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Stark-Heegner Points and P-Adic L-Functions View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Tesis Doctorals en Xarxa Stark-Heegner points and p-adic L-functions by Daniele Casazza ADVERTIMENT La consulta d’aquesta tesi queda condicionada a l’acceptació de les següents condicions d'ús: La difusió d’aquesta tesi per mitjà del repositori institucional UPCommons (http://upcommons.upc.edu/tesis) i el repositori cooperatiu TDX ( http://www.tdx.cat/) ha estat autoritzada pels titulars dels drets de propietat intel·lectual únicament per a usos privats emmarcats en activitats d’investigació i docència. No s’autoritza la seva reproducció amb finalitats de lucre ni la seva difusió i posada a disposició des d’un lloc aliè al servei UPCommons o TDX.No s’autoritza la presentació del seu contingut en una finestra o marc aliè a UPCommons (framing). Aquesta reserva de drets afecta tant al resum de presentació de la tesi com als seus continguts. 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WARNING On having consulted this thesis you’re accepting the following use conditions: Spreading this thesis by the institutional repository UPCommons (http://upcommons.upc.edu/tesis) and the cooperative repository TDX (http://www.tdx.cat/?locale- attribute=en) has been authorized by the titular of the intellectual property rights only for private uses placed in investigation and teaching activities. Reproduction with lucrative aims is not authorized neither its spreading nor availability from a site foreign to the UPCommons service. Introducing its content in a window or frame foreign to the UPCommons service is not authorized (framing). These rights affect to the presentation summary of the thesis as well as to its contents. In the using or citation of parts of the thesis it’s obliged to indicate the name of the author. Stark-Heegner points and p-adic L-functions A dissertation presented by Daniele Casazza to the departments Institute de Mathematiques´ Facultat de matematiques` i estad´ıstica Universite´ de Bordeaux, Universitat Politecnica` de Catalunya Talence, France Barcelona, Spain in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Mathematics October 2016 ii Contents Introduction v 1 Background material 1 1.1 Modular forms for Γ1(N).....................................1 1.2 Nebentype decomposition and Hecke operators . .2 1.3 Oldforms, newforms and basis . .4 1.4 Eisenstein series . .5 1.5 Algebraic modular forms . .7 1.6 Overconvergent modular forms and Coleman classicality result . .7 1.7 CM points and elliptic units . .9 2 Complex L-functions 11 2.1 Compatible systems of λ-adic representations . 12 2.2 Hecke characters and L-functions . 15 2.3 Double product L-function and Rankin's method . 16 2.3.1 Classical Rankin's L-function . 16 2.3.2 The Gross-Zagier formula . 18 2.4 Petersson product and L-functions: comparison of various formulae . 19 2.5 Triple product L-function and Garrett's method . 22 3 p-adic interpolation of classical L-functions 25 3.1 Complex multiplication and p-adic L-functions . 26 3.1.1 Periods and complex multiplication . 27 3.1.2 Katz p-adic L-function . 29 3.1.3 Bertolini-Darmon-Prasanna p-adic L-function . 30 3.2 Hida's work and p-adic L-functions . 32 3.2.1 Iwasawa algebrae and the weight space . 32 3.2.2 Λ-adic modular forms and Hida main result . 34 3.2.3 Castell`a's p-adic L-function . 37 3.2.4 p-adic interpolation of trilinear periods . 39 3.2.5 Hida-Rankin p-adic L-function . 40 3.2.6 Garrett-Hida p-adic L-function . 42 4 Proof of the main results 45 4.1 Elliptic Stark conjecture and p-adic Gross-Zagier formula . 45 4.2 Discussion on hypothesis C and C' . 47 4.3 Proof of theorem 0.6 . 47 4.4 Proof of theorem 0.7 and elliptic Stark conjecture when pjjNE ................. 51 iii iv CONTENTS Introduction The theory of L-functions plays a key role in modern number theory. After Iwasawa's new insight on this theory, p-adic L-functions became an interesting alternative to classical ones. A central feature of the theory of L-functions (classical or p-adic) is the study of their special values: this is where the analytic and the algebraic data meet. Let us describe the most basic example. Let K be a number field of discriminant DK and denote by OK its ring of integers. The seminal example is the Dedekind zeta function of the field K, denoted by ζK (s). It is defined for <(s) > 1 as an Euler product indexed by all prime ideals in OK as follows: Y 1 ζ (s) = : K 1 − NK (})s }⊂OK Q As was first proved by Hecke, this function admits a meromorphic continuation, and satisfies a functional equation of the form: s 1−s jDK j 2 · ζK;1(s) · ζK (s) = jDK j 2 · ζK;1(1 − s) · ζK (1 − s) where ζK;1(s) is an adequate product of Gamma factors. The integral point s = 0 lies outside the domain of definition and mathematicians were interested in understanding the nature of this value. × Let us recall Dirichlet's unit theorem: the group of units OK is a finitely generated abelian group, whose free part has rank r1 + r2 − 1, where r1 is the number of real embeddings K,! R and r2 is the number of complex embeddings K,! C up to conjugation, so that r1 + 2r2 = [K : Q]. The behaviour of ζK (s) at s = 0 is the content of the so-called analytic class number formula, which we state here as follows: Theorem 0.1 (Class number formula). The following are true: CNF0 The zeta function ζK (s) admits an analytic continuation and a functional equation relating values at s and 1 − s; × CNF1 The order of vanishing of ζK (s) at s = 0 equals the rank of OK , i.e. ords=0ζK (s) = r (analytic rank = algebraic rank); CNF2 The following formula holds true: −r hK RK lim s ζK (s) = − : s!0 !K where: { hK is the class number of K; { !K is the number of roots of unity of OK ; { RK is the complex regulator involving the logarithm of units of OK . v vi INTRODUCTION One generally refers to the analytic class number formula when speaking about the following result: r1 r2 2 (2π) hK RK Ress=1ζK (s) = p !K jDK j which can be easily proven to be equivalent to CFN1 and CFN2 together. In the 1970's Stark developed in [Stark] an equivariant conjectural refinement of the above formula, now known as Stark's conjectures. More precisely, for any field extension H j K, the Artin formalism allows one to decompose the zeta function ζH (s) as a product of L-functions over K twisted by irreducible Artin representations ρ of Gal (HjK). The Stark conjectures aim to describe vanishing properties and special values of these twists at s = 0 in terms of invariants linked to the objects involved in a Gal (HjK)- equivariant way. Not surprisingly, one can formulate a similar conjecture for elliptic curves, which will be the main motivation for the present thesis. More precisely, we consider: • an elliptic curve E defined over K. We denote by V (E) the compatible system of Galois represen- tations induced by its Tate module; • an Artin representation ρ : Gal (HjK) ! GLn(L), where L ⊂ C is the field of coefficients of ρ. We associate to it a compatible system of Galois representations V (ρ). Our main object of interest is the Hasse-Weil-Artin L-function of E twisted by ρ: L(E; ρ, s) := L(V (E) ⊗ V (ρ); s); This function is defined as an Euler product similar to that of ζK (s), which converges for <(s) > 3=2. When ρ = 1K , the twisted L-function L(E; ρ, s) is equal to the original one L(E=K; s). According to the Mordell-Weil theorem, E(K) is a finitely generated abelian group. The rank of its free part is called the algebraic rang of E over K, that we denote by r = r(E=K). In the early 60's, Birch and Swinnerton-Dyer formulated in [BSD] the following: Conjecture 0.2 (Birch and Swinnerton-Dyer). The following are true: BSD0 The zeta function L(E=K; s) admits an analytic continuation and a functional equation relating values at s and 2 − s; BSD1 The order of vanishing at s = 1 equals the rank E over K, i.e. ords=1L(E=K; s) = r (analytic rank = algebraic rank); BSD2 The following formula holds true: −r Y #ShaE=K lim s L(E=K; s) = ΩE=K · c} · RE=K s!0 #T 2 }⊂OK where: { ΩE=K is the complex period attached to E over K; { c} is the Tamagawa number of E at }; { ShaE=K is the Shafarevich-Tate group; { RE=K is the complex regulator involving N´eron-Tate heights of a basis of the free part of the Mordell-Weil group.
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