Research

Some of my publications come along with software or data. See the download section.

Note that the versions of my paper on arXiv or HAL usually do not coincide with published revisions, if not indicated otherwise. For some journals, I strongly recommend to use the arXiv version.

Research Publications

1. The maximal discrete extension of the Hermitian modular group.
   Joint with Aloys Krieg and Annalena Wernz.
   Doc. Math. 26 (2021) and arXiv 1910.12466 (2019).

2. Modular forms of virtually real-arithmetic type I.
   Joint with Michael Mertens.
   Math. Res. Lett. 28.2 (2021) and arXiv 1712.03004 (2017).

3. Non-Holomorphic Ramanujan-type Congruences for Hurwitz Class Numbers.
   Joint with Olivia Beckwith and Olav Richter.
   Proc. Nat. Acad. Sci. U.S.A. 117.36 (2020) and arXiv 2004.06886 (2020).

4. All modular forms of weight 2 can be expressed by Eisenstein series.
   Joint with Jiacheng Xia.
   Res. Number Theory 6.32 (2020) and arXiv 1908.03616 (2019).

5. The skew-Maass lift I.
   Joint with Olav Richter.
   Res. Math. Sci. 6.2 (2019) and arXiv 1810.06810 (2018).

6. Hyper-Algebras of Vector-Values Modular Forms.
   SIGMA 14.108 (2018) and HAL-01289143 (2016).

7. Indecomposable Harish-Chandra modules for Jacobi groups.
   Contributions in Mathematical and Computational Sciences 10 (2017).

8. On Direct Integration for Mirror Curves of Genus Two and an Almost Meromorphic Siegel Modular Form.
   Joint with Albrecht Klemm, Maximilian Poretschkin, and Thorsten Schimannek.
   Commun. Number Theory Phys. 10.4 (2016) and arXiv 1502.00557 (2015).

9. Almost holomorphic Poincare series corresponding to products of harmonic Siegel-Maass forms.
   Joint with Kathrin Bringmann and Olav Richter.
   Res. Math. Sci. 3.30 (2016) and arXiv 1604.05105 (2016).

10. Harmonic Weak Siegel Maass Forms I.
    Int. Math. Res. Not. (2016) and arXiv 1510.03342 (2015).

11. Products of Vector Valued Eisenstein Series.
    Forum Math. 29.1 (2017) and arXiv 1411.3877 (2014).

12. Harmonic Maaß-Jacobi forms of degree 1 with higher rank indices.
    Joint with Charles Conley
    Int. J. Number Theory 12.07 (2016) and arXiv 1012.2897 (2010).

13. Computing Genus 1 Jacobi Forms.
    Math. Comp. 85 (2016) and arXiv 1212.1834 (2012).

14. Spans of special cycles of codimension less than 5.
    J. Reine Angew. Math. 718 (2016) and arXiv 1302.1451 (2013).

15. Sturm Bounds for Siegel Modular Forms.
    Joint with Olav Richter.
    Res. in Number Theory (2015) 1.5 and arXiv 1501.07733 (2015).

16. Kudla’s Modularity Conjecture and Formal Fourier-Jacobi Series.
    Joint with Jan Hendrik Bruinier.
    Forum Math. Pi 3 (2015) and arXiv 1409.4996 (2014).

17. Formal Fourier Jacobi Expansions and Special Cycles of Codimension Two.
    Compos. Math. 151.12 (2015) and arXiv 1302.0880 (2013).

18. The Structure of Siegel Modular Forms modulo p and U(p) Congruences.
    Joint with Olav Richter.
    Math. Res. Lett. 22.3 (2015) and arXiv 1312.559 (2013).

19. Harmonic Maass-Jacobi forms with singularities and a theta-like decomposition.
    Joint with Kathrin Bringmann and Olav Richter.
    Trans. Amer. Math. Soc. 367.9 (2015) and arXiv 1207.5600 (2012).

20. H-harmonic Maass-Jacobi Forms of Degree 1: The Analytic Theory of Some Indefinite Theta Series.
    Res. Math. Sci. (2015) 2.12 and arXiv 1207.5603 (2012).

21. Holomorphic projections and Ramanujan’s mock theta functions.
    Joint with Ozlem Imamoglu and Olav Richter.
    Proc. Nat. Acad. Sci. U.S.A. 111.11 (2014) and arXiv 1306.3919 (2013).

22. M24-twisted Product Expansions are Siegel Modular Forms.
    Commun. Number Theory Phys. 7.3 (2013) and arXiv 1208.3453 (2012).

23. Mock period functions, sesquiharmonic Maass forms, and non-critical values of L-functions.
    Joint with Kathrin Bringmann and Nikolaus Diamantis.
    Advances in Math. 233 (2013) and arXiv 1107.0573 (2011).

24. Computing Borcherds Products.
    Joint with Dominic Gehre and Judith Kreuzer.
    LMS J. Comput. Math. 16 (2013) and arXiv 1111.5574 (2011).

25. The functional equation of the twisted spinor L-function in genus 2.
    Joint with Aloys Krieg.
    Abh. Math. Semin. Univ. Hambg. 83.1 (2013) and arXiv 0907.2767 (2009).

26. Kohnen’s limit process for real-analytic Siegel modular forms.
    Joint with Kathrin Bringmann and Olav Richter.
    Advances in Math. 231 (2012) and arXiv 1105.5482 (2011).

27. How to implement a modular form.
    J. Symb. Comp. 46.12 (2011) and MPI Preprint 582312 (2010).

28. Hecke algebras related to the unimodular and modular groups over quadratic field extensions and quaternion algebras.
    Proc. Amer. Math. Soc. 139.4 (2011) and arXiv 0907.2766 (2009).

29. Efficiently generated spaces of classical Siegel modular forms and the Boecherer conjecture.
    J. Aust. Math. Soc. 89.3 (2010) and arXiv 1002.3883 (2010).

Accepted

• Congruences of Hurwitz class numbers on square classes.
  Joint with Olivia Beckwith and Olav Richter.
  Accepted for Advances in Math. and arXiv 2203.11273 (2022).

• Relations among Ramanujan-Type Congruences I.
  Accepted for Advances in Math. and arXiv 2010.06272 (2020).

• Scarcity of congruences for the partition function.
  Joint with Scott Ahlgren and Olivia Beckwith.
  Accepted for publication in Am. J. Math. and arXiv 2006.07645 (2020).

Preprints

• Scalar-valued depth two Eichler-Shimura Integrals of Cusp Forms.
  Joint with Tobias Magnusson.
  arXiv 2209.00488 (2022).

• A classification of polyharmonic Maaß forms via quiver representations.
  Joint with Claudia Alfes-Neumann and Igor Burban.
  arXiv 2207.02278 (2022).

• The Bernstein-Gelfand Tensor Product Functor and the Weight-2 Eisenstein Series.
  arXiv 2205.08226 (2022).

• Eichler integrals and generalized second order Eisenstein series.
  Joint with Albin Ahlbäck and Tobias Magnusson.
  arXiv 2203.15462 (2022).

• On the Computation of General Vector-valued Modular Forms.
  Joint with Tobias Magnusson.
  arXiv 2202.06676 (2022).

• A classification of harmonic weak Maaß forms of half-integral weight.
  Joint with Claudia Alfes-Neumann.
  arXiv 2202.00744 (2022).

• Relations among Ramanujan-Type Congruences II.
  arXiv 2105.13170 (2021).

• Hyperelliptic Curves over Small Finite Fields and GPU Accelerators.
  arXiv 1805.11991 (2018).

Books

• Modular Forms are Everywhere: Celebration of Don Zagier’s 65th Birthday.
  Co-Edited with Ken Ono, Kathrin Bringmann, Maxim Kontsevich, Pieter Moree.
  Research in the Mathematical Sciences, Topical Collection.

• Siegel Modular Forms and Jacobi Forms.
  In preparation. Publishing agreement with Springer.

Various Publications

• K. Bringmann, A. Folsom, K. Ono, L. Rolen: “Harmonic Maass Forms and Mock Modular Forms: Theory and Applications”.
  Jahresber. Dtsch. Math. Ver. (2019).

• From Solutions of polynomial equations to the Langlands Program.
  Svenska Matematikersamfundet Bulletinen (Oct. 2018).

• Harmonic weak Siegel Maass forms.
  Oberwolfach Reports 2016.23.

• Analytic properties of some indefinite theta series.
  Oberwolfach Reports 2016.3.

• Appendix to Pinched hypersurfaces contract to round points by M. Franzen.
  arXiv 1502.07908 (2015).

• Symmetric Formal Fourier Jacobi Series and Kudla’s Conjecture.
  Oberwolfach Reports 2014.2.

• Dual weights in the theory of harmonic Siegel modular forms.
  Ph.D. Thesis, University of Bonn (2012).
  Local copy.

• Konstanten der Arithmetik: Perioden und ihre Relationen.
  Joint with Sven Raum.
  Annual report of the Max Planck Society, MPI for Mathematics (2012).

• The functional equation for certain twisted Rankin convolutions and an application to the spinor L-function.
  Diplomarbeit (former days’ master thesis), RWTH Aaachen University (2009).

Software

• ModularForms.jl.
  Joint with Albin Ahlbäck and Tobias Magnusson.
  GitLab (2022).

• Contributions to the computer algebraic system Oscar (since 2018).
  GitHub Merge Requests: AbstractAlgebra.
  GitHub Merge Requests: Nemo.
  GitHub Merge Requests: Hecke.

• QuickTest.jl.
  GitHub (2016).

• HyCu.
  GitHub (2016).

• Contributions to Sage (2011-2015).
  Sage Trac Server #19668, #14482, #13531, #11624, #11139, #10987, #10837, #8094, #7474, #6671, #6670, #6669, #5731, #4578.

• HLinear.
  GitHub (2015).

• HFlint.
  GitHub (2015).

• algebraic-structures.
  GitHub (2015).

• Genus 1 Jacobi Forms.
  Sage Trac Server #16448 (2014).

• Jacobi Forms.
  Sage Extension in Purple Sage (2012).

• Modular Forms Framework.
  Sage Extension in Purple Sage (2011).

• Computation of Siegel Modular Forms of Genus 2.
  Joint with Alex Ghitza, Nathan Ryan, Nils-Peter Skoruppa, and Gonzalo Tornaría.
  Sage Extension in Purple Sage (2011).

Preprints no longer intended for publication

• Congruences on square-classes for the partition function.
  arXiv 1911.04925 (2019).

  This is completely superseeded, and corrected, by my manuscripts on Relations among Ramanujan-type Congruences I && II.

• Indefinite Theta Series on Tetrahedral Cones.
  arXiv 1608.08874 (2016).

  This manuscript was intended as a quick response to work of Alexandrov-Banerjee-Manschot-Pioline. It turned our more hasty than quick (slightly stronger conditions on the rationality of the cone are required, cf. Roth’s theorem), but also taught me that for the time being I do not feel comfortable contributing to this field.

• HLinear: Exact Dense Linear Algebra in Haskell.
  Joint with Alexandru Ghitza.
  arXiv 1605.02532 (2016).

  The manuscript describes a preliminary implementation of echelon normal forms in Haskell. The general critique has been that it is only superior to state-of-the-art C code for matrices with less than 20 rows. While I am convinced this can be mitigated, I can no longer justify the effort, after moving my modular forms computation from Haskell to Julia/Nemo.

• Explicit computations of Siegel modular forms of degree two.
  Joint with Nathan Ryan and Gonzalo Tornaría.
  arXiv 1205.6255 (2012).

  This manuscript was intended as a survey of ad-hoc constructions for Fourier expansions of Siegel modular forms in conjunction with an explanation of multiplication in rings of invariant Fourier expansions. While the latter was already achieved in my paper “How to implement a modular form”, the former suffered from irreconcilably different taste of the authors. This was also echoed by various referees.

• Elementary divisor theory for the modular group over quadratic field extensions and quaternion algebras.
  arXiv 0907.2762 (2009).

  This manuscript was intended as a companion paper to “Hecke algebras related to the unimodular and modular groups over quadratic field extensions and quaternion algebras”, summarizing some of the normal form theory needed in as explicit form as possible, following Siegel’s style. The theory was already developed in less explicit form by Shimura. This was repeatedly brought up against this manuscript. In this light, it seemed reasonable to no longer pursue its publication.

Advising

PostDocs

• Adam Keilthy.
  Since Sep. 2022.
  Chalmers University of Technology.

• Caihua Luo.
  Mar. 2019 - Jan. 2021.
  Chalmers University of Technology.
  Now postdoc at Bar-Ilan University.

• Sebastián Herrero.
  Dec. 2016 - Dec. 2018.
  Chalmers University of Technology.
  Now professor at Pontificia Universidad Católica de Valparaíso.

PhD Students

• Jan Feldmann.
  Co-adviser not yet appointed.
  Since Jun. 2022.
  Chalmers University of Technology.

• Tobias Magnusson.
  Co-adviser is Anders Södergren.
  Since Aug. 2018.
  Chalmers University of Technology.

• Jiacheng Xia.
  Co-adviser was Dennis Eriksson.
  Graduated July 2021.
  Some cases of Kudla’s Modularity Conjecture for unitary Shimura varieties.
  Chalmers University of Technology.
  Now postdoc at Laval University.

• Manh Hung Tran.
  Co-adviser. Principal adviser was Dennis Eriksson. Previous adviser was Per Sahlberger.
  The density of rational points and invariants of genus one curves.
  Graduated Jan. 2020.
  Chalmers University of Technology.

• Andreas Freh.
  Co-advised with Aloys Krieg.
  Dimension Formulas for Spaces of Hermitian Modular Forms.
  Graduated 2017.
  RWTH Aachen University.

Master Students

• Alfred Arvidsson and Jakob Hendén. Examiner.
  Real-time route prediction of emergency vehicles.
  Graduated 2021.
  Chalmers University of Technology.

• Tomas Forssmark. Examiner.
  Lowering Levels of Modular Forms.
  Graduated 2021.
  Gothenburg University.

• Erik von Brömssen. Examiner.
  Computing persistent homology in parallel with a functional language.
  Graduated 2021.
  Chalmers University of Technology.

• David Elinder. Examiner.
  Investigating abelian categories in univalent type theory.
  Graduated 2021.
  Chalmers University of Technology.

• Martin Gulliksson. Examiner.
  A genetic programming approach to finding discrepancies in Android log files.
  Graduated 2021.
  Chalmers University of Technology.

• Merve Göksu. Examiner.
  Die maximal diskrete Erweiterung der Hermiteschen Modulgruppe über Q(√−1) und Q(√−3).
  Graduated 2020.
  RWTH Aachen University.

• Felix Rydell. Examiner.
  Three Perspectives of Schiemann’s Theorem.
  Graduated 2020.
  Chalmers University of Technology.

• Dundar Göc.
  Vectorized FFT for mulitplication of polynomials modulo a prime in FLINT.
  Graduated 2019.
  Chalmers University of Technology.

• Erik Nilsson. Examiner.
  Can one hear the shape of of a flat torus?
  Graduated 2019.
  Chalmers University of Technology.

• Albert Zeyer.
  Dimension Formulas for Spaces of Hermitian Modular Forms.
  Graduated 2013.
  RWTH Aachen University.

Teaching

I teach a course on High Performance Computing (TMA881, MMA620) during the 1st term 2021/22.

Given Courses

• Algebraic Number Theory. Lecture.
  Chalmers University of Technology, 2021/22, 3th Term.

• High Performance Computing. Lecture.
  Chalmers University of Technology, 2021/22, 1th Term.

• High Performance Computing. Lecture.
  Chalmers University of Technology, 2020/21, 1th Term.

• Algebraic Number Theory. Lecture.
  Chalmers University of Technology, 2019/20, 3th Term.

• High Performance Computing. Lecture.
  Chalmers University of Technology, 2019/20, 1th Term.

• High Performance Computing. Lecture.
  Chalmers University of Technology, 2018/19, 1th Term.

• High Performance Computing. Lecture.
  Chalmers University of Technology, 2016/17, 4th Term.

• Algebraic Number Theory. Lecture.
  Chalmers University of Technology, 2016/17, 2th Term.

• High Performance Computing. Lecture.
  Chalmers University of Technology, 2015/16, 4th Term.

• Modular Forms and Generating Series. Lecture.
  Chalmers University of Technology, 2015/16, 2nd/3rd Term.

• Jacobi Forms: Applications and Computations. Lecture.
  ETH Zurich, 2013, Winter Term.

• Sage and Quaternion Modular Forms. Project group.
  RWTH Aachen University, 2011, Summer Term.

• Sage and Quaternion Modular Forms. Seminar.
  RWTH Aachen University, 2010, Winter Term.

• Automorphic Forms for GL(2). Seminar.
  RWTH Aachen University, 2010, Summer Term.

• Elliptic Curves and Applications to Crytography. Seminar.
  RWTH Aachen University, 2009, Winter Term.

• Algebraic Number Theory II. Lecture as teaching assistent.
  RWTH Aachen University, 2009, Summer Term.

• Algebraic Number Theory II. Lecture as regular substitute lecturer.
  RWTH Aachen University, 2009, Summer Term.

• Algebraic Number Theory I. Lecture as teaching assistent.
  RWTH Aachen University, 2008, Winter Term.

Downloads

• Polyharmonic Maaßforms.

  An implementation to compute polyharmonic Maaß forms from spectral families used in the paper “A classification of polyharmonic Maaß forms via quiver representations” with Claudia Alfes-Neumann and Igor Burban.

• Hurwitz class numbers.

  An implementation to compute Hurwitz class numbers modulo given 64-integers, including a set of scripts to analyse the results.

• Scarcity of congruences for the partition function.

  The scripts used in the paper “Scarcity of congruences for the partition function” with Scott Ahlgren and Olivia Beckwith.

• The maximal discrete extension of the Hermitian modular group.

  The script to compute generators of the maximal discrete extension and the associated fields of definition.

• Modular forms of real-arithmetic types.

  The scripts to compute Hecke operators and to compute saturations.

• HLinear.

  The build script and the stack-yaml file.

  • Install the latest version of stack.
  • Install GHC or configure stack to use an existing installation.
  • Move the above two files into a new directory, and run the install script.
  • cd haskell/hlinear.
  • stack ghci hlinear to obtain a ghci instance with hlinear loaded.

• On Direct Integration for Mirror Curves of Genus Two and an Almost Meromorphic Siegel Modular Form.

  The Sage scripts used to compute Fourier expansions, and the resulting data.

• Computing Genus 1 Jacobi Forms.

  The expansions of vector valued modular forms that I computed.

• M24 - Twisted Product Expansions are Siegel Modular Forms.

  The Sage code used to compute the results, and the resuls in machine readable form.

• The modular forms framework.

  A recent version of the modular forms framework, integrated into PSage, is available on GitHub. It does not, however, include all available implementations. PSage by now is dead. The modular forms framework does not compile with recent Sage anymore.

• Computing Borcherds Products.

  Example code and results that was used in this publication. Use my hermitian branch of Purple Sage.

• Kohnen’s limit process for real-analytic Siegel modular forms.

  Sage files containing the computer assisted proofs.

  • Remark in Section 2.1
  • Proof of Theorem 3, Equation (15)
  • Proof of Theorem 3, T = 0
  • Proof of Theorem 3, T has rank 2

• Efficiently generated spaces of modular forms and the Böcherer conjecture.

  Data coming with this paper.

Contact

My office is

• Chalmers University of Technology
  Department of Mathematical Sciences
  H4028
  Chalmers Tvärgata 3
  SE-412 96 Göteborg

My post address is

• Chalmers tekniska högskola
  Institutionen för Matematiska vetenskaper
  Martin Raum
  SE-412 96 Göteborg, Sweden

You can send me mail at raum@chalmers.se.