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 I_i,...,I_H invertible left O-ideals representing the classes. Fix an integer m. Let e_j denote the number of units in the right order of I_j and let B_ij denote the number of α in I_j⁻¹I_i with reduced norm N(α) equal to mN(I_i)/N(I_j). The Brandt matrix B(m) is the H×H matrix with entries B_ij. Up to conjugation by a permutation matrix it is independent of the choice of representatives I_j; 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 (ed.), Modular functions of one variable I, Lecture Notes in Mathematics, vol. 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. (eds.), 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, vol. 7, Providence, RI: American Mathematical Society, pp. 159–178, ISBN 0-8218-0880-X, Zbl 0914.05051