Serre and Stark found a basis for the space of modular forms of weight 1/2 in terms of theta series. In this paper, we generalize their result - under certain mild restrictions on the level and character - to the case of weight 1/2 Hilbert modular forms over a totally real field of narrow class number 1. The methods broadly follow those of Serre-Stark; however we are forced to overcome technical difficulties which arise when we move out of Q.
We have discovered conjectural near-addition formulas for Somos sequences. We have preliminary evidence suggesting the existence of modular theta functions.
In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta-theta functions by constructing mock modular forms from the eta-theta functions with even characters, such that the shadows of these mock modular forms are given by the eta-theta functions with odd characters. In addition, we prove that our mock modular forms are quantum modular forms. As corollaries, we establish simple finite hypergeometric expressions which may be used to evaluate Eichler integrals of the odd eta-theta functions, as well as some curious algebraic identities.
We consider the action of Hecke-type operators on Hilbert-Siegel theta series attached to lattices of even rank. We show that average Hilbert-Siegel theta series are eigenforms for these operators, and we explicitly compute the eigenvalues.
For a prime $p$ larger than $7$, the Eisenstein series of weight $p-1$ has some remarkable congruence properties modulo $p$. Those imply, for example, that the $j$-invariants of its zeros (which are known to be real algebraic numbers in the interval $[0,1728]$), are at most quadratic over the field with $p$ elements and are congruent modulo $p$ to the zeros of a certain truncated hypergeometric series. In this paper we introduce "theta modular forms" of weight $k \geq 4$ for the full modular group as the modular forms for which the first dim$(M_k)$ Fourier coefficients are identical to certain…
Let $F$ be a totally real number field and $\mathfrak{o}$ the ring of integers of $F$. We study theta functions which are Hilbert modular forms of half-integral weight for the Hilbert modular group $\mathrm{SL}_2(\mathfrak{o})$. We obtain an equivalent condition that there exists a multiplier system of half-integral weight for $\mathrm{SL}_2(\mathfrak{o})$. We determine the condition of $F$ that there exists a theta function which is a Hilbert modular form of half-integral weight for $\mathrm{SL}_2(\mathfrak{o})$. The theta function is defined by a sum on a fractional ideal $\mathfrak{a}$ of $…
We show that the meromorphic modular forms recently considered by Bringmann and Kane can be obtained as images of regularized theta lifts of Poincaré series under weight raising operators. We use this fact in order to simplify the evaluation of the regularized pairing, also defined by Bringmann and Kane, of these functions with other meromorphic modular forms.
Given a characteristic, we define a character of the Siegel modular group of level 2, the computations of their values are also obtained. By using our theorems, some key theorems of Igusa [1] can be recovered.
We prove that the coefficients of certain mock theta functions possess no linear congruences modulo 3. We prove similar results for the moduli 2 and 3 for a wide class of weakly holomorphic modular forms and discuss applications. This extends work of Radu on the behavior of the ordinary partition function modulo 2 and 3.
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 mock theta conjectures are ten identities involving Ramanujan's fifth-order mock theta functions. The conjectures were proven by Hickerson in 1988 using q-series methods. Using methods from the theory of harmonic Maass forms, specifically work of Zwegers and Bringmann-Ono, Folsom reduced the proof of the mock theta conjectures to a finite computation. Both of these approaches involve proving the identities individually, relying on work of Andrews-Garvan. Here we give a unified proof of the mock theta conjectures by realizing them as an equality between two nonholomorphic vector-valued modu…
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 investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we prove that v-adic modular forms in the sense of Serre, as defined by Goss and Vincent, are preserved under hyperdifferentiation. Moreover, upon multiplication by a Carlitz factorial, hyperdifferentiation preserves v-integrality.
Generalizing a result of~\cite{Z1991} for modular forms of level~one, we give a closed formula for the sum of all Hecke eigenforms on $Γ_0(N)$, multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level $N$. We also show that for $N=2$,~3 and~5 this formula completely determines the Fourier expansions of all Hecke eigenforms of all weights on $Γ_0(N)$.
The mod $p$ kernel of the theta operator is the set of modular forms whose image of the theta operator is congruent to zero modulo a prime $p$. In the case of Siegel modular forms, the authors found interesting examples of such modular forms. For example, Igusa's odd weight cusp form is an element of mod 23 kernel of the theta operator. In this paper, we give some examples which represent elements in the mod $p$ kernel of the theta operator in the case of Hermitian modular forms of degree 2.
Let $Γ_n(\mathcal{\scriptstyle{O}}_{\mathbb{K}})$ denote the Hermitian modular group of degree $n$ over an imaginary quadratic number field $\mathbb{K}$ and $Δ_{n,\mathbb{K}}^*$ its maximal discrete extension in the special unitary group $SU(n,n;\mathbb{C})$. In this paper we study the action of $Δ_{n,\mathbb{K}}^*$ on Hermitian theta series and Maass spaces. For $n=2$ we will find theta lattices such that the corresponding theta series are modular forms with respect to $Δ_{2,\mathbb{K}}^*$ as well as examples where this is not the case. Our second focus lies on studying two different Maass sp…
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.
We construct almost holomorphic and holomorphic modular forms by considering theta series for quadratic forms of signature $(n-1,1)$. We include homogeneous and spherical polynomials in the definition of the theta series (generalizing a construction of the second author) to obtain holomorphic, almost holomorphic and modular theta series. We give a criterion for these series to coincide, enabling us to construct almost holomorphic and holomorphic cusp forms on congruence subgroups of the modular group. Further, we provide numerous explicit examples.
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.
We investigate Siegel theta series for quadratic forms of signature $(m-1,1)$. On the one hand, we construct a holomorphic series that does not transform like a modular form. On the other hand, we construct a non-holomorphic series that transforms like a Siegel modular form of weight $m/2$. Moreover, the holomorphic series describes almost everywhere the holomorphic part of the modular series.
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 …
Siegel modular forms in the space of the mod $p$ kernel of the theta operator are constructed by the Eisenstein series in some odd-degree cases. Additionally, a similar result in the case of Hermitian modular forms is given.
We describe the construction of vector valued modular forms transforming under a given congruence representation of the modular group SL$(\bold Z)$ in terms of theta series. We apply this general setup to obtain closed and easily computable formulas for conformal characters of rational models of $W$-algebras.
Sander Zwegers showed that Ramanujan's mock theta functions are $q$-hypergeometric series, whose $q$-expansion coefficients are half of the Fourier coefficients of a non-holomorphic modular form. George Andrews, Henri Cohen, Freeman Dyson, and Dean Hickerson found a pair of $q$-hypergeometric series each of which contains half of the Fourier coefficients of Maass waveform of eigenvalue $1/4$. This series of papers shows that a $q$-series construction, called ``renormalization'', yields the other half of the Fourier coefficients from a series which contains half of them. This construction unifi…
Ramanujan in his second notebook recorded total of seven $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 5 and 15. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving his theta--functions $φ(q)$ and $ψ(-q)$ with moduli of orders 1, 3, 5 and 15. As a consequence, several values of quotients of theta--function and a continued fraction of order 12 are explicitly evaluated.
We study the reduction of Picard modular surfaces modulo an inert prime, mod p and p-adic modular forms. We construct a theta operator on such modular forms and study its poles and its effect on Fourier-Jacobi expansions.
In a series of papers we have been studying the geometric theta correspondence for non-compact arithmetic quotients of symmetric spaces associated to orthogonal groups. It is our overall goal to develop a general theory of geometric theta liftings in the context of the real differential geometry/topology of non-compact locally symmetric spaces of orthogonal and unitary groups which generalizes the theory of Kudla-Millson in the compact case. In this paper we study in detail the geometric theta lift for Hilbert modular surfaces. In particular, we will give a new proof and an extension (to all f…
In this paper, we compute the Zwegers's modification of the mock theta functions $Φ^{[m,0] \, \ast}$ and study the modular transformation properties of the indefinite modular forms which appear in the explicit formula for the modified functions $\tildeΦ^{[m,0] \, \ast}$.
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.
In this paper, we use regularized theta liftings to construct weak Maass forms weight 1/2 as lifts of weak Maass forms of weight 0. As a special case we give a new proof of some of recent results of Duke, Toth and Imamoglu on cycle integrals of the modular j-invariant and extend these to any congruence subgroup. Moreover, our methods allow us to settle the open question of a geometric interpretation for periods of j along infinite geodesics in the upper half plane. In particular, we give the `central value' of the (non-existing) `L-function' for j. The key to the proofs is the construction of …
The modular transformations of Ramanujan's tenth order mock theta functions are computed, beginning from Choi's Hecke-type identites and using Zwegers' results on indefinite theta series. Explicit completions and shadows are found as an intermediate step. Our result for the modular transformations confirms numerical work by Gordon and McIntosh, and a recent conjecture by Cheng, Duncan, and Harvey.
We give some relations between the weights and the prime $p$ of elements of the mod $p$ kernel of the generalized theta operator $Θ^{[j]}$. In order to construct examples of the mod $p$ kernel of $Θ^{[j]}$ from any modular form, we introduce new operators $A^{(j)}(M)$ and show the modularity of $F|A^{(j)}(M)$ when $F$ is a modular form. Finally, we give some examples of the mod $p$ kernel of $Θ^{[j]}$ and the filtrations of some of them.
We consider the classical theta operator $θ$ on modular forms modulo $p^m$ and level $N$ prime to $p$ where $p$ is a prime greater than 3. Our main result is that $θ$ mod $p^m$ will map forms of weight $k$ to forms of weight $k+2+2p^{m-1}(p-1)$ and that this weight is optimal in certain cases when $m$ is at least 2. Thus, the natural expectation that $θ$ mod $p^m$ should map to weight $k+2+p^{m-1}(p-1)$ is shown to be false. The primary motivation for this study is that application of the $θ$ operator on eigenforms mod $p^m$ corresponds to twisting the attached Galois representations with the …
Utilizing a classification due to Lemke Oliver of eta-quotients which are also theta functions (here called eta-theta functions), Folsom, Garthwaite, Kang, Treneer, and the fourth author constructed a catalog of mock modular forms $V_{mn}$ having weight $3/2$ eta-theta function shadows and showed that these mock modular forms when viewed on certain sets of rationals, transform as quantum modular forms under the action of explicit subgroups. In this paper, we introduce an infinite class of functions that generalizes one row of the catalog, namely the $V_{m1}$, and show that the functions in thi…
Theta series for indefinite quadratic lattices were introduced by Zwegers, for signature (m-1,1), Alexandrov, Banerjee, Manschot and Pioline, for signature (m-2,2), and Nazaroglu, for signature (m-q,q). These series are modular modular completions, defined by means of generalized error functions, of certain non-modular holomorphic generating series associated to lattice vectors in positive cones. We show that these modular forms arise as integrals of the theta forms, defined in work of Millson and the second author, over certain singular q-cubes. We also give an explicit formula for the integr…
We describe the effect of the differential operators defined by Boecherer-Nagaoka, Flander-Ghitza and Yamauchi on the Galois representations (conjecturally) attached to Siegel modular eigenforms.
Using results from Ramanujan's lost notebook, Zudilin recently gave an insightful proof of a radial limit result of Folsom, Ono, and Rhoades for mock theta functions. Here we see that the author's previous work on the dual nature of Appell--Lerch sums and partial theta functions and on constructing bilateral $q$-series with mixed mock modular behaviour is well-suited for such radial limits. We present five more radial limit results which follow from mixed mock modular bilateral $q$-hypergeometric series. We also obtain the mixed mock modular bilateral series for a universal mock theta function…
Unary theta functions have played a significant role in the theory of holomorphic modular forms and modular $L$-functions. A partial theta functions is defined analogously, but the sum is over part of the integer lattice. Such sums fail to have modular properties. We give a construction which unifies these partial theta functions with the mock theta function introduced by Ramanujan. The modularity of Ramanujan's mock theta functions has only recently been understood by the work of Sander Zwegers.
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.
In the proposed work, we establish a total of six new $P$--$Q$ modular equations involving theta--function $f(-q)$ with moduli of orders 1, 3, 7 and 21.These equations can be regarded as modular identities in the alternate theory of signature 3. As a consequence, several values of quotients of theta--function are evaluated.
In his second notebook, Ramanujan recorded total of 23 P-Q modular equations involving theta-functions $f(-q)$, $φ(q)$ and $ψ(q)$. In this paper, modular equations analogous to those recorded by Ramanujan are obtained involving $f(-q)$. As a consequence, values of certain quotients of theta-function are evaluated.
We prove an algebraic property of the elements defining Hecke operators on period polynomials associated with modular forms, which implies that the pairing on period polynomials corresponding to the Petersson scalar product of modular forms is Hecke equivariant. As a consequence of this proof, we derive two indefinite theta series identities which can be seen as analogues of Jacobi's formula for the theta series associated with the sum of four squares.
We introduce and investigate an infinite family of functions which are shown to have generalised quantum modular properties. We realise their "companions" in the lower half plane both as double Eichler integrals and as non-holomorphic theta functions with coefficients given by double error functions. Further, we view these Eichler integrals in a modular setting as parts of certain weight two indefinite theta series.
We prove that all mod $p^m$ singular forms of level $N$, degree $n+r$, and $p$-rank $r$ with $n\ge r$ are congruent mod $p^m$ to linear combinations of theta series of degree $r$ attached to quadratic forms of some level. Moreover, we prove that, the levels of theta series are of the form ``$p\mbox{-power}\times N$''. Additionally, in some cases of mod $p$ singular forms with smallest possible weight, we prove that the levels of theta series should be $p$.
Inspired by the original definition of mock theta functions by Ramanujan, a number of authors have considered the question of explicitly determining their behavior at the cusps. Moreover, these examples have been connected to important objects such as quantum modular forms and ranks and cranks by Folsom, Ono, and Rhoades. Here we solve the general problem of understanding Ramanujan's definition explicitly for any weight $\frac12$ mock theta function, answering a question of Rhoades. Moreover, as a side product, our results give a large, explicit family of modular forms.
False theta functions closely resemble ordinary theta functions, however they do not have the modular transformation properties that theta functions have. In this paper, we find modular completions for false theta functions, which among other things gives an efficient way to compute their obstruction to modularity. This has potential applications for a variety of contexts where false and partial theta series appear. To exemplify the utility of this derivation, we discuss the details of its use on two cases. First, we derive a convergent Rademacher-type exact formula for the number of unimodal …
We construct many examples of level one Siegel modular forms in the kernel of theta operators mod $p$ by using theta series attached to positive definite quadratic forms.
We propose to generalize the work of Régis Dupont for computing modular polynomials in dimension $2$ to new invariants. We describe an algorithm to compute modular polynomials for invariants derived from theta constants and prove under some heuristics that this algorithm is quasi-linear in its output size. Some properties of the modular polynomials defined from quotients of theta constants are analyzed. We report on experiments with our implementation.
Properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobi's quartic theta function identity and counterparts of Jacobi's Principles of Duplication, Dimidiation and Change of Sign Formulas. The resulting library of quintic transformation formulas is used to describe series multisections for modular forms in terms of simple matrix operations. These efforts culminate in a formal technique for deducing congruences modulo powers of five for a variety of combinator…
Ramanujan studied the analytic properties of many $q$-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious $q$-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have $q$-expansions resembling modular theta functions, is not well understood. Here we consider families of $q$-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, includi…
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.
