The classical modular equations involve bivariate polynomials that can be seen to be univariate with coefficients in the modular invariant $j$. Kiepert found modular equations relating some $η$-quotients and the Weber functions $γ_2$ and $γ_3$. In the present work, we extend this idea to double $η$-quotients and characterize all the parameters leading to this kind of equation. We give some properties of these equations, explain how to compute them and give numerical examples.
Cheng constructed product expansions from twists of elliptic genera of symmetric powers of K3 surfaces that are related to M_24 moonshine. We study which of them are Siegel modular forms. If the predicted level is non-composite, they are modular, and their powers can be represented as products of rescaled Borcherds products.
Recently, D. Choi obtained a description of the coefficients of the infinite product expansions of meromorphic modular forms over $Γ_0(N)$. Using this result, we provide some bounds on these infinite product coefficients for holomorphic modular forms. We give an exponential upper bound for the growth of these coefficients. We show that this bound is also a lower bound in the case that the genus of the associated modular curve $X_0(N)$ is $0$ or $1$.
We extend to positive real weights Haberland's formula giving a cohomological description of the Petersson scalar product of modular cusp forms of positive even weight. This relation is based on the cup product of an Eichler cocycle and a Knopp cocycle. We also consider the cup product of two Eichler cocycles attached to modular forms. In the classical context of integral weights at least $2$ this cup product is uninteresting. We show evidence that for real weights this cup product may very well be non-trivial. We approach the question whether the cup product is a non-trivial coinvariant by du…
Analytic continuation and functional equation of a Dirichlet series constructed from two (not necessarily cuspidal) holomorphic modular forms is discussed, where either weights of the modular forms or characters are not necessarily equal to each other. The applications to quadratic forms are also given.
Ming-Lun Hsieh constructed three-variable triple product p-adic L-functions attached to triples of primitive Hida families and proved interpolation formulas. We generalize his result in the unbalanced case and construct a three-variable triple product p-adic L-function attached to a primitive Hida family and two more general p-adic families of modular forms.
We consider a natural extension of the Petersson scalar product to the entire space of modular forms of integral weight $k\ge 2$ for a finite index subgroup of the modular group. We show that Hecke operators have the same adjoints with respect to this inner product as for cusp forms, and we show that the Petersson product is nondegenerate for $Γ_1(N)$ and $k>2$. For $k=2$ we give examples when it is degenerate, and when it is nondegenerate.
It was shown in previous works that the measure associated to holomorphic newforms of weight $k$ and level $q$ will tend weakly to the Haar measure on modular curve of level 1, as $qk\rightarrow \infty$. In this paper we proved that this phenomenon is also true on modular curves of general level N.
We prove that amongst all real quadratic fields and all spaces of Hilbert modular forms of full level and of weight $2$ or greater, the product of two Hecke eigenforms is not a Hecke eigenform except for finitely many real quadratic fields and finitely many weights. We show that for $\mathbb Q(\sqrt 5)$ there are exactly two such identities.
In this paper we study generalizations of quadratic form Poincaré series, which naturally occur as outputs of theta lifts. Integrating against them yields evaluations of higher Green's functions. For this we require a new regularized inner product, which is of independent interest.
We develop a regularization for Petersson inner products of arbitrary weakly holomorphic modular forms, generalizing several known regularizations. As one application, we extend work of Duke, Imamoglu, and Toth on regularized inner products of weakly holomorphic modular forms of weights $0$ and $3/2$. These regularized inner products can be evaluated in terms of the coefficients of holomorphic parts of harmonic Maass forms of dual weights. Moreover, we study the errors of modularity of the holomorphic parts of such a harmonic Maass forms and show that they induce cocyles in the first parabolic…
We define graded hyper-algebras of vector-valued Siegel modular forms, which allow us to study tensor products of the latter. We also define vector-valued Hecke operators for Siegel modular forms at all places of ${\mathbb Q}$, acting on these hyper-algebras. These definitions bridge the classical and representation theoretic approach to Siegel modular forms. Combining both the product structure and the action of Hecke operators, we prove in the case of elliptic modular forms that all cusp forms of sufficiently large weight can be obtained from products involving only two fixed Eisenstein seri…
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 $Γ_0(N)$.
We explicitly write down the Eisenstein elements inside the space of modular symbols for Eisenstein series with integer coefficients for the congruence subgroups Γ_0 (pq) with p and q distinct odd primes, giving an answer to a question of Merel in these cases. We also compute the winding elements explicitly for these congruence subgroups. Our results are explicit versions of the Manin-Drinfeld Theorem.
In this paper we study regularized Petersson products between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight 1 modular form with integral Fourier coefficients. In our recent work \cite{Via CM Green} motivated by the conjecture of B. Gross and D. Zagier on the CM values of higher Green's functions we have discovered that such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. The main result of the present paper is the explicit factori…
We discuss the relationship between different notions of "integrality" in motivic cohomology/K-theory which arise in the Beilinson and Bloch-Kato conjectures, and prove their equivalence in some cases for products of curves (used in the authors' previous paper in this series), as well as obtaining a general result, first proved by Jannsen (unpublished), which reduces their equivalence to standard conjectures in arithmetic algebraic geometry.
We p-adically interpolate the relative de Rham cohomology of the universal elliptic curve over strict neighbourhoods of the ordinary locus of modular curves, together with the Hodge filtration and Gauss-Manin connection. Sections of these sheaves provide the so called nearly overconvergent modular forms. This extends previous work of Andreatta, Iovita and Pilloni where we p-adically interpolate powers of the Hodge bundle and in that case the sections coincide with Coleman overconvergent modular forms. We also show that, under suitable assumptions, one can p-adically interpolate the Gauss-Manin…
We show that certain spaces of vector valued modular forms are isomorphic to spaces of scalar valued modular forms whose Fourier coefficients are supported on suitable progressions. As an application we give a very explicit description of Borcherds products on Hilbert modular surfaces.
We give theoretical and practical information on the Pari/GP modular forms package available since the spring of 2018. Thanks to the use of products of two Eisenstein series, this package is the first which can compute Fourier expansions at any cusps, evaluate modular forms near the real axis, evaluate L-functions of non-eigenforms, and compute general Petersson scalar products.
We show, for levels of the form $N = p^a q^b N'$ with $N'$ squarefree, that in weights $k \geq 4$ every cusp form $f \in \mathcal{S}_k(N)$ is a linear combination of products of certain Eisenstein series of lower weight. In weight $k=2$ we show that the forms $f$ which can be obtained in this way are precisely those in the subspace generated by eigenforms $g$ with $L(g, 1) \neq 0$. As an application of such representations of modular forms we can calculate Fourier expansions of modular forms at arbitrary cusps and we give several examples of such expansions in the last section.
We investigate Poincaré series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincaré series are almost holomorphic as well. In general this is not the case. The main point of this paper is the study of Siegel-Poincaré series of degree $2$ attached to products of terms of Fourier series of harmonic Siegel-Maass forms and holomorphic Siegel modular forms. …
We compute the p-adic Abel-Jacobi map of the product of a Hilbert modular surface and a modular curve at a null-homologous (modified) embedding of the modular curve in this product, evaluated on differentials associated to a Hilbert cuspidal form f of weight (2,2) and a cuspidal form of weight 2. We generalize this computation to suitable null-homological cycles in the fibre products of the universal families on the surface and the curve, evaluated at differentials associated to f and g of higher weights. We express the values of the p-adic Abel-Jacobi map at these weights in terms of a p-adic…
In his striking 1995 paper, Borcherds found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant $-d$ evaluated at the modular $j$-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, $A(n,d)$, are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of $A(n,d)$ modulo a prime $\ell$ by relatin…
Let X be a product of Drinfeld modular curves over a general base ring A of odd characteristic. We classify those subvarieties of X which contain a Zariski-dense set of CM points. This is an analogue of the André-Oort conjecture. As an application, we construct non-trivial families of higher Heegner points on modular elliptic curves over global function fields.
We use Borcherds products and their restrictions to Hirzebruch-Zagier curves to determine generators and relations for the graded rings of Hilbert modular forms for the fields $\mathbb{Q}(\sqrt{29})$ and $\mathbb{Q}(\sqrt{37})$. These seem to be the first cases where the graded ring can be computed despite obstructions to the existence of Borcherds products with arbitrary divisors.
We construct an Euler system in the cohomology of the tensor product of the Galois representations attached to two modular forms, using elements in the higher Chow groups of products of modular curves. We use this Euler system to prove a finiteness theorem for the strict Selmer group of the Galois representation when the associated p-adic Rankin--Selberg L-function is non-vanishing at s = 1.
We provide an exact formula for the complex exponents in the modular product expansion of the modular units, and deduce a characterization of the modular units in terms of the growth of these exponents, answering a question of W. Kohnen.
We show that every elliptic modular form of integral weight greater than $1$ can be expressed as linear combinations of products of at most two cusp expansions of Eisenstein series. This removes the obstruction of nonvanishing central $\mathrm{L}$-values present in all previous work. For weights greater than $2$, we refine our result further, showing that linear combinations of products of exactly two cusp expansions of Eisenstein series suffice.
Let $f\in S_{k+1/2}(N,χ)$ be a Hecke eigenform of half integral weight $k+1/2\,(k\geq 2)$ and the real nebentypus $χ=\pm 1$ where the Fourier coefficients $a(n)$ are reals. We prove that the sequence $\{χ(p^ν)a(tp^{2ν})\}_{ν\in\N}$ has infinitely many sign changes for almost all primes $p$ where $t$ is a squarefree integer such that $a(t)\neq 0$. The same result holds for the sequences of Fourier coefficients $\{a(tp^{2(2ν+1)})\}_{ν\in\N}$ and $\{a(tp^{4ν})\}_{ν\in\N}$.
The main goal of this paper is to give a modular type representation for the infinite product $(1-x)(1-xq)(1-xq^2)(1-xq^3)...$. It is shown that this representation essentially contains the well-known modular formulae either for Dedekind's eta function, Jacobi theta function or for certain Lambert series. Thus a new and unified approach is outlined for the study of elliptic and modular functions and related series.
We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic curves over certain real quadratic fields are modular.
Let $p$ be a prime for which the congruence group $Γ_0(p)^*$ is of genus zero, and $j_p^*$ be the corresponding Hauptmodul. Let $f$ be a nearly holomorphic modular form of weight 1/2 on $Γ_0(4p)$ which satisfies some congruence condition on its Fourier coefficients. We interpret $f$ as a vector valued modular form. Applying Borcherds lifting of vector valued modular forms we construct infinite products associated to $j_p^*$ and extend Zagier's trace formula for singular values of $j_p^*$. Further we investigate the twisted traces of sigular values of $j_p^*$ and construct Borcherds products re…
We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two. Moreover, we determine the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and study Faltings heights of arithmetic Hirzebruch-Zagier divisors.
Let $X_0(I)$ be the Drinfeld's modular curve with level $I$ structure, where $I$ is a monic square-free ideal in $\F_{q}[T]$. In this paper we show the existence of an element in the motivic cohomology group $H^3_{\M}(X_0(I) \times X_0(I),\Q(2))$ whose regulator is related to a special value of a Ranking-Selberg convolution $L$-function. This result is the function field analogue of a theorem of Beilinson for the self product of a modular curve.
We classify the simple even lattices of square free level and signature (2,n) for n > 3. A lattice is called simple if the space of cusp forms of weight 1+n/2 for the dual Weil representation of the lattice is trivial. For a simple lattice every formal principal part obeying obvious conditions is the principal part of a vector valued modular form. Using this, we determine all holomorphic Borcherds products of singular weight (arising from vector valued modular forms with non-negative principal part) for the simple lattices. We construct the corresponding vector valued modular forms by eta prod…
Beilinson obtained a formula relating the special value of the L-function of H^2 of a product of modular curves to the regulator of an element of a motivic cohomology group - thus providing evidence for his general conjectures on special values of L-functions. In this paper we prove a similar formula for the L-function of the product of two Drinfeld modular curves providing evidence for an analogous conjecture in the case of function fields.
We use spectral method to prove a joint equidistribution of primitive rational points and the same along expanding horocycle orbits in the products of the circle and the unit cotangent bundle of the modular surface. This result explicates the error bound in a recent work of Einsiedler, Luethi, and Shah \cite[Theorem $1.1$]{ELS}. The error is sharp upon the best known progress towards the Ramanujan conjecture at the finite places for the modular surface.
We show that a weakly holomorphic modular function can be written as a sum of modular units of higher level. We further find a necessary and sufficient condition for a Siegel modular function of degree $g$ to have neither zero nor pole on the domain when restricted to certain subset of the Siegel upper half-space $\mathbb{H}_g$.
We find some modularity criterion for a product of Klein forms of the congruence subgroup $Γ_1(N)$ and, as its application, construct a basis of the space of modular forms for $Γ_1(13)$ of weight $2$. In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form and prove it conditionally by applying Hecke operators.
In recent work, Bacher and de la Harpe define and study conjugacy growth series for finitary permutation groups. In two subsequent papers, Cotron, Dicks, and Fleming study the congruence properties of some of these series. We define a new family of conjugacy growth series for the finitary alternating wreath product that are related to sums of modular forms of integer and half-integral weights, the so-called \textit{mixed weight modular forms}. The previous works motivate the study of congruences for these series. We prove that congruences exist modulo powers of all primes $p \geq 5$. Furthermo…
We prove effective equidistribution of primitive rational points and of primitive rational points defined by monomials along long horocycle orbits in products of the torus and the modular surface. This answers a question posed in joint work by the first and the last named author with Shahar Mozes and Uri Shapira. Under certain congruence conditions we prove the joint equidistribution of conjugate rational points in the two-torus and the modular surface.
Borisov and Gunnells observed in 2001 that certain linear relations between products of two holomorphic weight 1 Eisenstein series had the same structure as the relations between periods of modular forms; a similar phenomenon exists in higher weights. We give a conceptual reason for this observation in arbitrary weight. This involves an unconventional way of expanding the Rankin-Selberg convolution of a cusp form with an Eisenstein series. We also prove a partial result towards understanding the action of a Hecke operator on a product of two Eisenstein series.
In this note we consider a question of Ono, concerning which spaces of classical modular forms can be generated by sums of $η$-quotients. We give some new examples of spaces of modular forms which can be generated as sums of $η$-quotients, and show that we can write all modular forms of level $Γ_0(N)$ as rational functions of $η$-products.
We use the main theorem of Boxer-Calegari-Gee-Pilloni (arXiv:1812.09269) to give explicit examples of modular abelian surfaces $A$ over $\mathbf{Q}$ without extra endomorhpisms such that $A$ has good reduction outside the primes 2, 3, 5, and 7.
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 study modular forms of some congruence subgroups. In this paper, we treat the cases level is 2-power, 3-power or 5. Structures of graded rings and many identities of infinite sum or infinite product are given. Theory of rational (1/3, 1/4, 1/5 etc) weight or more formally weight modular form are introduced.
In this paper we construct 16 free algebras of modular forms on symmetric domains of type IV for some reflection groups related to the eight lattices $A_1(2)$, $A_1(3)$, $A_1(4)$, $2A_1(2)$, $A_2(2)$, $A_2(3)$, $A_3(2)$, $D_4(2)$. As a corollary, we prove the modularity of formal Fourier--Jacobi expansions for these reflection groups.
We give generators and relations for the graded rings of Hermitian modular forms of degree two over the rings of integers in $\mathbb{Q}(\sqrt{-7})$ and $\mathbb{Q}(\sqrt{-11})$. In both cases we prove that the subrings of symmetric modular forms are generated by Maass lifts. The computation uses a reduction process against Borcherds products which also leads to a dimension formula for the spaces of modular forms.
In this paper, we consider modular forms for finite index subgroups of the modular group whose Fourier coefficients are algebraic. It is well-known that the Fourier coefficients of any holomorphic modular form for a congruence subgroup (with algebraic coefficients) have bounded denominators. It was observed by Atkin and Swinnerton-Dyer that this is no longer true for modular forms for noncongruence subgroups and they pointed out that unbounded denominator property is a clear distinction between modular forms for noncongruence and congruence modular forms. It is an open question whether genuine…
