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Martin Raum

Publications and source records attributed to Martin Raum.

At least 19 recordsLinked to original sources
arXiv2026

Pseudodifferential Jacobi forms and Geometric Rankin-Cohen Brackets

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 functions on the Poincar\'e upper half plane. We investigate pseudodifferential operators on the Jacobi upper half space with respect to the elliptic variable instead of the modular variable typically considered. We introduce a family of actions of the Jacobi group and show that a space of invariant pseudodifferential operators is isomorphic to the space of Jacobi forms by producing an equivariant map. Our construction arises from the explicit action of a Casimir operator for the complexified Lie algebra of the real Jacobi Lie group. As an application, we identify new families of Rankin-Cohen brackets with geometric origin indexed by a complex parameter. In particular, we isolate a subvariety of lines of Rankin-Cohen brackets in each degree of expected dimension $1$ reflecting the geometry of the Jacobi upper half space.

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arXiv2026

Recursions for Mock Theta Functions

We establish weighted recursions for the coefficients of Ramanujan's third order mock theta functions $f$ and $\omega$. Specifically, we apply a holomorphic projection operator to vector-valued Rankin-Cohen brackets of completed mock theta series and their shadows. By employing a vector-valued framework, we exploit the vanishing of certain spaces of vector-valued cusp forms. Our proof is AI-assisted and prioritizes accessibility, allowing for straightforward customization and replication within the broader research community.

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arXiv2026

Weighted Recursions for Hurwitz Class Numbers

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. Our main tool is the vector-valued holomorphic projection operator applied to mock modular forms. We invoke representation theory to connect the relevant spaces of vector-valued modular forms to spaces of classical new and old forms. We thereby leverage the vanishing of spaces of vector-valued cusp forms not available in the scalar case.

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arXiv2026

The Geometric Unitary Kudla Conjecture

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 form. As an application, we show that the Chow-valued Kudla generating series of special cycles on unitary Shimura varieties for Hermitian lattices over CM fields of signature $(p,1)$ at one infinite place and $(p+1,0)$ at all others is modular of weight $p+1$ for a Weil representation, establishing the geometric unitary Kudla Conjecture in arbitrary codimension. This removes the modularity hypothesis from the arithmetic inner product formula by Li-Liu.

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arXiv2026

Theta Cycles of Modular Forms Modulo $p^2$

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. Here we completely determine the theta cycle of a weight $k < p$ modular form modulo $p^2$ on the initial segment of length $p$ and we prove exact values or nontrivial bounds for the weight filtrations on $p-2$ further segments of length $p - k + 1$. In particular, asymptotically as $p \to \infty$ we establish 50% of the theta cycle exactly, and we provide nontrivial bounds for 100% of it. We determine the first two low points exactly and $\left\lfloor \frac{p - k + 1}{2} \right\rfloor$ further low points at regular positions. Moreover, we detect low points at exceptional positions which solve a quadratic equation modulo $p$, and which disturb the otherwise regular structure in the segments that we exhibit.

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arXiv2025

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

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 result, our isomorphisms are given by holomorphic covariant differential operators. In contrast to previous work, however, we avoid explicit calculations, which we replace by general differential geometric arguments. In the process, we obtain a structure theorem on nearly holomorphic functions on the Jacobi upper half space.

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arXiv2025

Eisenstein series modulo $p^2$

We study congruences for Eisenstein series on $\mathrm{SL}_2(\mathbb{Z})$ modulo $p^2$, where $p \geq 5$ is prime. It is classically known that all Eisenstein series of weight at least $4$ are determined modulo $p^2$ by those of weight at most $p^2-p+2$. We prove that up to powers of $E_{p-1}$, each such Eisenstein series is in fact determined modulo $p^2$ by a modular form of weight at most $2p-4$. We also determine $E_2$ modulo $p^2$ in terms of a modular form of weight $p+1$.

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arXiv2024

A classification of polyharmonic Maaß forms via quiver representations

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 exponential growth allowed at the cusps. This extends results of Bringmann--Kudla in the harmonic case. While in the harmonic setting there are nine cases, our classification comprises ten; A new case arises in weights $k > 1$. To obtain the classification we introduce quiver representations into the topic and show that those associated with polyharmonic Maaß forms are cyclic, indecomposable representations of the two-cyclic or the Gelfand quiver. A classification of these transfers to a classification of polyharmonic weak Maaß forms. To realize all possible cases of Harish-Chandra modules we develop a theory of weight shifts for Taylor coefficients of vector-valued spectral families. We provide a comprehensive computer implementation of this theory, which allows us to provide explicit examples.

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arXiv2024

Formal Siegel modular forms for arithmetic subgroups

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. Assuming an upper bound on the affine covering number of the Siegel modular variety associated with $Γ$, we prove that all formal Siegel modular forms are given by Fourier-Jacobi expansions of classical holomorphic Siegel modular forms. We also show that the required upper bound is always met if $2\leq n \leq 4$. As an application we consider the case of the paramodular group of squarefree level and genus $2$.

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arXiv2024

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

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 of differentials on Riemannian surfaces. This space carries actions of the foliated Laplacian derived from the SL(2,R)-action as well as various differential operators related to relative period translations. In the paper we give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows to employ techniques from representation theory and global analysis. Even for this simple stratum exhibiting all Siegel-Veech transforms requires novel configurations of saddle connections. We also show that the contiunuous spectrum of the foliated Laplacian is much larger than the space of Siegel-Veech transforms, as opposed to the case of the modular curve. This defect can be remedied by using instead a compound Laplacian involving relative period translations.

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arXiv2024

Formal deformations of modular forms and multiple L-values

We relate analytically defined deformations of modular curves and modular forms from the literature to motivic periods via cohomological descriptions of deformation theory. Leveraging cohomological vanishing results, we prove the existence and essential uniqueness of deformations, which we make constructive via established Lie algebraic arguments and a notion of formal logarithmic deformations. Further, we construct a canonical and a totally holomorphic canonical universal family of deformations of modular forms of all weights, which we obtain from the canonical cocycle associated with periods on the moduli space $\mathcal{M}_{1,1}$. Our uniqueness statement shows that non-critical multiple $\mathrm{L}$-values, which appear in our deformations but are a priori non-geometric, are genuinely linked to deformations. Our work thus suggests a new geometric perspective on them.

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arXiv2024

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

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 origin and which of them are mere artifacts of the connection with modular forms. Ranks and generalized ranks have been proposed as a tool to discern this question. We formalize this idea as "explainable Ramanujan-type congruences" with reference to Jacobi forms with singularities at torsion points, and then associated with them subspaces in a specific complex representation of the modular group. The resulting representation theoretic perspective allows us to prove that explainable Ramanujan-type congruences occur on square-classes $M \mathbb{Z} + u^2 β$ of arithmetic progressions.

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arXiv2023

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

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 ${\int_τ^{i\infty}f(z)(X-z)^{k-2}I_g(z;Y)\mathrm{d}z}$, where $I_g$ is the Eichler integral of $g$ and $X,Y$ are formal variables. We provide an explicit vector-valued modular form whose top components are given by $I_{f,g}$. We show that this vector-valued modular form gives rise to a scalar-valued iterated Eichler integral of depth two, denoted by $\mathcal{E}_{f,g}$, that can be seen as a higher-depth generalization of the scalar-valued Eichler integral $\mathcal{E}_f$ of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol-Popa. We show that $\mathcal{E}_{f,g}$ can be expressed in terms of sums of products of components of vector-valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form $Δ$. This allows for effective computation of $\mathcal{E}_{f,g}$.

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arXiv2023

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

Given an odd prime $\ell$ and finite set of odd primes $S_+$, we prove the existence of an imaginary quadratic field whose class number is indivisible by $\ell$ and which splits at every prime in $S_+$. Notably, we do not require that $p \not\equiv -1 \pmod{\ell}$ for any of the split primes $p$ that we impose. Our theorem is in the spirit of a result by Wiles, but we introduce a new method. It relies on a significant improvement of our earlier work on the classification of non-holomorphic Ramanujan-type congruences for Hurwitz class numbers.

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arXiv2023

On the Computation of General Vector-valued Modular Forms

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 $2$ associated with representations whose kernel is a congruence subgroup. It complements two available algorithms that are limited to inductions of Dirichlet characters and to Weil representations, thus covering further applications like Moonshine or Jacobi forms for congruence subgroups. We examine the calculation of invariants in specific representations via techniques from permutation groups, which greatly aids runtime performance. We explain how a generalization of cusp expansions of classical modular forms enters our implementation. After a heuristic consideration of time complexity, we relate the formulation of our algorithm to the two available ones, to highlight the compromises between level of generality and performance that each them makes.

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arXiv2023

Relations among Ramanujan-Type Congruences II

We show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra and provide several structure results for them. We discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on arithmetic progressions with cube-free periods. The scarcity of the latter was investigated recently. We explain that they provide congruences among algebraic parts of twisted central $\mathrm{L}$-values. We specialize our results to partition congruences, for which we investigate the proofs of partition congruences by Atkin and by Ono, and develop a heuristic that suggests that their approach by Hecke operators acting diagonally modulo $\ell$ on modular forms is optimal. In an extended example, we showcase how to employ our conclusions to benefit experimental work.

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arXiv2022

Scarcity of congruences for the partition function

The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+β)\equiv 0\pmod\ell$ for the primes $\ell=5, 7, 11$, and it is known that there are no others of this form. On the other hand, for every prime $\ell\geq 5$ there are infinitely many examples of congruences of the form $p(\ell Q^m n+β)\equiv 0\pmod\ell$ where $Q\geq 5$ is prime and $m\geq 3$. This leaves open the question of the existence of such congruences when $m=1$ or $m=2$ (no examples in these cases are known). We prove in a precise sense that such congruences, if they exist, are exceedingly scarce. Our methods involve a careful study of modular forms of half integral weight on the full modular group which are related to the partition function. Among many other tools, we use work of Radu which describes expansions of such modular forms along square classes at cusps of the modular curve $X(\ell Q)$, Galois representations and the arithmetic large sieve.

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