Holomorphic Modular Forms Research Articles

Asymptotic formulas for coefficients of Kac–Wakimoto Characters

AbstractWe study the coefficients of Kac and Wakimoto's character formulas for the affine Lie superalgebrassℓ(n+1|1)∧. The coefficients of these characters are the weight multiplicities of irreducible modules of the Lie superalgebras, and their asymptotic study begins with Kac and Peterson's earlier use of modular forms to understand the coefficients of characters for affine Lie algebras. In the affine Lie superalgebra setting, the characters are products of weakly holomorphic modular forms and Appell-type sums, which have recently been studied using developments in the theory of mock modular forms and harmonic Maass forms. Using our previously developed extension of the Circle Method for products of mock modular forms along with the Saddle Point Method, we find asymptotic series expansions for the coefficients of the characters with polynomial error.

The Lie theory of certain modular form and arithmetic identities

Modular form identities lying in the framework of Shimura’s theory of nearly holomorphic modular forms are obtained by Lie theoretic means as consequences of identities relating the Maass–Shimura operator and the Rankin–Cohen brackets, which in turn follow from change-of-basis formulae in the theory of Verma modules. The Lie theoretic origin of known van der Pol and Lahiri-type arithmetic identities is thus unveiled, and similar new ones are derived in a systematic way. These identities relate divisor functions, Ramanujan’s τ-function and functions defined by the Fourier coefficients of other cusp forms and involve hybrid coefficients, drawn from Lie theory and number theory, given explicitly by formulae combining the arithmetic Clebsch–Gordan coefficients and the Bernoulli numbers. A few side results, interesting in their own right, such as Leibniz-type rules satisfied by the Rankin–Cohen brackets, are also obtained.

Zeros of weakly holomorphic modular forms of levels 2 and 3

Let $M_k^\sharp(N)$ be the space of weakly holomorphic modular forms for $\Gamma_0(N)$ that are holomorphic at all cusps except possibly at $\infty$. We study a canonical basis for $M_k^\sharp(2)$ and $M_k^\sharp(3)$ and prove that almost all modular forms in this basis have the property that the majority of their zeros in a fundamental domain lie on a lower boundary arc of the fundamental domain.

A fast algorithm to computeL(1/2,f×χq)

Let f be a fixed (holomorphic or Maass) modular cusp form. Let χ q be a Dirichlet character mod q . We describe a fast algorithm that computes the value L ( 1 / 2 , f × χ q ) up to any specified precision. In the case when q is smooth or highly composite integer, the time complexity of the algorithm is given by O ( 1 + | q | 5 / 6 + o ( 1 ) ) .

Basis for the space of weakly holomorphic modular forms in higher level cases

Let k be an even integer. We find a canonical basis for the space of weakly holomorphic modular forms of weight k for Γ 0 + ( p ) and investigate its properties.

On the asymptotic behavior of Kac-Wakimoto characters

Recently, Kac and Wakimoto established specialized character formulas for irreducible highest weight s ℓ ( m , 1 ) ∧ s\ell (m,1)^\wedge modules and established a main exponential term in their asymptotic expansions. By different methods, we improve upon the Kac-Wakimoto asymptotics for these characters, obtaining an asymptotic expansion with an arbitrarily large number of terms beyond the main term. More specifically, it is well known that in the case of holomorphic modular forms, asymptotic information may be obtained using modular transformation properties. However, here this is not the case due to the analytic nature of the Kac-Wakimoto series as discovered recently by the first author and Ono. We first “complete” these series by adding to them certain integrals, obtaining functions that exhibit suitable modular transformation laws, at the expense of the completed objects being nonholomorphic. We then exploit this mock-modular behavior of the Kac-Wakimoto series to obtain our asymptotic expansion. In particular, we show that beyond the main term, the asymptotic behavior is dictated by the nonholomorphic part of the completed Kac-Wakimoto characters, which is a priori invisible. Euler numbers (equivalently, zeta-values) appear as coefficients.

Linear relations among Poincaré series via harmonic weak Maass forms

We discuss the problem of the vanishing of Poincare series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan’s mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincare series.

Boundary behaviour of special cohomology classes arising from the Weil representation

AbstractIn our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

On Kaufhold’s Whittaker functions

In this paper we give an integral representation for a Whittaker function of an non holomorphic Eisenstein series which is a non holomorphic Sigel modular form of degree 2. Our integral representation is very useful to the theory of the theta lifting of automorphic forms.

Logarithmic vector-valued modular forms and polynomial-growth estimates of their Fourier coefficients

We establish (Theorem 3.6) polynomial-growth estimates for the Fourier coefficients of holomorphic logarithmic vector-valued modular forms.

One-loop BPS amplitudes as BPS-state sums

AbstractRecently, we introduced a new procedure for computing a class of one-loop BPS-saturated amplitudes in String Theory, which expresses them as a sum of one-loop contributions of all perturbative BPS states in a manifestly T-duality invariant fashion. In this paper, we extend this procedure toallBPS-saturated amplitudes of the form ∫FΓd+k,dΦ, withΦbeing a weak (almost) holomorphic modular form of weight − k/2. We use the fact that any suchΦcan be expressed as a linear combination of certain absolutely convergent Poincaré series, against which the fundamental domain F can be unfolded. The resulting BPS-state sum neatly exhibits the singularities of the amplitude at points of gauge symmetry enhancement, in a chamber-independent fashion. We illustrate our method with concrete examples of interest in heterotic string compactifications.

Eichler–Shimura theory for mock modular forms

We use mock modular forms to compute generating functions for the critical values of modular \(L\)-functions, and we answer a generalized form of a question of Kohnen and Zagier by deriving the “extra relation” that is satisfied by even periods of weakly holomorphic cusp forms. To obtain these results we derive an Eichler–Shimura theory for weakly holomorphic modular forms and mock modular forms. This includes two “Eichler–Shimura isomorphisms”, a “multiplicity two” Hecke theory, a correspondence between mock modular periods and classical periods, and a “Haberland-type” formula which expresses Petersson’s inner product and a related antisymmetric inner product on \(M_{k}^{!}\) in terms of periods.

NON-VANISHING OF DERIVATIVES OF L-FUNCTIONS ATTACHED TO HILBERT MODULAR FORMS

This paper is to show a non-vanishing property of the derivative of certain L-functions. For certain primitive holomorphic Hilbert modular forms, if the central critical value of the standard L-function does not vanish, then neither does its derivative. This is a generalization of a result by Gun, Murty and Rath in the case of elliptic modular forms. Some applications in transcendental number theory deduced from this result are discussed as well.

Rapid Computation of L-functions for Modular Forms

Let $f$ be a fixed (holomorphic or Maass) modular cusp form, with $L$-function $L(f,s)$. We describe an algorithm that computes the value $L(f,1/2+ iT)$ to any specified precision in time $O(1+|T|^{7/8})$.

Galois representations for holomorphic Siegel modular forms

We prove local–global compatibility (up to a quadratic twist) of Galois representations associated to holomorphic Hilbert–Siegel modular forms in many cases (induced from Borel or Klingen parabolic), and as a corollary we obtain a conjecture of Skinner and Urban. For Siegel modular forms, when the local representation is an irreducible principal series we get local–global compatibility without a twist. We achieve this by proving a version of rigidity (strong multiplicity one) for GSp(4) using, on the one hand the doubling method to compute the standard L-function, and on the other hand the explicit classification of the irreducible local representations of GSp(4) over p-adic fields; then we use the existence of a globally generic Hilbert–Siegel modular form weakly equivalent to the original and we refer to Sorensen (Mathematica 15:623–670, 2010) for local–global compatibility in that case.

Linear relations among Poincaré series

We discuss the problem of the vanishing of Poincar\'e series. This problem is known to be related to the existence of weakly holomorphic forms with prescribed principal part. The obstruction to the existence is related to the pseudomodularity of Ramanujan's mock theta functions. We embed the space of weakly holomorphic modular forms into the larger space of harmonic weak Maass forms. From this perspective we discuss the linear relations between Poincar\'e series and the connection to Ramanujan's mock theta functions.

Linear dependence of certain L -values of half-integral weight modular forms

Let h be a cusp form of half-integral weight, and ε 2 / 3 ( χ ) be a certain (not necessarily holomorphic) modular form of weight 3 2 associated with a Dirichlet character χ. We then prove linear dependence of the values R ( l , h , ε 2 / 3 ( χ ) ) of the Rankin–Selberg convolution products of h and ε 2 / 3 ( χ ) when we fix an integer l and vary χ. A main idea for the proof is to express such a convolution product as the twisted Koecher–Maass series L(s, M(h), χ) for the Maass lift M(h) of h, whose values at integers were investigated by Choie and Kohnen. Moreover, by using such an expression, we obtain an algebraicity result for the values of L(s, M(h), χ) at half-integers, which was not considered by Choie and Kohnen.

Two-divisibility of the coefficients of certain weakly holomorphic modular forms

We study a canonical basis for spaces of weakly holomorphic modular forms of weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a relation between the Fourier coefficients of modular forms in this canonical basis and a generalized Ramanujan tau-function, and use this to prove that these Fourier coefficients are often highly divisible by 2.

$p$-adic properties of coefficients of basis for the space of weakly holomorphic modular forms of weight 2

We observe properties of coefficients of certain basis elements for the space of weakly holomorphic modular forms of weight 2 for $SL_{2}(\mathbf{Z})$. Moreover we show that these coefficients are often highly divisible by the primes 2, 3, 5, 7, 11.

The sixth moment of the family of Γ1(q)-automorphic L-functions

In this paper we investigate the sixth moment of the family of L-functions associated to holomorphic modular forms on GL 2 with respect to a congruence subgroup Γ1(q). The bound for central values averaged over the family, consistent with the Lindelof hypothesis, is obtained for prime levels q.