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Stark's Conjectures Stark’s Conjectures by Samit Dasgupta a thesis presented to the Department of Mathematics in partial fulfillment of the requirements for the degree of Bachelor of Arts with Honors Harvard University Cambridge, Massachusetts April 5, 1999 Contents 1 Introduction 4 2 Basic notation 7 3 The non-abelian Stark conjecture 9 3.1 The Dedekind zeta-function . 9 3.2 Artin L-functions . 11 3.3 The Stark regulator . 14 3.4 Stark’s original example . 16 3.5 The non-abelian Stark conjecture . 18 3.6 Changing the isomorphism f ............................... 19 3.7 Independence of the set S ................................. 21 4 The cases r(χ) = 0 and r(χ) = 1 25 4.1 The case r(χ)=0 ..................................... 25 4.2 The case r(χ)=1 ..................................... 27 4.3 The rank one abelian Stark conjecture . 31 5 A numerical confirmation 36 5.1 The example . 36 0 5.2 Calculating ζS(0, γ)..................................... 37 5.2.1 The functional equation . 37 5.2.2 The auxiliary function ΛS ............................. 38 5.2.3 The functional equation for ΛS .......................... 38 5.2.4 The main proposition for the calculation . 40 5.2.5 How to calculate the relevant integrals . 41 5.2.6 Number of integrals and residues to calculate . 42 0 5.2.7 The results for ζS(0, γ)............................... 42 5.3 Finding the Stark unit ..................................√ 42 5.3.1 The polynomial of over k( −3) . 43 5.3.2 Bounds on the ti .................................. 43 5.4 Creating a finite list of possibilities . 44 5.5 The results . 45 6 Tate’s reformulation of the rank one abelian Stark conjecture 48 6.1 Definitions for Tate’s reformulation . 48 6.2 Arithmetic preliminaries for Tate’s reformulation . 50 6.3 Tate’s reformulation of Stark’s Conjecture . 53 6.4 Tate’s version of the proofs from section 4.3 . 54 7 Galois groups of exponent two 56 7.1 Basic results . 56 7.2 Relative quadratic extensions . 57 7.3 General K/k of exponent 2 . 60 1 8 The Brumer-Stark conjecture 63 8.1 Statement of the Brumer-Stark conjecture . 63 8.2 Galois groups of exponent 2 . 65 8.2.1 Relative quadratic extensions . 65 8.2.2 General K/k of exponent 2 . 67 9 Proof of Stark’s Conjecture for rational characters 69 9.1 A decomposition theorem for rational characters . 69 9.2 Definition of the invariant q(M, f) ............................ 70 9.3 Proof of Stark’s Conjecture for rational characters . 72 9.4 Facts from group cohomology . 74 9.5 Proof of Theorem 9.2.3 . 75 9.6 The category MG...................................... 79 A Background 83 A.1 The Dedekind zeta-function . 83 A.2 Facts from representation theory . 83 A.3 Global class field theory . 84 A.4 Statements from local class field theory . 88 A.5 Finite sets of primes S ................................... 88 A.6 Abelian L-functions . 89 A.7 Non-abelian Artin L-functions . 90 A.8 The functional equation for Artin L-functions . 91 A.9 Group cohomology . 92 A.10 The Herbrand Quotient . 93 A.11 Basic field theory . 95 A.12 Galois descent and Schur indices . 95 B Specifics of the numerical confirmation 99 B.1 Basic propositions about the Γ-function . 99 B.2 The integrals F (t) and G(t)................................ 99 2 Acknowledgments There are many people without whom I could never have completed this endeavor. First and foremost, I wish to thank my parents, who raised my brother Sandip and me in a loving environment in which we could flourish. Any successes I may achieve in my life are directly traceable to their support of my interests and devotion to my development. In addition to my parents, there are many other individuals who have supported me as I worked on this project and throughout my time at Harvard. In particular, I would certainly be lost without the love and friendship of Lisa Zorn. There are many individuals who have shaped my still growing view of mathematics. Primary among these are Darlyn Counihan, who introduced me to the beauty of mathematics, Professor Arnold Ross, who introduced me to number theory, and Eric Walstein and Professor Lawrence Washington, who supported my work in high school. I am also greatly indebted to Professors Benedict Gross and Noam Elkies, whose instruction has defined my current view of modern math- ematics. With regards to this thesis, Professor Brian Conrad deserves an indescribable amount of thanks for the incredible number of hours he spent advising me. From helping me learn the basics of class field theory, to giving me an intuition for Stark’s Conjectures, to editing the final paper, Professor Conrad went far beyond his duty as an advisor. I am greatly indebted to him for his patience and enthusiasm. I would also like to thank Professor Gross for suggesting this thesis topic to me. Finally, I am grateful to Adam Logan, Daniel Biss, and Jacob Lurie for their assistance in my work. I dedicate this thesis to my parents, who taught me by example the values of love, education, and love of education. 3 1 Introduction One fascinating and mysterious aspect of modern number theory is the interaction between the analytic and the algebraic points of view. The most fundamental example of this interaction is the relationship between the Dedekind zeta-function ζk(s) of a number field k and certain algebraic invariants of the field k. The definition of the complex-valued function ζk(s) is Y 1 ζ (s) = k 1 − Np−s p for Re(s) > 1, where the product runs over all nonzero prime ideals p of k. Through the process of analytic continuation, we obtain a meromorphic function ζk(s) defined on the complex plane. Interestingly, the analytic behavior of the function ζk allows one to prove purely algebraic facts about the number field k. For example, Dirichlet was able to exploit the fact that the meromorphic function ζQ(s) has a simple pole at s = 1 in order to prove that there are infinitely many primes in every arithmetic progression of the form {a, a+b, a+2b, ···} where a and b have no common factors (e.g., there are infinitely many primes of the form 12n + 5). To prove this theorem, Dirichlet also had to define a generalization of the function ζk(s), called a Dirichlet L-function. His arguments relied on the subtle analytic fact that certain such L-functions do not have a zero or a pole at s = 1. Even more, Dirichlet proved the celebrated “class number formula,” which gives an explicit formula for the residue of ζk(s) at s = 1: r1 r2 2 (2π) hkRk Res(ζk; 1) = p , (1) |Dk|ek where r1 and r2 are the number of real and complex places of k respectively, and hk, Rk, Dk, and ek are the class number, regulator, absolute discriminant, and number of roots of unity of k, respectively. This is a remarkable formula because the definition of ζk(s) uses only local information about k (i.e. the prime ideal structure) and analytic continuation, but the residue Res(ζk; 1) at 1 involves global invariants of k, such as hk and Rk. Using the functional equation for ζk(s), we can reformulate equation (1) as saying that the first non-zero term in the Taylor series of ζk(s) at s = 0 is given by h R − k k sr1+r2−1. (2) ek The term −hkRk/ek is the ratio between the (presumably) transcendental number Rk and the algebraic number −ek/hk. Furthermore, the regulator Rk is the determinant of (r1 + r2 − 1)- dimensional square matrix whose entries are logarithms of the archimedean valuations of units belonging to k. In the first half of this century, Artin, Hecke, Tate, and others established a general theory of L-functions and functional equations, partially extending some of the results obtained by Dirichlet. However, despite all of this work, there was no satisfactory generalization of the above Dirichlet class number formula in the context of arbitrary Artin L-functions. In the 1970’s, H. M. Stark attempted to find such a formula. He was guided by the following: Motivating Question. If K/k is a finite Galois extension with Galois group G and χ is the character of an irreducible, finite-dimensional, complex representation of G, is there a formula, analogous to (2), for the first non-zero coefficient in the Taylor expansion of LK/k(s, χ) at s = 0? More precisely, if r(χ) is the order of LK/k(s, χ) at s = 0, can this coefficient be expressed as the 4 product of an algebraic number and the determinant of an r(χ) × r(χ) matrix whose entries are linear forms in logarithms of archimedean valuations of units belonging to K? Answering this question would provide a generalization of the Dirichlet class number formula in two different ways. First, for any Galois extension K/k with Galois group G, the L-function LK/k(s, 1G) corresponding to the trivial character 1G is precisely the Dedekind zeta-function ζk(s). More importantly, however, there is the formula Y χ(1) ζK (s) = LK/k(s, χ) χ where the product runs over all irreducible characters χ of G. Therefore, providing a formula for the first non-zero Taylor coefficient of LK/k(s, χ) for all irreducible χ would strengthen the statement of the Dirichlet class number formula by showing how the leading term of ζK (s) factors into pieces, one for each irreducible representation of G.
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