Quadratic forms and quaternion algebras: algorithms and arithmetic by John Michael Voight BS

138 Pages·2008·0.85 MB·English

Quadratic forms and quaternion algebras: algorithms and arithmetic
by
John Michael Voight
B.S. (Gonzaga University) 1999
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Mathematics
in the
Graduate Division
of the
University of California at Berkeley
Committee in charge:
Professor Hendrik Lenstra, Chair
Professor Bjorn Poonen
Professor Roger Purves
Spring 2005
The dissertation of John Michael Voight is approved:
Chair Date
Date
Date
University of California at Berkeley
Spring 2005
Quadratic forms and quaternion algebras: algorithms and arithmetic
Copyright 2005
by
John Michael Voight
1
Abstract
Quadratic forms and quaternion algebras: algorithms and arithmetic
by
John Michael Voight
Doctor of Philosophy in Mathematics
University of California at Berkeley
Professor Hendrik Lenstra, Chair
This thesis comes in two parts which can be read independently of one another.
In the first part, we prove a result concerning representation of primes by quadratic
forms. Jagy and Kaplansky exhibited a table of 68 pairs of positive definite binary quadratic
forms that represent the same odd primes and conjectured that their list is complete outside of
“trivial”pairs. Weconfirmtheirconjecture,andinfactfindallpairsofsuchformsthatrepresent
the same primes outside of a finite set.
In the second part, we investigate a constellation of results concerning algorithms for
quaternionalgebrasandtheirapplicationtoShimuracurves. LetAbeaquaternionalgebraover
a number field F. We discuss the computational complexity and, in many cases, give effective
algorithms to solve the following problems:
Determine if A=M (F), and if so, exhibit an isomorphism;
• ∼ 2
Find a maximal order A; and
• O ⊂
Determine if a right ideal I is principal, and if so, exhibit a generator ξ.
• ⊂O
Wethenpresentfastmethodsforcomputingthevalueofhypergeometricseriestolargeprecision.
Putting these together, we are able to compute special values of the map j :Γ H P1 for Γ a
\ → C
2
compacttrianglegroup,whichwemayrecognizeasputativealgebraicnumbersbyalsocomputing
their Galois conjugates. We apply this to construct the canonical polynomial Φ (x,y) for the
N
curve X (N) and to find nontorsion points on some elliptic curves over number fields.
0
i
To the memory of those who stood by me
and are now gone
ii
Contents
I Quadratic forms 1
1 Primes represented by quadratic forms 2
1.1 Fields of definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Ring class fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Characterizing equivalence via class groups . . . . . . . . . . . . . . . . . . . . . 20
2 Finding the list of forms 26
2.1 Forms with the same fundamental discriminant . . . . . . . . . . . . . . . . . . . 26
2.2 Bounding class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Computing class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Finding the pairs of quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 Tables 39
II Quaternion algebras 49
4 Quaternion algebras 50
4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Fundamental algorithms for quaternion algebras . . . . . . . . . . . . . . . . . . 58
4.3 Computing a maximal order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4 Class group computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 Shimura curves 84
5.1 Triangle groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2 Computing hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3 CM points and Shimura reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4 Examples and applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6 Tables 117
Bibliography 125
iii
Acknowledgements
I am very grateful to my thesis adviser, Hendrik Lenstra, for patiently guiding my intellectual
developmentin innumerablewaysandprovidingsupportwhile writing this dissertation. I would
like to thank Bjorn Poonen and Bernd Sturmfels for teaching me so much; Peter Stevenhagen,
Pete Clark, and Jared Weinstein for their helpful comments on portions of this thesis; Samit
Dasgupta and Noam Elkies for their incredible insights and encouragement; Ronald van Luijk
forhisconstantcompanionship;andWilliamSteinandtheMECCAHclusterforcomputertime.
SurvivingtheprocessofwritingthisdissertationwasonlymadepossiblebyPaulBerg. Finally,I
wouldlike to givemy sincerestgratitudeto AaronMinnis, DavidMichaels,KristinOlson,Missy
Longshore, Mary Weaver, my mother, and the rest of my family and friends for their love and
understanding.
1
Part I
Quadratic forms
2
Chapter 1
Primes represented by quadratic
forms
The forms x2 +9y2 and x2 +12y2 represent the same set of prime numbers, namely,
thoseprimespwhichcanbe writtenp=12n+1. Whatotherlikepairsofformsexist? Jagyand
Kaplansky[27] performeda computer searchfor pairs that representthe same set of odd primes
andfound certain“trivial”pairswhichoccur infinitely often andlisted othersporadic examples.
They conjecture that their list is complete.
In this part, using the tools of class field theory we give a provably complete list of
such pairs. By a form Q we mean an integral positive definite binary quadratic form Q =
ax2 +bxy +cy2 Z[x,y]; the discriminant of Q is b2 4ac = D = df2 < 0, where d is the
∈ −
discriminant of Q(√D), the fundamental discriminant, and f 1. We will often abbreviate
≥
Q= a,b,c .
h i
Throughout, we look for forms that represent the same primes outside of a finite set—
we say then that they represent almost the same primes. A form represents the same primes as