## Publications of PD Dr. Markus Kirschmer

Note that the preprints on this page might differ from the published versions.

- Chow groups of orders in number fields (with Jürgen Klüners).
- Appendix to Moduli of elliptic K3 surfaces: monodromy and Shimada root lattice strata (Klaus Hulek and Michael Lönne), Algebraic Geometry 9 (2022) no. 2, 159--213. DOI
- Spanning the isogeny class of a power of an ordinary elliptic curve over a finite field. Application to the number of rational points of curves of genus ≤ 4 (with Fabien Narbonne, Christophe Ritzenthaler, Damien Robert), Mathematics of Computation 91 (2022), 401--449. DOI
- Binary hermitian lattices over number fields (with Gabriele Nebe), accepted to Exp. Math. DOI
- Determinant groups of hermitian lattices over local fields, Archiv der Mathematik, 113 (2019), no. 4, 337--347. DOI
- Automorphisms of even unimodular lattices over number fields, Journal of Number Theory, 197 (2019), 121--134. DOI
- Quaternary quadratic lattices over number fields (with Gabriele Nebe), International Journal of Number Theory, 15 (2019), 309--325. DOI
- One class genera of exceptional groups over number fields, J. Théorie Nombres Bordeaux, 30 (2018), no. 3, 847--857. DOI
- One class genera of lattice chains over number fields (with Gabriele Nebe), In Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, eds. G. Böckle, W. Decker and G. Malle, Springer, 2018, 503--532. DOI
- The constructive membership problem for discrete two-generator subgroups of SL(2,R) (with Marion Rüther), J. Algebra 480 (2017), 519--548. DOI
- Ternary quadratic forms over number fields with small class number (with David Lorch), Journal of Number Theory 161 (2016), 343--361. DOI

A list of all quaternion orders of type or class number at most two is available here. - One-class genera of maximal integral quadratic forms. Journal of Number Theory 136C (2014), 375--393. DOI A list of these genera is available here.
- The constructive membership problem for discrete free subgroups of rank 2 of SL(2,R) (with Bettina Eick and Charles R. Leedham-Green), LMS J. Comput. Math. 17 (2014), no. 1, 345--359. DOI Magma code
- Computing with subgroups of the modular group (with Charles R. Leedham-Green), Glasgow Mathematical Journal 57 (2014), no. 1, 173--180. DOI Magma code
- Single-class genera of positive integral lattices (with David Lorch), LMS J. Comput. Math. 16 (2013), 172--186. DOI
- On an Analogue to the Lucas-Lehmer-Riesel test using Elliptic Curves (with Michael H. Mertens), INTEGERS 13 (2013), A15. DOI
- A normal form for definite quadratic forms over F_q[t]. Math. Comp. 81 (2012), 1619--1634. DOI
- Finite symplectic matrix groups. Exp. Math. 20 (2011) no. 2, 217--228. DOI
- Algorithmic enumeration of ideal classes for quaternion orders (with John Voight), SIAM J. Comput. (SICOMP) 39 (2010), no. 5, 1714--1747.

## Theses

- Definite quadratic and hermitian forms with small class number. Habilitationsschrift. RWTH Aachen University, 2016

The lattices of class number one and two are available here. - Finite symplectic matrix groups. Dissertation. RWTH Aachen University, 2009

The invariant forms given in the appendix are available here. - Konstruktive Idealtheorie in Quaternionenalgebren. Diplomarbeit. Universität Ulm, 2005

## Magma Code

Most of the algorithms of my papers are shipped with Magma. Below are some packages that have not been included in Magma (yet).

- Quadratic and hermitian lattices over number fields
- Two-generator subgroups of (P)SL(2,R)
- Algorithms for the modular group