M. Raum

Pseudodifferential Jacobi forms and Geometric Rankin-Cohen Brackets

Martin Raum, Anne V. Shepler

Cohen, Manin, and Zagier recovered the Rankin-Cohen bracket for modular forms from an action of the modular group on pseudodifferential operators whose coefficients are holomorphic…

Recursions for Mock Theta Functions

Matthew Ortiz, Martin Raum, Olav K. Richter

We establish weighted recursions for the coefficients of Ramanujan's third order mock theta functions $f$ and $ω$ . Specifically, we apply a holomorphic projection operator to vecto…

Weighted Recursions for Hurwitz Class Numbers

Matthew Ortiz, Martin Raum, Olav K. Richter

We establish new recursions for Hurwitz class numbers with polynomial weights. In contrast to previous recursions, our results decouple class numbers of even and odd discriminants.…

Theta Cycles of Modular Forms Modulo $p^{2}$

Scott Ahlgren, Martin Raum, Olav K. Richter

The theta cycle of a modular form modulo a prime $p \geq 5$ is well understood. By contrast, the theta cycle modulo a power of $p$ is still mysterious and experimentally erratic. He…

The Geometric Unitary Kudla Conjecture

Martin Raum

We prove that, over an arbitrary CM field, every symmetric formal Fourier-Jacobi series converges and equals the Fourier-Jacobi expansion of a genuine Hermitian Hilbert modular for…

math.NT2025

Jacobi Forms of Affine Weight in Higher Cogenus and Nearly Holomorphic Functions

Jan Feldmann, Martin Raum

We describe Jacobi forms of vector-valued weights in terms of classical ones, extending previous results by Ibukiyama and Kyomura to the case of arbitrary cogenus. As in their resu…

math.NT2025

Correction to "Kudla's Modularity Conjecture and Formal Fourier-Jacobi Series"

Jan Hendrik Bruinier, Martin Raum

We correct an error in Lemma 4.4 and its application in Theorem 4.5 in our paper ``Kudla's Modularity Conjecture and Formal Fourier-Jacobi Series''.

math.NT2025

Eisenstein series modulo $p^{2}$

Scott Ahlgren, Michael Hanson, Martin Raum +1

We study congruences for Eisenstein series on $SL_{2} (Z)$ modulo $p^{2}$ , where $p \geq 5$ is prime. It is classically known that all Eisenstein series of weight at l…

Spectral decomposition and Siegel-Veech transforms for strata: The case of marked tori

Jayadev Athreya, Jean Lagacé, Martin Möller +1

Generalizing the well-known construction of Eisenstein series on the modular curves, Siegel-Veech transforms provide a natural construction of square-integrable functions on strata…

Formal deformations of modular forms and multiple L-values

Adam Keilthy, Martin Raum

We relate analytically defined deformations of modular curves and modular forms from the literature to motivic periods via cohomological descriptions of deformation theory. Leverag…

Explainable Ramanujan-type Congruences on Square-Classes of Arithmetic Progressions

Martin Raum

While examples of Ramanujan-type congruences are amply available via their relation to Hecke operators, it remains unclear which of them should be considered of combinatorial origi…

Formal Siegel modular forms for arithmetic subgroups

Jan Hendrik Bruinier, Martin Raum

The notion of formal Siegel modular forms for an arithmetic subgroup $Γ$ of the symplectic group of genus $n$ is a generalization of symmetric formal Fourier-Jacobi series. Assumin…

math.NT2023

Imaginary quadratic fields with $ℓ$ -torsion-free class groups and specified split primes

Olivia Beckwith, Martin Raum, Olav Richter

Given an odd prime $ℓ$ and finite set of odd primes $S_{+}$ , we prove the existence of an imaginary quadratic field whose class number is indivisible by $ℓ$ and which splits at…

Scalar-valued depth two Eichler-Shimura Integrals of Cusp Forms

Tobias Magnusson, Martin Raum

Given cusp forms $f$ and $g$ of integral weight $k \geq 2$ , the depth two holomorphic iterated Eichler-Shimura integral $I_{f, g}$ is defined by ${\int_τ^{i\infty}f(z)(X-z)^{k-2}I_g…

A classification of polyharmonic Maaß forms via quiver representations

Claudia Alfes-Neumann, Igor Burban, Martin Raum

We give a classification of the Harish-Chandra modules generated by the pullback to~$\SL{2}(\RR)$ of \emph{poly}harmonic Maaß forms for congruence subgroups of~$\SL{2}(\ZZ)$ with e…

The Bernstein-Gelfand Tensor Product Functor and the Weight-2 Eisenstein Series

Martin Raum

The Bernstein-Gelfand tensor product functors are endofunctors of the category of Harish-Chandra modules provided by tensor products with finite dimensional modules. We provide an…

Eichler integrals and generalized second order Eisenstein series

Albin Ahlbäck, Tobias Magnusson, Martin Raum

We show that all Eichler integrals, and more generally all "generalized second order modular forms" can be expressed as linear combinations of corresponding generalized second orde…

Congruences of Hurwitz class numbers on square classes

Olivia Beckwith, Martin Raum, Olav Richter

We extend a holomorphic projection argument of our earlier work to prove a novel divisibility result for non-holomorphic congruences of Hurwitz class numbers. This result allows us…

On the Computation of General Vector-valued Modular Forms

Tobias Magnusson, Martin Raum

We present and discuss an algorithm and its implementation that is capable of directly determining Fourier expansions of any vector-valued modular form of weight at least $2$ assoc…