#### Access Type

Open Access Dissertation

#### Date of Award

January 2022

#### Degree Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics

#### First Advisor

Luca Candelori

#### Second Advisor

Andrew Salch

#### Abstract

The aim of this work is to determine for which commutative rings integral representations of SL_2(Z/nZ) exist and to explicitly compute them. We start with R = Z/pZ and then consider Z=p^\lambda Z. A new approach will be used to do this based on the Weil representation. We then consider general finite rings Z/nZ by extending methods described in [26]. We make extensive use of group theory, linear representations of finite groups, ring theory, algebraic geometry, and number theory. From number theory we will employ results regarding modular forms, Legendre symbols, Hilbert symbols, and quadratic forms. We consider the works of Andre Weil[38], Alexandre Nobs[23][24] and Udo Riese[26]. We explicitly compute the irreducible representations for several odd primes using Nobs and Wolfart's methods. Then we will explore Riese's[26] construction of the integral representations of SL2(A_\lambda) and explicitly compute them as the paper only proves the existence. We will use integrality results of the Weil representations by Luca Candelori, Shaul Zemel, and Yilong Wang (to appear). Then we will extend Reise's results to construct integral representations for rings that are not of the form A_\lambda.

#### Recommended Citation

Patel, Yatin Dinesh, "Integral Representations Of Sl_2(z/nz)" (2022). *Wayne State University Dissertations*. 3647.

https://digitalcommons.wayne.edu/oa_dissertations/3647