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L-Functions in Number Theory by Yichao Zhang a Thesis Submitted In L-functions in Number Theory by Yichao Zhang A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright c 2010 by Yichao Zhang Abstract L-functions in Number Theory Yichao Zhang Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2010 As a generalization of the Riemann zeta function, L-function has become one of the central objects in Number Theory. The theory of L-functions, which produces a large family of consequences and conjectures in a unified way, concerns their zeros and poles, functional equations, special values and the connections between objects in different fields. Although most generalizations are largely conjectural, there are many existing results that provide us the evidence. In this thesis, we shall consider some L-functions and look into some problems men- tioned above. More explicitly, for the L-functions associated to newforms of fixed square- free level, we will consider an average version of the fourth moments problem. The final bound is proven by considering definite rational quaternion algebras and divisor functions in them, generalizing Maass Converse Theorem and one of Duke's result and eventually applying the solution to Basis Problem. 1 We then consider the problem of expressing the special value at 2 of the Rankin- Selberg L-function associated to two newforms in terms of the Pertersson inner product, where one of the newforms is twisted by the derivative of some Eisenstein series. Finally, we consider the Artin L-functions attached to irreducible 4-dimensional S5- Galois representations and deal with the modularity problem. One sufficient condition on the modularity is given, which may help to find an affirmative example for Strong Artin Conjecture in this case. ii Acknowledgements The author is greatly indebted to his advisor, Professor Henry H. Kim, who suggested all the problems treated in this thesis. His constant guidance and support have been invaluable. The author would also like to thank Professor Stephen Kudla for many valuable and inspiring conversations and comments and Professor James Arthur and Professor Valentin Blomer for the help during the work of this thesis. The author is also grateful to Professor William Duke for agreeing to be the external examiner for this thesis. The author thanks all of the faculty and the staff of the Department of Mathematics at University of Toronto for their wonderful help. Special thanks go to Ms. Ida Bulat, our graduate administrator, without whom the life at University of Toronto would never be easy and joyful for the author. Lastly, the author would like to thank his wife, Hailing Liu, for all the sacrifices she has made these years. Without her company and support, this thesis could not have been completed. iii List of Symbols C the field of complex numbers R the field of real numbers + R the field of complex numbers N the set of non-negative rational integers Z the ring of rational integers + Z the set of positive rational integers AF the ring of adeles of F × AF the group of ideles of F OF the ring of integers of F Fv the completion of F at v OF;v the completion of OF at v R× the group of units in the ring R × GL2(R) the group of 2 × 2 matrices over the ring R with determinant in R SL2(R) the group of 2 × 2 matrices over the ring R of determinant one + GL2(R) elements of GL2(R) with positive determinant iv Γ(N) fγ 2 SL2(Z): γ ≡ I2×2 mod Ng Γ1(N) fγ 2 SL2(Z): aγ ≡ dγ ≡ 1 mod N; cγ ≡ 0 mod Ng Γ0(N) fγ 2 SL2(Z): cγ ≡ 0 mod Ng H the upper half plane of C cχ the conductor of the character χ f d f d(z) = f(dz) + fjkγ weight k action of γ 2 GL2(R) q e(z) = exp(2πiz) Mk(Γ) the space of modular forms of weight k for the group Γ Mk(N; χ) the subspace of Mk(Γ1(N)) that Γ0(N) acts by Dirichlet character χ Sk(Γ) the space of cusp forms of weight k for the group Γ Sk(N; χ) Sk(Γ1(N)) \Mk(N; χ) new Sk (N; χ) the subspace of newforms in Sk(N; χ) old Sk (N; χ) the subspace of oldforms in Sk(N; χ) Tk(n) n-th Hecke operator of weight k T; Tk the Hecke algebra of fixed weightk ζ(s) the Riemann zeta function ζO(s) the zeta function associated to an order O ζF (s) the zeta function associated to the number field F L(s; χ) the L-function attached to χ L(s; f) the L-function attached to the modular form f L(s; ρ) the L-function attached to the Galois representation ρ v Lv(s; ·) the local factor at v of the L-function L(s; ·) AF a quaternion algebra over the field F (a; b)F the quaternion algebra generated as a vector space by f1; i; j; kg; whose ring structure is given by i2 = a; j2 = b; ij = −ji = k Av the localization of A at v; that is A ⊗F Fv AA the ring of adeles of A A× the group of ideles of A A A× the group of finite ideles of A A;f Ol(M) the left order of M Or(M) the right order of M av the localization of a at v; that is a ⊗OF OF;v b jl a b divides a from the left a ∼l b a and b are left-equivalent H; H(A) the class number of A α the conjugate of α T r; T rA=F the reduced trace of AF N; NA=F the reduced norm of AF d(a) jf(b; c): a = bcgj; the divisor function a(n) the number of integral ideals (of fixed left order) of norm n 2 Cn+1 the Clifford algebra generated by fi1; ··· ; ing subject to ih = −1; and ihik = −ikih(k 6= h) n+1 V fx = x0 + x1i1 + ··· + xnin : xi 2 Rg ⊂ Cn+1 vi n+1 n+1 H fx 2 V : xn > 0g n n+1 X @ 1−n @ ∆n+1 xn (xn ) @xh @xh h=0 Pm the space of spherical harmonics of degree m (in n variables) X∗ the summation over any orthonormal basis of Pm Pm The following symbols are used in limiting process, say, as x ! a: f g when x is close enough to a, f(x) ≤ Ajg(x)j for some positive constant A f = O(g) f g f(x) f = o(g) lim = 0 x!a g(x) f g when x is close enough to a, A1jg(x)j ≤ f(x) ≤ A2jg(x)j for some positive constants A1 and A2 f ∼ g f − g = o(g) vii Contents Abstract ii Acknowledgements iii List of Symbols iv 1 Introduction 1 1.1 Generalized Lindel¨ofHypothesis and Moments Problem . 1 1.2 Special Values and Non-vanishing Properties . 4 1.3 Strong Artin Conjecture . 5 2 L-functions 7 2.1 L-functions . 7 2.2 Modular Forms and L-functions . 9 2.2.1 Modular Forms . 9 2.2.2 The Theory of Hecke Operators . 11 2.2.3 The Theory of Newforms . 13 2.2.4 L-functions Attached to Modular Forms . 14 2.2.5 Automorphic Representations and L-functions . 15 2.3 Galois Representations and Artin L-functions . 17 3 Divisor Function 21 viii 3.1 Introduction . 21 3.1.1 General Notions . 21 3.1.2 Local and Global Theory . 24 3.2 Divisor Function over Quaternion Algebras . 27 4 Fourth Moments 39 4.1 Introduction . 39 4.2 Brandt Matrices and Newforms of Level N . 40 4.3 Maass Correspondence Theorem . 45 4.4 Application to Fourth Moments of L-functions . 50 5 Critical Values 57 5.1 Introduction . 57 5.2 General Equality . 58 5.3 Applications . 65 6 Modularity 67 6.1 Introduction . 67 6.2 Modularity of ρ and σ ............................ 68 6.3 An Open Problem . 70 Appendix: Character Tables 72 Bibliography 77 ix Chapter 1 Introduction 1.1 Generalized Lindelof¨ Hypothesis and Moments Problem Conjecture 1.1.1 (Generalized Lindelof¨ Hypothesis) Assume L(s) is an L-function (over Q) which admits meromorphic continuation to the whole s-plane and a functional equation under s $ 1 − s. Then for any > 0, 1 L( + it) = O(t): 2 1 Lindel¨ofconjectured this for the Riemann zeta function, namely, ζ( 2 + it) = O(t ). Note that the Riemann Hypothesis, which asserts that all of the non-trivial zeros of ζ(s) 1 lie on the vertical line Re(s) = 2 , implies the Lindel¨ofHypothesis. Actually the former is much stronger. Define µ(σ) = inffa > 0 : ζ(σ + it) = O(ta)g; 1 and Lindel¨ofHypothesis states µ( 2 ) = 0. In 1908, Lindel¨of proved that µ is convex and 1 1 it follows that µ( 2 ) ≤ 4 , which is called the convexity bound. On the other hand, one can show that the Lindel¨ofHypothesis for Riemann zeta 1 Chapter 1. Introduction 2 function is equivalent to the statement that the 2k-th integral moment Z T 1 jζ( + it)j2kdt = O(T 1+); as T ! 1; 0 2 for all positive integers k and all positive real numbers . The latter is known for k = 1; 2, but for k ≥ 3, it is still open. For a general L-function L(s), moments problems are the problems of providing non- trivial bounds for the moments of L(s), namely, a result would take the form Z T 1 jL( + it)j2kdt = O(T a): 0 2 In Chapter 3 and 4, we will consider an average version of fourth moments problem for L-functions attached to newforms.
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