Software Images icon An illustration of two photographs. Images Donate icon An illustration of a heart shape Donate Ellipses icon An illustration of text ellipses. EMBED for wordpress. Want more? Advanced embedding details, examples, and help! The first systematic theory of generalized functions also known as distributions was created in the early s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics.
The six-volume collection, Generalized Functions, written by I. Average Value Theorems in Function Fields. Back Matter Pages About this book Introduction Elementary number theory is concerned with arithmetic properties of the ring of integers. Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting, for example, analogues of the theorems of Fermat and Euler, Wilsons theorem, quadratic and higher reciprocity, the prime number theorem, and Dirichlets theorem on primes in an arithmetic progression.
After presenting the required foundational material on function fields, the later chapters explore the analogy between global function fields and algebraic number fields. A variety of topics are presented, including: the ABC-conjecture, Artins conjecture on primitive roots, the Brumer-Stark conjecture, Drinfeld modules, class number formulae, and average value theorems.
The first few chapters of this book are accessible to advanced undergraduates. The later chapters are designed for graduate students and professionals in mathematics and related fields who want to learn more about the very fruitful relationship between number theory in algebraic number fields and algebraic function fields. In this book many paths are set forth for future learning and exploration.
Bray Teaching Award in We will then discuss algebraic function fields, valuations and primes, the dictionary between function fields and algebraic curves, differentials and divisors, Riemann-Roch, zeta functions, Galois theory of function fields, class groups, elliptic curves and elliptic function fields, cyclotomic function fields, L -functions, S -units, and other related topics as interest dictates. We will finish up with a treatment of the Weil conjectures for curves, culminating in a proof of the abc conjecture for function fields along with a proof of the Riemann hypothesis.
Textbook: I will generally follow Rosen's "Number Theory in Function Fields", but may supplement the discussion with various papers and material from other books, depending on student interest. If you have difficulty obtaining a copy of the textbook, please contact the instructor.
Background: There are no formal prerequisites, but students should have comfort with algebra and number theory at the level of Math or or I will freely refer to some results from elementary number theory, commutative algebra, algebraic geometry, Galois theory, and complex analysis, but the goal is to make the course as self-contained as possible.
Depending on student and instructor interest, there may also be student presentations on course-related topics during the semester. Homework 1.
0コメント