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Wissam Raji

Publications and source records attributed to Wissam Raji.

17 recordsLinked to original sources
arXiv2026

Euler-type Recurrence Relations for Partition Functions with Congruence Conditions

We study partition functions $p_{δ,g}(n)$ counting partitions into parts congruent to $0$ or $\pm g \pmodδ$. Using generalized Dedekind eta functions and Rankin-Cohen brackets, we derive infinite families of Euler-type recurrences involving divisor sums and Fourier coefficients of cusp forms. We also obtain an explicit recurrence for $δ=5$, which, as a corollary, gives a Ramanujan-type congruence. As a corollary of our method of proof, we obtain a Rademacher-type formula involving Kloosterman sums and Bessel functions.

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arXiv2026

Vector-Valued Period Polynomials and Zeta Values of Quadratic Fields

Let $k\ge 2$ and $N\ge 1$ be integers. Let $D$ be a positive integer that is congruent to a square modulo $4N$, and fix $ρ$ with $ρ^2\equiv D\pmod{4N}$. In this paper, we consider two weight $2k$ cusp forms $f^{\pm}_{k,N,D,ρ}$ on $Γ_0(N)$ defined by sums over binary quadratic forms, and investigate the vector-valued period polynomial arising from these forms. Our first main result gives a closed formula for this vector-valued period polynomial. The identity component of this formula is particularly explicit: it separates as the sum of a finite \textit{algebraic part} coming from some binary forms and a \textit{zeta part} involving the values at $s=k$ of certain zeta functions. Using this formula together with a symmetry of vector-valued period polynomials, we explicitly evaluate, for odd $k$, the difference between the zeta values corresponding to the two choices of square root of $D$ modulo $4N$, in terms of Bernoulli numbers and a finite quadratic-form sum. Finally, under a vanishing condition on Fricke-invariant cusp forms at lower levels, we obtain a finite divisor-sum formula for the Dedekind zeta values $ζ_{\mathbb{Q}(\sqrt{D})}(k)$ at even integers $k$.

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arXiv2026

Rankin-Cohen Bracket for Vector-Valued Modular Forms

In this paper, we explore the relationship between Rankin-Cohen brackets for vector-valued modular forms and Petersson's inner products, deriving an explicit description of the adjoint map for the bracket operator. The study extends to the cases of Jacobi forms and skew-holomorphic Jacobi forms, establishing connections between their respective Rankin-Cohen brackets and those defined for vector-valued modular forms through an isomorphism. Adjoint maps for these extended bracket operators are also examined.

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arXiv2024

Period-like polynomials for $L$-series associated with half-integral weight cusp forms

Given the L-series of a half-integral weight cusp form, we construct a cohomology class with coefficients in a finite dimensional vector space in a way that parallels the Eichler cohomology in the integral weight case. We also define a lift of half-integral weight cusp forms to integral weight modular forms that is compatible with the $L$-series of the respective forms.

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arXiv2023

Results on the Non-Vanishing of Derivatives of L-Functions of Vector-Valued Modular Forms

We show a non-vanishing result for the averages of the derivatives of $L$-functions associated with the orthogonal basis of the space of vector-valued cusp forms of weight $k\in \frac12 \mathbb{Z}$ on the full group in the critical strip. We also show the existence of at least one basis element whose $L$-function does not vanish under certain conditions. As an application, we generalize our result to Kohnen's plus space and prove an analogous result for Jacobi forms.

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arXiv2023

Non-Vanishing of L-Functions of Vector-Valued Modular Forms

We show a non-vanishing result for the averages of L-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms on the full group. We also show the existence of at least one basis element whose L-function does not vanish under certain conditions.

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arXiv2023

The Riemann Hypothesis for period polynomials of cusp forms

We consider the period polynomials $r_f(z)$ associated with cusp forms $f$ of weight $k$ on all of $\mathrm{SL}_2\left( \mathbb{Z} \right)$, which are generating functions for the critical $L$-values of the modular $L$-function associated to $f$. In 2014, El-Guindy and Raji proved that if $f$ is an eigenform, then $r_f(z)$ satisfies a ``Riemann hypothesis" in the sense that all its zeros lie on the natural boundary of its functional equation. We show that this phenomenon is not restricted to eigenforms, and we provide large natural infinite families of cusp forms whose period polynomials almost always satisfy the Riemann hypothesis. For example, we show that for weights $k \geq 120$, linear combinations of eigenforms with positive coefficients always have unimodular period polynomials.

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arXiv2019

Class Numbers and Self-Conjugate 7-Cores

We investigate $sc_7(n)$, the number of self-conjugate $7$-core partitions of size $n$. It turns out that $sc_7(n)=0$ for $n\equiv 7\pmod 8$. For $n\equiv 1, 3, 5\pmod 8$, with $n\not \equiv 5\pmod 7,$ we find that $sc_7(n)$ is essentially a Hurwitz class number. Using recent work of Gao and Qin, we show that $$ sc_7(n) = 2^{-\varepsilon(n)-1}\cdot H(-D_n), $$ where $-D_n:=-4^{\varepsilon(n)}(7n+14)$ and $\varepsilon(n):=\frac{1}{2}\cdot(1+(-1)^{\frac{n-1}{2}})$. This fact implies several corollaries which are of interest. For example, if $-D_n$ is a fundamental discriminant and $p\not \in \{2, 7\}$ is a prime with $ord_p(-D_n)\leq 1$, then for every positive integer $k$ we have $$ sc_7\left((n+2)p^{2k}-2\right)=sc_7(n)\cdot \left(1+\frac{p^{k+1}-p}{p-1}-\frac{p^k-1}{p-1}.\left(\frac{-D_n}{p}\right)\right), $$ where $\left(\frac{-D_n}{p}\right)$ is the Legendre symbol.

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arXiv2016

Periods of modular forms and identities between Eisenstein series

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.

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arXiv2013

Eichler integrals for Maass cusp forms of half-integral weight

In this paper, we define and discuss Eichler integrals for Maass cusp forms of half-integral weight on the full modular group. We discuss nearly periodic functions associated to the Eichler integrals, introduce period functions for such Maass cusp forms, and show that the nearly periodic functions and the period functions are closely related. Those functions are extensions of the periodic functions and period functions for Maass cusp forms of weight 0 on the full modular group introduced by Lewis and Zagier.

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arXiv2012

Eichler Cohomology of Generalized Modular Forms of Real Weights

In this paper, we prove the Eichler cohomology theorem of weakly parabolic generalized modular forms of real weights on subgroups of finite index in the full modular group. We explicitly establish the isomorphism for large weights by constructing the map from the space of cusp forms to the cohomology group.

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arXiv2012

Generalized Maass Wave Forms

We initiate the study of generalized Maass wave forms, those Maass wave forms for which the multiplier system is not necessarily unitary. We then prove some basic theorems inherited from the classical theory of modular forms with a generalization of some examples from the classical theory of Maass forms.

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arXiv2010

On generalized modular forms supported on cuspidal and elliptic points

In this paper, we extend previous results to prove that generalized modular forms with rational Fourier expansions whose divisors are supported only at the cusps and certain other points in the upper half plane are actually classical modular forms. We discuss possible limitations to this extension and pose questions about possible zeroes for modular forms of prime level.

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