Brandt matrix
In mathematics, Brandt matrices are matrices, introduced by Brandt (1943), that are related to the number of ideals of given norm in an ideal class of a definite quaternion algebra over the rationals, and that give a representation of the Hecke algebra. Eichler (1955) calculated the traces of the Brandt matrices.
Let O be an order in a quaternion algebra with class number H, and Ii,...,IH left O-ideals representing the classes. Fix an integer m. Let ej denote the number of units in the right order of Ij and let Bij denote the number of α in Ij−1Ii with reduced norm N(α) equal to mN(Ii)/N(Ij). The Brandt matrix B(m) is the H×H matrix with entries Bij. Up to conjugation by a permutation matrix it is independent of the choice of representatives Ij; it is dependent only on the level of the order O.
References
- Brandt, Heinrich (1943), "Zur Zahlentheorie der Quaternionen", Jahresbericht der Deutschen Mathematiker-Vereinigung, 53: 23–57, ISSN 0012-0456, MR 0017775, Zbl 0028.10802
- Eichler, Martin (1955), "Zur Zahlentheorie der Quaternionen-Algebren", Journal für die reine und angewandte Mathematik (in German), 195: 127–151, doi:10.1515/crll.1955.195.127, ISSN 0075-4102, MR 0080767, Zbl 0068.03303
- Eichler, Martin (1973), "The basis problem for modular forms and the traces of the Hecke operators", in Kuyk, Willem, Modular functions of one variable I, Lecture Notes in Mathematics, 320, Springer-Verlag, pp. 75–151, ISBN 3-540-06219-X, Zbl 0258.10013
- Pizer, Arnold K. (1998), "Ramanujan graphs", in Buell, D.A.; Teitelbaum, J.T., Computational perspectives on number theory. Proceedings of a conference in honor of A. O. L. Atkin, Chicago, IL, USA, September 1995, AMS/IP Studies in Advanced Mathematics, 7, Providence, RI: American Mathematical Society, pp. 159–178, ISBN 0-8218-0880-X, Zbl 0914.05051