On the Cohomology of GL2 and SL2 over Imaginary Quadratic Fields
(2024)
Journal Article
Gangl, H., Gunnells, P. E., Hanke, J., & Yasaki, D. (online). On the Cohomology of GL2 and SL2 over Imaginary Quadratic Fields. Experimental Mathematics, https://doi.org/10.1080/10586458.2024.2379797
Professor Herbert Gangl's Outputs (18)
On the Goncharov depth conjecture and polylogarithms of depth two (2024)
Journal Article
Charlton, S., Gangl, H., Radchenko, D., & Rudenko, D. (2024). On the Goncharov depth conjecture and polylogarithms of depth two. Selecta Mathematica (New Series), 30(2), Article 27. https://doi.org/10.1007/s00029-024-00918-6
Functional equations of polygonal type for multiple polylogarithms in weights 5, 6 and 7 (2023)
Journal Article
Charlton, S., Gangl, H., & Radchenko, D. (2023). Functional equations of polygonal type for multiple polylogarithms in weights 5, 6 and 7. Pure and Applied Mathematics Quarterly, 19(1), 85-93. https://doi.org/10.4310/pamq.2023.v19.n1.a5
On two conjectures of Sun concerning Apéry-like series (2023)
Journal Article
Charlton, S., Gangl, H., Lai, L., Xu, C., & Zhao, J. (2023). On two conjectures of Sun concerning Apéry-like series. Forum Mathematicum, 35(6), 1533-1547. https://doi.org/10.1515/forum-2022-0325We prove two conjectural identities of Z.-W. Sun concerning Apéry-like series. One of the series is alternating, whereas the other one is not. Our main strategy is to convert the series and the alternating series to log-sine-cosine and log-sinh-cosh... Read More about On two conjectures of Sun concerning Apéry-like series.
On functional equations for Nielsen polylogarithms (2021)
Journal Article
Charlton, S., Gangl, H., & Radchenko, D. (2021). On functional equations for Nielsen polylogarithms. Communications in Number Theory and Physics, 15(2), 363-454. https://doi.org/10.4310/cntp.2021.v15.n2.a4We derive new functional equations for Nielsen polylogarithms. We show that, when viewed moduloLi5 and products of lower weight functions, the weight 5 Nielsen polylogarithm S3,2 satisfies the dilogarithm five-term relation. We also give some functio... Read More about On functional equations for Nielsen polylogarithms.
Hyperbolic tessellations and generators of K₃ for imaginary quadratic fields (2021)
Journal Article
Burns, D., de Jeu, R., Gangl, H., Rahm, A. D., & Yasaki, D. (2021). Hyperbolic tessellations and generators of K₃ for imaginary quadratic fields. Forum of Mathematics, Sigma, 9, Article e40. https://doi.org/10.1017/fms.2021.9We develop methods for constructing explicit generators, modulo torsion, of the K₃ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic 3 -space or on direct calculations in suitable pre-Bloch gr... Read More about Hyperbolic tessellations and generators of K₃ for imaginary quadratic fields.
Homophonic quotients of linguistic free groups: German, Korean, and Turkish (2018)
Journal Article
Gangl, H., Karaali, G., & Lee, W. (2019). Homophonic quotients of linguistic free groups: German, Korean, and Turkish. Involve, 12(3), https://doi.org/10.2140/involve.2019.12.463
On the topological computation of K_4 of the Gaussian and Eisenstein integers (2018)
Journal Article
Gangl, H., Dutour Sikiriˇc, M., Gunnells, P., Hanke, J., Schuermann, A., & Yasaki, D. (2019). On the topological computation of K_4 of the Gaussian and Eisenstein integers. Journal of Homotopy and Related Structures, 14, 281-291. https://doi.org/10.1007/s40062-018-0212-8In this paper we use topological tools to investigate the structure of the algebraic K-groups K4(R) for R=Z[i] and R=Z[ρ] where i:=−1−−−√ and ρ:=(1+−3−−−√)/2. We exploit the close connection between homology groups of GLn(R) for n≤5 and those of rela... Read More about On the topological computation of K_4 of the Gaussian and Eisenstein integers.
On the cohomology of linear groups over imaginary quadratic fields (2016)
Journal Article
Dutour Sikirić, M., Gangl, H., Gunnells, P. E., Hanke, J., Schürmann, A., & Yasaki, D. (2016). On the cohomology of linear groups over imaginary quadratic fields. Journal of Pure and Applied Algebra, 220(7), 2564-2589. https://doi.org/10.1016/j.jpaa.2015.12.002Let Γ be the group GLN(OD), where OD is the ring of integers in the imaginary quadratic field with discriminant D<0. In this paper we investigate the cohomology of Γ for N=3,4 and for a selection of discriminants: D≥−24 when N=3, and D=−3,−4 when N=4...
Perfect forms, K-theory and the cohomology of modular groups. (2013)
Journal Article
Elbaz-Vincent, P., Gangl, H., & Soule, C. (2013). Perfect forms, K-theory and the cohomology of modular groups. Advances in Mathematics, 245, 587-624. https://doi.org/10.1016/j.aim.2013.06.014
Functional equations and ladders for polylogarithms (2013)
Journal Article
Gangl, H. (2013). Functional equations and ladders for polylogarithms. Communications in Number Theory and Physics, 7(3), 397-410. https://doi.org/10.4310/cntp.2013.v7.n3.a1We give a number of S3-symmetric functional equations for polylogarithms up to weight 7. This allows one to obtain the first proven ladder relations, à la Lewin, of weight 6 and 7.
On special elements in higher algebraic K-theory and the Lichtenbaum-Gross Conjecture (2012)
Journal Article
Burns, D., de Jeu, R., & Gangl, H. (2012). On special elements in higher algebraic K-theory and the Lichtenbaum-Gross Conjecture. Advances in Mathematics, 230(3), 1502-1529. https://doi.org/10.1016/j.aim.2012.03.014
From polygons and symbols to polylogarithmic expressions (2012)
Journal Article
Duhr, C., Gangl, H., & Rhodes, J. (2012). From polygons and symbols to polylogarithmic expressions. Journal of High Energy Physics, 2012(10), Article 075. https://doi.org/10.1007/jhep10%282012%29075
Goncharov's trilogarithm relation on pictures. (2007)
Journal Article
Gangl, H. (2007). Goncharov's trilogarithm relation on pictures. Journal of Number Theory, 124(1), 17-25. https://doi.org/10.1016/j.jnt.2006.05.021
Generators and Relations for K_2 O_F (2004)
Journal Article
Belabas, K., & Gangl, H. (2004). Generators and Relations for K_2 O_F. K-Theory, 31(3), 195 - 231. https://doi.org/10.1023/b%3Akthe.0000028979.91416.00Tate's algorithm for computing K_2 O_F for rings of integers in a number field has been adapted for the computer and gives explicit generators for the group and sharp bounds on their order---the latter, together with some structural results on the p-... Read More about Generators and Relations for K_2 O_F.
Functional equations for higher logarithms (2003)
Journal Article
Gangl, H. (2003). Functional equations for higher logarithms. Selecta Mathematica (New Series), 9(3), 361 - 377. https://doi.org/10.1007/s00029-003-0312-zFollowing earlier work by Abel and others, Kummer gave in 1840 functional equations for the polylogarithm function Li_m(z) up to m = 5, but no example for larger m was known until recently. We give the first genuine 2-variable functional equation for... Read More about Functional equations for higher logarithms.
Quelques calculs de la cohomologie de GL_N(Z) et de la K-theorie de Z (2002)
Journal Article
Elbaz-Vincent, P., Gangl, H., & Soulé, C. (2002). Quelques calculs de la cohomologie de GL_N(Z) et de la K-theorie de Z. Comptes Rendus Mathématique, 335(4), 321-324. https://doi.org/10.1016/s1631-073x%2802%2902481-0Voronoi cell decomposition is used to determine most of the structure of the homology of GL_N(Z) for N=5, 6. This determines K_5(Z) and gives that K_6(Z) is a 3-group.
On poly(ana)logs I (2002)
Journal Article
Elbaz-Vincent, P., & Gangl, H. (2002). On poly(ana)logs I. Compositio Mathematica, 130(2), 161-214. https://doi.org/10.1023/a%3A1013757217319We investigate a connection between the differential of polylogarithms (as considered by Cathelineau) and a finite variant of them. This allows to answer a question raised by Kontsevich concerning the construction of functional equations for the fini... Read More about On poly(ana)logs I.