E) is a natural constant. These bounds are effective, and they offer improvements to known lower bounds for many discriminants.
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We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for n>1 we prove that τ(n)∉{± 1, ± 3, ± 5, ± 7, ± 691}. This result is an example of general theorems for... more
We consider natural variants of Lehmer's unresolved conjecture that Ramanujan's tau-function never vanishes. Namely, for n>1 we prove that τ(n)∉{± 1, ± 3, ± 5, ± 7, ± 691}. This result is an example of general theorems for newforms with trivial mod 2 residual Galois representation, which will appear in forthcoming work of the authors with Wei-Lun Tsai. Ramanujan's well-known congruences for τ(n) allow for the simplified proof in these special cases. We make use of the theory of Lucas sequences, the Chabauty-Coleman method for hyperelliptic curves, and facts about certain Thue equations.
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The denominator formula for the Monster Lie algebra is the product expansion for the modular function j(z)-j(τ) in terms of the Hecke system of SL_2(Z)-modular functions j_n(τ). This formula can be reformulated entirely... more
The denominator formula for the Monster Lie algebra is the product expansion for the modular function j(z)-j(τ) in terms of the Hecke system of SL_2(Z)-modular functions j_n(τ). 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 τ 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 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.
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In recent work, Sun constructed two q-series, and he showed that their limits as q→1 give new derivations of the Riemann-zeta values ζ(2)=π^2/6 and ζ(4)=π^4/90. Goswami extended these series to an infinite family of q-series, which he... more
In recent work, Sun constructed two q-series, and he showed that their limits as q→1 give new derivations of the Riemann-zeta values ζ(2)=π^2/6 and ζ(4)=π^4/90. Goswami extended these series to an infinite family of q-series, which he analogously used to obtain new derivations of the evaluations of ζ(2k)∈Q·π^2k for every positive integer k. Since it is well known that Γ(1/2)=√(π), it is natural to seek further specializations of these series which involve special values of the Γ-function. Thanks to the theory of complex multiplication, we show that the values of these series at all CM points τ, where q:=e^2π iτ, are algebraic multiples of specific ratios of Γ-values. In particular, classical formulas of Ramanujan allow us to explicitly evaluate these series as algebraic multiples of powers of Γ(1/4)^4/π^3 when q=e^-π, e^-2π.