It is proved that the theta series of an even lattice whose level is a power of a prime $\ell$ is congruent modulo $\ell$ to an elliptic modular form of level~1. The proof uses arithmetic and algebraic properties of lattices rather than methods from the theory of modular forms. The methods presented here may therefore be especially pleasing to those working in the theory of quadratic forms, and they admit generalizations to more general types of theta series as they occur e.g. in the theory of Siegel or Hilbert modular forms.
We investigate congruence relations of the form $p_r(\ell m n + t) \equiv 0 \pmod{\ell}$ for all $n$, where $p_r(n)$ is the number of $r$-colored partitions of $n$ and $m,\ell$ are distinct primes.
Let $f(q)$ denote Ramanujan's mock theta function \[f(q) = \sum_{n=0}^{\infty} a(n) q^{n} := 1+\sum_{n=1}^{\infty} \frac{q^{n^{2}}}{(1+q)^{2}(1+q^{2})^{2}\cdots(1+q^{n})^{2}}.\] It is known that there are many linear congruences for the coefficients of $f(q)$ and other mock theta functions. We prove that if the linear congruence $a(mn+t) \equiv 0 \pmod{\ell}$ holds for some prime $\ell \geq 5$, then $\ell | m$ and $(\frac{24t-1}{\ell}) \neq (\frac{-1}{\ell})$. We prove analogous results for the mock theta function $ω(q)$ and for a large class of weakly holomorphic modular forms which includes …
Generalizing math.AG/0003076 we express universal triple Massey products between line bundles on elliptic curves in terms of Hecke's indefinite theta series. We show that all Hecke's indefinite theta series arise in this way.
Let M(q)=\sum c(n) q^n be one of Ramanujan's mock theta functions. We establish the existence of infinitely many linear congruences of the form c(An+B) \equiv 0 (mod \ell^j), where A is a multiple of \ell and an auxiliary prime p. Moreover, we give an effectively computable upper bound on the smallest such p for which these congruences hold. The effective nature of our results is based on prior works of Lichtenstein and Treneer.
In this paper, we study the diagonal restrictions of certain Hilbert theta series for a totally real field $F$, and prove that they span the corresponding space of elliptic modular forms when the $F$ is quadratic or cubic. Furthermore, we give evidence of this phenomenon when $F$ is quartic, quintic and sextic.
The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vignéras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series.
For any natural number $\ell $ and any prime $p\equiv 1 \pmod{4}$ not dividing $\ell $ there is a Hermitian modular form of arbitrary genus $n$ over $L:=\Q [\sqrt{-\ell}]$ that is congruent to 1 modulo $p$ which is a Hermitian theta series of an $O_L$-lattice of rank $p-1$ admitting a fixed point free automorphism of order $p$. It is shown that also for non-free lattices such theta series are modular forms.
Let $\ell>0$ be a square-free integer congruent to 3 mod 4 and $Ø_K$ the ring of integers of the imaginary quadratic field $K=Q(\sqrt{-\ell})$. Codes $C$ over rings $Ø_K / p Ø_K$ determine lattices $Λ_\ell (C) $ over $K$. If $ p \nmid \ell$ then the ring $\R:=Ø_K / p Ø_K$ is isomorphic to $\F_{p^2}$ or $\F_p \times \F_p$. Given a code $C$ over $\R$, theta functions on the corresponding lattices are defined. These theta series $θ_{Λ_{\ell}(C)}$ can be written in terms of the complete weight enumerator of $C$. We show that for any two $\ell < \ell^\prime$ the first $\frac {\ell + 1} 4$ terms of …
We continue our study of Yoshida's lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.
We explain how the work of Johnson-Leung and Roberts on lifting Hilbert modular forms for real quadratic fields to Siegel modular forms can be adapted to imaginary quadratic fields. For this we use archimedean results from Harris, Soudry, Taylor and replace the global arguments of Roberts by the non-vanishing result of Takeda. As an application of our lifting result, we exhibit an abelian surface $B$ defined over $\mathbb{Q}$, which is not restriction of scalars of an elliptic curve and satisfies the Brumer-Kramer Paramodularity Conjecture.
Let t be any integer and fix an odd prime ell. Let Phi(x) = T_ell^n(x)-t denote the n-fold composition of the Chebyshev polynomial of degree ell shifted by t. If this polynomial is irreducible, let K = bbq(theta), where theta is a root of Phi. A theorem of Dedekind's gives a condition on t for which K is monogenic. For other values of t, we apply the Montes algorithm to obtain a formula for the discriminant of K and to compute basis elements for the ring of integers O_K.
In this paper, we consider the relation between the simplest types of arithmetic theta series, those associated to the cycles on the moduli space $\Cal C$ of elliptic curves with CM by the ring of integers $\OK$ in an imaginary quadratic field $\kay$, on the one hand, and those associated to cycles on the arithmetic surface $\M$ parametrizing 2-dimensional abelian varieties with an action of the maximal order $O_B$ in an indefinite quaternion algebra $B$ over $\Q$, on the other. We show that the arithmetic degree of the pullback to $Cal C$ of the arithmetic theta function of weight 3/2 valued …
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 $E/\mathbb{Q}$. 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…
We prove several asymptotic results for partial and false theta functions arising from Jacobi forms, as the modular variable $τ$ tends to $0$ along the imaginary axis, and the elliptic variable $z$ is unrestricted in the complex plane. We observe that these functions exhibit Stokes' phenomenon - the asymptotic behavior of these functions sharply differs depending on where the elliptic variable $z$ is located within the complex plane. We apply our results to study the asymptotic expansions of regularized characters and quantum dimensions of the $(1,p)$-singlet vertex operator algebra coming fro…
For an elliptic curve E over an abelian extension k/K with CM by K of Shimura type, the L-functions of its [k:K] Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a modular curve to E pulls back the 1-forms on E to give the Hecke theta functions. This article refines the study of our earlier work and shows that certain class of chiral correlation functions in Type II string theory with [E]_C (E as real analytic manifold) as a target space yield the same Hecke theta functions as objects on the modular curve. The Kahler paramet…
It was shown in previous work that the one-variable $\widehatμ$-function defined by Zwegers (and Zagier) and his indefinite theta series attached to lattices of signature $(r\!+\!1,1)$ are both Heisenberg harmonic Maaß-Jacobi forms. We extend the concept of Heisenberg harmonicity to Maaß-Jacobi forms of arbitrary many elliptic variables, and produce indefinite theta series of "product type" for non-degenerate lattices of signature $(r\!+\!s,s)$. We thus obtain a clean generalization of $\widehatμ$ to these negative definite lattices. From restrictions to torsion points of Heisenberg harmonic M…
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, Chevyr…
In this paper, we consider natural geometric objects coming from Lagrangian Floer theory and mirror symmetry. Lau and Zhou showed that some of the explicit Gromov-Witten potentials computed by Cho, Hong, Kim, and Lau are essentially classical modular forms. Recent work by Zwegers and two of the authors determined modularity properties of several simpler pieces of the last, and most mysterious, function by developing several identities between functions with properties generalizing those of the mock modular forms in Zwegers' thesis. Here, we complete the analysis of all pieces of Cho, Hong, Kim…
In this paper, we study modularity of several functions which naturally arose in a recent paper of Lau and Zhou on open Gromov-Witten potentials of elliptic orbifolds. They derived a number of examples of indefinite theta functions, and we provide modular completions for several such functions which involve more complicated objects than ordinary modular forms. In particular, we give new closed formulas for special indefinite theta functions of type $(1,2)$ in terms of products of mock modular forms. This formula is also of independent interest.
We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals $K$ and $E$. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.
We prove that if $k$ and $\ell$ are sufficiently large, then all the zeros of the weight $k+\ell$ cusp form $E_k(z) E_{\ell}(z) - E_{k+\ell}(z)$ in the standard fundamental domain lie on the boundary. We moreover find formulas for the number of zeros on the bottom arc with $|z|=1$, and those on the sides with $x = \pm 1/2$. One important ingredient of the proof is an approximation of the Eisenstein series in terms of the Jacobi theta function.
We show that Hida's families of $p$-adic elliptic modular forms generalize to $p$-adic families of Jacobi forms. We also construct $p$-adic versions of theta lifts from elliptic modular forms to Jacobi forms. Our results extend to Jacobi forms previous works by Hida and Stevens on the related case of half-integral weight modular forms.
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/…
