Similarity Classes of 3 × 3 Matrices Over a Local Principal Ideal Ring

Abstract

Abstract. In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.

Split abelian surfaces over finite fields and reductions of genus-2 curves

“…Theorem 1.2. We have 1 (log q) 3 (log log q) 4 ≪ #A 2,split (F q ) q 5/2 ≪ (log q) 4 (log log q) 2 for all q.…”

“…This convention unfortunately conflicts with the standard use in analytic number theory of the letter p as a generic prime, for instance when writing Euler product representations of arithmetic functions. In such situations in this paper (see for example equation (4) in Section 4), we will instead use ℓ to denote a generic prime, and we explicitly allow the possibility that ℓ = p.…”

“…We call this the ordinary split nonisotypic case. 4 (log log q) 2 for all q, unconditionally, q 5/2 (log q) 2 (log log q) 4 for all q, under GRH.…”

On representations of general linear groups over principal ideal local rings of length two

“…Several authors, for instance, Frobenius [6], Rohrbach [23], Kloosterman [15,16], Tanaka [30], Nobs and Wolfart [21], Nobs [20], Kutzko [17], Nagornyȋ [18], and Stasinski [27] studied the representations of the groups SL 2 (O ) and GL 2 (O ). Nagornyȋ [19] obtained partial results regarding the representations of GL 3 (O ) and Onn [22] constructed all the irreducible representations of the groups G ( 1 , 2 ) . Recently, Avni, Klopsch, Onn, and Voll [2] have announced results about the representation theory of the groups SL 3 (Z p ).…”

“…The groups GL n (O 2 ), for distinct rings of integers O, are not necessarily isomorphic, even when the residue fields are isomorphic. For example, for a natural number n and a prime p, the group GL n (F p [[t]]/t 2 ) is a semi-direct product of the groups M n (F p ) and GL n (F p ), but on the other hand GL n (Z p /p 2 Z p ) is not unless n = 1 or (n, p) = (2, 2), (2, 3) or (3, 2) (Sah [24, p. 22], Ginosar [8]). …”