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1. Shulman, Andrew. Elementary divisors of reductions of generic Drinfeld modules.

Degree: 2012, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/8268

Let q be a power of an odd prime, A := F_{q}[T], and k := F_{q}(T). Let ψ be a Drinfeld A-
module over a fi nite extension K of k of rank r ≥ 2. Let \wp be a prime of K, of good reduction for ψ, \F_{\wp} the residue fi eld at \wp, and consider the reduced Drinfeld A-module ψ \otimes \wp over \F_{\wp}. The A-module action of ψ \otimes \wp on \F_{\wp}, denote \F_{\wp}, makes \F_{\wp} isomorphic, as an A-module, to A / d_{1,\wp}(ψ)A × … × A / d_{r,\wp}(ψ) A for uniquely determined monic polynomials
d_{1,\wp}(ψ),…,d_{r,\wp}(ψ), depending on ψ and \wp, such that d_{1,\wp}(ψ) | … | d_{r,\wp}(ψ). The elements d_{1,\wp}(ψ),…,d_{r,\wp}(ψ) are called the elementary divisors of ψ modulo \wp. In this thesis, we study the growth of the largest elementary divisor, d_{r,\wp}(ψ), as the prime \wp varies, in analogy
with a result by W. Duke pertaining to elliptic curves. We also consider the distribution of
the smallest elementary divisor, d_{1,\wp}(ψ), again as \wp varies, in analogy with work started by J.-P. Serre related to Lang and Trotter's elliptic curve formulation of Artin's primitive root conjecture. One of our main results is that for a density 1 of primes \wp of K, the infinity norm of d_{r,\wp}(ψ) is as large as possible. More precisely, we show that for any function f de fined on the primes of K with values in A such that f grows very slowly as the degree of the primes in K increases to infinity, then almost all primes \wp of K satisfy |d_{r,\wp}(ψ)|_{\ifnty} > \frac{|\wp|_{∞}}{|f(\wp)|_{∞}.
*Advisors/Committee Members: Cojocaru, Alina Carmen (advisor), Coskun, Izzet (committee member), Popa, Mihnea (committee member), Talkoo-Bighash, Ramin (committee member), Zieve, Michael (committee member).*

Subjects/Keywords: Drinfeld modules; elliptic curves; reduction; elementary divisors

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APA (6^{th} Edition):

Shulman, A. (2012). Elementary divisors of reductions of generic Drinfeld modules. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/8268

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Shulman, Andrew. “Elementary divisors of reductions of generic Drinfeld modules.” 2012. Thesis, University of Illinois – Chicago. Accessed July 12, 2020. http://hdl.handle.net/10027/8268.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Shulman, Andrew. “Elementary divisors of reductions of generic Drinfeld modules.” 2012. Web. 12 Jul 2020.

Vancouver:

Shulman A. Elementary divisors of reductions of generic Drinfeld modules. [Internet] [Thesis]. University of Illinois – Chicago; 2012. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/10027/8268.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Shulman A. Elementary divisors of reductions of generic Drinfeld modules. [Thesis]. University of Illinois – Chicago; 2012. Available from: http://hdl.handle.net/10027/8268

Not specified: Masters Thesis or Doctoral Dissertation