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University of California Santa Cruz Stark's Conjectures UNIVERSITY OF CALIFORNIA SANTA CRUZ STARK'S CONJECTURES FOR P -ADIC L-FUNCTIONS A dissertation submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS by Joseph W. Ferrara June 2018 The Dissertation of Joseph W. Ferrara is approved: Professor Samit Dasgupta, Chair Professor Robert Boltje Professor Martin H. Weissman Tyrus Miller Vice Provost and Dean of Graduate Studies Copyright c by Joseph W. Ferrara 2018 Table of Contents Abstract v Dedication vi Acknowledgments vii Intoduction 1 1 The rank one abelian Stark conjecture for quadratic fields 10 1.1 The archimedean conjecture . 10 1.2 The imaginary quadratic case . 17 1.3 Ideas for a p-adic L-function in this case . 21 2 Background for definition of the p-adic L-function 25 2.1 Conventions for modular forms . 25 2.2 Weight space . 30 2.3 Modular symbols . 32 2.4 The p-adic L-function of an ordinary weight k ≥ 2 modular form . 33 2.5 Overconvergent modular symbols . 36 2.6 Families of overconvergent modular symbols . 42 2.7 Two-variable p-adic L-function . 49 3 Definition of a p-adic L-function and p-adic Stark Conjectures 58 3.1 Definition of one and two-variable p-adic L-functions . 59 3.2 p-adic Conjectures . 68 4 Proof of the conjecture when F is imaginary quadratic and p splits in F 83 4.1 Katz's p-adic L-function . 83 4.2 Definition and properties of the periods (Ωp; Ω1)............. 88 4.3 The CM Hida family . 92 iii 4.4 Two-variable p-adic L-function of the CM family . 100 4.5 Two-variable specialization of Lp;Katz ................... 102 4.6 Choice of periods and comparison . 108 4.7 Proof of the conjecture in this case . 112 5 Conjecture 3.2.7 when F is real quadratic, χ is a mixed signature character, and Ind χ = Ind for a character of an imaginary quadratic field in which p splits 117 5.1 Notation . 118 5.2 Archimedean Stark conjecture . 120 5.3 p-adic Stark conjecture . 126 6 Numerical evidence 132 p p p 6.1 F = Q( 17), K = Q( 4 + 17), p =5 .................. 132 7 Appendix 141 7.1 Rigid analytic geometry . 141 7.2 Topological rings, modules, and completed tensor product . 154 7.3 Hecke characters . 154 7.4 Hecke L-functions . 162 Bibliography 165 iv Abstract Stark's Conjectures for p-adic L-functions by Joseph W. Ferrara We give a new definition of a p-adic L-function for a mixed signature character of a real quadratic field and for a nontrivial ray class character of an imaginary quadratic field. We then state a p-adic Stark conjecture for this p-adic L-function. We prove our conjecture in the case when p is split in the imaginary quadratic field by relating our construction to Katz's p-adic L-function. We also prove our conjecture in the real quadratic setting for one special case and give numerical evidence in one specific example. v Dedicated to my parents. vi Acknowledgments I must first thank my advisor Professor Samit Dasgupta for his unending guid- ance, patience, and support. Without him this thesis would not be possible. Professor Dasgupta explained to me the fundamental ideas that led to the work done in this thesis, and he has had patience with me as I've gone through graduate school trying to figure out how to do research in number theory. Not only that, he has also been extremely supportive in all that I have tried accomplish during graduate school. The example he sets of perseverance and a positive attitude in doing research are two of many essential qualities he's taught me about what it takes to be a successful researcher. Making it through graduate school without peers to work with and learn from would be next to impossible. I thank Shawn Tsosie for his support in learning number theory together. I thank Ander Steele for his never ending encouragement that I needed to finish this thesis. I also thank Ryan Drury for his patience and knowledge in helping me with coding issues. I have to thank Professors Robert Boltje, Junecue Suh, and Marty Weisssman. Their patience in teaching me, mathematical guidance, and professional wisdom has helped me tremendously throughout graduate school. Without my family I would not be anywhere. I thank my parents for their unending love and support. I thank my mom for being an example of compassion. My dad for showing me how to follow my passion. My brother for inspiration and for continually reminding me why I'm doing this in the first place. vii I thank my friends and mentors Patrick Allmann, Kirsten Carraway, Steven Flynn, Joe Frantz, Bob Hingtgen, Connor Jackman, Yukichi Kawada, Julie Kimball, Lluvia Rios, and Sisi Song for their support throughout the past five years, and with whom I absolutely would not have made it through graduate school without. Finally I greatly thank Ralph Greenberg for sharing his work with Nike Vatsal ([15]) with me, and for reading and helping with an earlier version of this thesis. His input has been essential in finishing this project. I also thank Rob Pollack and Rob Harron for letting me use their SAGE code for computing overconvergent modular symbols. The use of their code was and is an essential part of this project. viii Introduction Let F be a number field and let × χ : GF −! C be a continuous one dimensional representation of the absolute Galois group of F . Let K be the fixed field of the kernel of χ. Let p be an odd prime number and fix embeddings (for the rest of this thesis) ι1 : Q ,! C and ιp : Q ,! Cp so we may view any algebraic number as a complex or p-adic number. Via the Artin map, to χ we may associate the complex Hecke L-function, L(χ, s), defined by the series X χ(a) L(χ, s) = N as a⊂OF for Re(s) > 1. The function L(χ, s) has a meromorphic continuation to the whole complex plane. In the late 1970s, in a series of papers, Stark made precise conjectures concerning the leading term of the Taylor series expansion at s = 0 of L(χ, s) ([29], [30], [31]). Stark's conjectures relate the leading term of L(χ, s) at s = 0 to the determinant of a matrix of linear combinations of logarithms of units in K. His conjectures refine 1 Dirichlet's class number formula. Stark proved his conjectures when the field F is equal to Q or to an imaginary quadratic field. In general the conjectures are open. Around the same time that Stark made his conjectures, p-adic L-functions were constructed interpolating the critical values of complex Hecke L-functions for gen- eral number fields. This vastly generalized Kubota and Leopoldt's work on the p-adic × Riemann zeta function. When F is a totally real field and χ : GF ! C is a totally even character, Cassou-Nogues, and Deligne{Ribet ([6], [11]) defined a p-adic meromorphic function Lp(χ, s): Zp −! Cp determined by the following interpolation property: for all n 2 Z≤0, we have Y n−1 −n n−1 Lp(χ, n) = (1 − χω (p)N p )L(χω ; n) (0.1) pjp where ! is the Teichm¨ullercharacter. Siegel and Klingen ([27]) showed that the values n−1 n−1 L(χω ; n) lie in the field obtained by adjoining the values of χω to Q. The equality (0.1) takes place in Q. Now let F be a CM field with maximal totally real subfield E. A prime p is called ordinary for F if every prime above p in E splits in F . For such primes p, Katz ([18],[19]) defined a p-adic L-function associated to any ray class character × χ : GF ! C . Katz's p-adic L-function interpolates the values of complex L-functions of algebraic Hecke characters with nonzero infinity type. To specify the interpolation property we specialize to the case that F is imaginary quadratic. Let p be a rational prime that is split in F . Let λ be a Hecke character of infinity type (1; 0). Then Katz 2 constructed a p-adic meromorphic function Lp(χ, t; s): Zp × Zp −! Cp determined by the following interpolation property: for all k; j 2 Z such that 1 ≤ j ≤ k − 1, we have k−1 Lp(χ, k; j) L(χλ ; j) k−1 = Ep(χ, k; j) k−1 : Ωp Ω1 × × In the formula above, Ep(χ, k; j) is an explicit complex number and Ωp 2 Cp ,Ω1 2 C are p-adic and complex periods that make both sides of the above equality algebraic. In the two cases F is totally real and F is imaginary quadratic, p-adic Stark conjectures have been made for Lp(χ, s) and Lp(χ, t; s), and some progress has been made on these conjectures. When F is totally real and χ is totally odd Gross ([17]) stated a conjecture for the order of vanishing of Lp(χω; s) at s = 0 and the leading term of the Taylor series of Lp(χω; s) at s = 0. Progress has been made on the order of vanishing, and recently the formula for the leading term was proved in [9] building off of earlier work in [8]. When F is totally real and χ is totally even there is a conjecture for the value Lp(χ, 1) known as the Serre-Solomon-Stark conjecture ([28], [32]).
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