# PhD Dissertation Defense for Yatin Patel

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Speaker: Yatin Patel

Title: INTEGRAL REPRESENTATIONS OF SL2(Z/NZ)

Abstract: The aim of this work is to determine for which commutative rings integral representations of SL2(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

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## Contact

Department of Mathematics

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