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

• All modular forms of weight 2 can be expressed by Eisenstein series.
  Joint with Jiacheng Xia.
  Res. Number Theory 6.32 (2020) and arXiv 1907.03616 (2019).

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

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

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

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

5. 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).

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

7. Products of Vector Valued Eisenstein Series.
   Forum Math. 29.1 (2017), pp. 157–186 and arXiv 1411.3877 (2014).

8. 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).

9. Computing Genus 1 Jacobi Forms.
   Math. Comp. 85 (2016) pp. 931-960 and arXiv 1212.1834 (2012).

10. Spans of special cycles of codimension less than 5.
    J. Reine Angew. Math. 718 (2016), pp. 39–57. and arXiv 1302.1451 (2013).

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

12. Kudla’s Modularity Conjecture and Formal Fourier-Jacobi Series.
    Joint with Jan Hendrik Bruinier.
    Forum Math. Pi 3 (2015), 30 pp. and arXiv 1409.4996 (2014).

13. Formal Fourier Jacobi Expansions and Special Cycles of Codimension Two.
    Compos. Math. 151.12 (2015) pp. 2187-2211 and arXiv 1302.0880 (2013).

14. The Structure of Siegel Modular Forms modulo p and U(p) Congruences.
    Joint with Olav Richter.
    Math. Res. Lett. 22.3 (2015), pp. 899-928 and arXiv 1312.559 (2013).

15. 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), pp. 6647-6670 and arXiv 1207.5600 (2012).

16. 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).

17. 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), pp. 3961-3967 and arXiv 1306.3919 (2013).

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

19. 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), pp. 115-134 and arXiv 1107.0573 (2011).

20. Computing Borcherds Products.
    Joint with Dominic Gehre and Judith Kreuzer.
    LMS J. Comput. Math. 16 (2013), pp. 200-215 and arXiv 1111.5574 (2011).

21. The functional equation of the twisted spinor L-function in genus 2.
    Joint with Aloys Krieg.
    Abh. Math. Semin. Univ. Hambg. 83.1 (2013), pp. 29-52 and arXiv 0907.2767 (2009).

22. Kohnen’s limit process for real-analytic Siegel modular forms.
    Joint with Kathrin Bringmann and Olav Richter.
    Advances in Math. 231 (2012), pp. 1100-1118 and arXiv 1105.5482 (2011).

23. How to implement a modular form.
    J. Symb. Comp. 46.12 (2011), pp. 1336-1354 and MPI Preprint 582312 (2010).

24. Hecke algebras related to the unimodular and modular groups over quadratic field extensions and quaternion algebras.
    Proc. Amer. Math. Soc. 139.4 (2011), pp. 1321-1331 and arXiv 0907.2766 (2009).

25. Efficiently generated spaces of classical Siegel modular forms and the Boecherer conjecture.
    J. Aust. Math. Soc. 89.3 (2010), pp. 393-405 and arXiv 1002.3883 (2010).

Accepted

• Modular forms of virtually real-arithmetic type I.
  Joint with Michael Mertens.
  Accepted for publication in Math. Res. Lett. and arXiv 1712.03004 (2017).

Preprints

• Scarcity of congruences for the partition function
  Joint with Scott Ahlgren and Olivia Beckwith.
  arXiv 2006.07645 (2020).

• Non-Holomorphic Ramanujan-type Congruences for Hurwitz Class Numbers
  Joint with Olivia Beckwith and Olav Richter.
  arXiv 2004.06886 (2020).

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

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

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

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).

Software

• HyCu.
  Github, https://github.com/martinra/hycu(2016).

• HLinear.
  Github, https://github.com/martinra/hlinear (2015).

• HFlint.
  Github, https://github.com/martinra/hflint (2015).

• algebraic-structures.
  Github, https://github.com/martinra/algebraic-structures (2015).

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

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

• Computation 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

• 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-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

• Caihua Luo.
  Started Mar. 2019.
  Chalmers University of Technology.

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

PhD Students

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

• Jiacheng Xia.
  Co-adviser is Dennis Eriksson.
  Started Sep. 2016.
  Chalmers University of Technology.

• 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.

Teaching

I teach a course on High Performance Computing (TMA881, MMA620) during the 1st term 2020/21.

Given Courses

• 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 Theoary. 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.

• Autormorphic 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

• Hurwitz class numbers.

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

• The maximal discrete extension of the Hermitian modular group.

  The script to compuate 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.