Date of Award
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Master of Arts (MA)
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This paper has two primary goals. The first is to prove that the so-called "twisted" L-functions of modular forms have analytic continuation to the entire complex plane. On the way to obtaining this result, we will undertake an examination of many of the elementary properties of modular forms, and derive many standard results about them. The second goal is to show that the completed L-function and completed twisted L-functions of a modular form can be expressed as adelic integrals. This is effectively a reexamination of the first goal from a cleaner perspective that will yield new results. Modular forms are an important, albeit somewhat esoteric, tool of current mathematical research. Their serious investigation dates back to initial work by Jacobi and Eisenstein in the latter half of the nineteenth century. Jacobi used modular forms in connection with the study of quadratic forms. In particular, he sought to find a solution to the classical problem of counting the number of ways to represent a given integer n as the sum of k squares. To do so, he manipulated the generating function for counting these representations into the now-famous theta function (or theta series)—a modular form. While many of the earliest results on modular forms, such as Jacobi's, were obtained as the need arose, mathematics eventually progressed to a point where a more general theory became both useful and necessary. The explicit, abstract study of modular forms has occupied a central role in number theory since the 1950's, due in large part to the work of Taniyama, Shimura, Jacquet, Langlands, and others. As a prominent example, the study of modular forms played a large role in Wiles's famous proof of Fermat's last theorem. L-functions are constructed from modular forms, and are of interest for the number-theoretic information they encode. Dirichlet was the first to work seriously with L-functions in the 1830's while completing his work on primes in arithmetic progression. Their investigation has proven quite fruitful in the meantime, although much of the work remains conjectural. This paper will define modular forms and L-functions, and then prove the analytic continuation and functional equations of L-functions of cusp forms. We will then introduce character functions, examine twisting L-functions by Dirichlet characters, and indicate one of the applications of doing so. From there we will summarize many basic results about p-adic numbers, adeles, and cusp forms. We will then construct an adelic twisting operator in order to realize both twisted and untwisted L-functions as adelic integrals. We will conclude by indicating one advantage of the adelic approach by expressing the L-function as an Euler product. We hope that even the novice reader will find the majority of this paper intelligible and enjoyable. We have listed many useful references where elementary results are omitted, but the majority of the earlier chapters are virtually self-contained, requiring only some complex analysis.
Jackson, John James, "Twisted L-Functions" (2011). Electronic Theses and Dissertations. 1661.