Artin L-functions for abelian extensions of imaginary quadratic fields.

69 Pages·2005·0.37 MB·English

Artin L-functions for abelian extensions of
imaginary quadratic fields.
Thesis by
Jennifer Michelle Johnson
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California
2005
(Defended May 26, 2005)
ii
(cid:13)c 2005
Jennifer Michelle Johnson
All Rights Reserved
iii
In memory of my grandfather
Lyle Albert Bean
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Acknowledgements
It is not knowledge, but the act of learning, not possession but the act of getting there,
which grants the greatest enjoyment.
–Karl Friedrich Gauss
Thanks is due first to my advisor Matthias Flach who accepted the task of turning
a chemist into a mathematician. I am grateful for his guidance and his patience, and
I have taken to heart his approach that “the best way to see it is to be as abstract
as possible.” If I understand this subject at all it is in no small part thanks to many
conversations with Matthew Gealy. I am lucky to have him as a colleague and blessed
to call him my friend. I have also benefited greatly from my interactions with Dinakar
Ramakrishnan. He has been my “second advisor,” and I will sorely miss having the
giggles during seminar. The Caltech Math Department has been a wonderful place
to study. Many thanks are due to the administrative staff, past and present, and
especially to Stacey Croomes whose smile keeps the place bright.
Seraj Muhammed has often told me that I have raised two babies, my son and
my PhD. Such a feat would be impossible without the help and support of many
friends: Claudine Chen, Niki Zacharias, Liz Boon, Jennie Stephens, Janet Pavelich,
Irina Nenciu, Daniel Katz, and Christopher Lee. I especially want to thank Stephanie
Rogers for welcoming me to motherhood, Matthew Wright, Gary Lorden and David
Lutzer for encouraging me to take the plunge into mathematics, David Whitehouse,
Vladimir Baranovsky, and David Gabai for helping me learn to swim.
I am grateful for the financial support of the ARCS Foundation, the Dolores
Zohrab Liebmann Fund and the NSF. I also appreciate the support of the Dean of
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Graduate Studies, Mike Hoffmann, and his staff, Rosa Carrasco and Natalie Gilmore.
This would have been much harder without them.
I cannot thank my parents enough for their love, their example and their prayers.
My son, Kaleb, has kept my feet on the ground over the last seven years. Story
time has been my refuge. As a six-year old he has been wonderfully patient and
flexible while his mom has written her thesis. I could not have asked for more.
Finally, I thank my fianc´e, Fok-Yan Leung, for keeping the faith even when I had
none.
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Abstract
Let F be an abelian extension of an imaginary quadratic field K with Galois group G.
We form the Galois-equivariant L-function of the motive M = h0(SpecF)(j) where
the Tate twists j are negative integers. The leading term in Taylor expansion at
s = 0 decomposes over the group algebra Q[G] into a product of Artin L-functions
indexed by the characters of G. We construct a motivic element ξ via the Eisenstein
symbol and relate the L-value to periods of ξ via regulator maps. Working toward
the equivariant Tamagawa number conjecture, we prove that the L-value gives a basis
in ´etale cohomology which coincides with the basis given by the p-adic L-function
according to the main conjecture of Iwasawa theory.
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Contents
Acknowledgements iv
Abstract vi
1 Introduction 1
1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Modern directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Current Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Related Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 A word on motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 The Main Theorem 7
2.1 Statement of the conjecture . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 A conjecture for any smooth projective variety over Q . . . . . . . . . 13
3 Formulas for L-values 15
3.1 Analytic Computation . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 ‘-adic Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Iwasawa Main Conjecture 37
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Iwasawa theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 A Theorem of Bley . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
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5 Comparison of Integral Lattices 46
5.1 The image of the L-value in Det RΓ (Z[1],M ) . . . . . . . . . . . . 46
A‘ c S ‘
5.2 Descent from the 2-variable main conjecture . . . . . . . . . . . . . . 50
Bibliography 58
1
Chapter 1
Introduction
Number theorists are like lotus-eaters – having once tasted of this food they can never
give it up.
–Leopold Kronecker
1.1 History
Euler’s eighteenth century solution to the Basel problem on finding a closed form for
the infinite sum
∞
X 1
ζ(2) = = π2/6
n2
n=1
led to his interest in the zeta function and subsequent discovery of the Euler product
Y 1
ζ(s) =
1−p−s
p
andsowedtheseedforthestudyofspecialvaluesofL-functions. Thisseedgerminated
inthe mid-nineteenth century withtheworkofDirichlet, RiemannandDedekind who
recognizedrelationshipsbetweenzetafunctionsandthestructureofintegers. Inorder
to study the density of the prime numbers, Riemann considered ζ(s) as a function
on the entire complex plane with a simple pole at s = 1. While Dedekind, after
formulating the theory of ideals, was able to define for any number field F the zeta
2
function
X Y
ζ (s) = Na−s = (1−Np−s)−1.
F
a⊆OF p
Much of modern number theory is rooted in these discoveries, though for the present
investigation we shall focus on the following pair of results.
Theorem 1.1.1. Unit Theorem (Dirichlet)
Let S be a finite set of places of F containing the infinite ones, Y = L Z =
F,S v∈S
{P n ·v : n ∈ Z}, and O denote the ring of S-integers of F. The regulator is
v∈S v v F,S
the map
λ :O× → Y ⊗ R
F,S F,S F,S Z
X
u 7→ log|u| ·v.
v
v∈S
(cid:16) P (cid:17)
Setting X := ker Y → Z , the following properties hold:
F,S F,S
a) ker(λ ) is a finite group.
F,S
b) im(λ ) is a discrete lattice in X ⊗ R.
F,S F,S Z
c) λ induces an isomorphism O× ⊗ R ’ X ⊗ R. Therefore,
F,S F,S Z F,S Z
O× ’ W ×Z|S|−1
F,S F,S
where W is a finite group.
F,S
We formulate the class number formula in the case that S is comprised only of
the inifinite places.
Theorem 1.1.2. Analytic Class Number Formula (Dedekind)
2r1+r2πr2
lim(s−1)ζ (s) = R h
F p F F
s→1 w |d |
F F
where r (r ) is the number of real (complex) places of F, h is the class number, d
1 2 F F
is the discriminant, w is the order of the group of roots of unity and the regulator
F
R is the covolume of the lattice determined by the units.
F