DOCSLIB.ORG

L-FUNCTIONS and CYCLOTOMIC UNITS 1. Introduction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

L-FUNCTIONS AND CYCLOTOMIC UNITS TOM WESTON, UNIVERSITY OF MICHIGAN 1. Introduction Let K be a number field with r1 real embeddings and r2 pairs of complex con- jugate embeddings. If ζK (s) is the Dedekind zeta function of K, then ζK (s) has a zero of order r1 + r2 1 at s = 0, and the value of the first non-zero derivative is given by the Dirichlet− class number formula: (r1+r2 1) hK RK ζK − (0) = : − wK Here RK is the regulator of K, wK is the number of roots of unity in K and hK is the class number of K. This formula is a striking connection between arithmetic and analysis, and there have been many attempts to generalize it to other L-functions: one expects that the value of a \motivic" L-function at an integer point should involve a transcendental factor, a boring rational factor, and an interesting rational factor. In the case of the Dedekind zeta function, these roles are played by RK , wK , and hK , respectively. In the general case one expects that the interesting rational factor is the order of a certain Selmer group. 2. Selmer groups and Kolyvagin systems Let us recall the definition of the Selmer group of a p-adic Galois representation. Let T be a free Zp-module with a continuous action of the absolute Galois group ¯ GK = Gal(K=K). We define the Cartier dual of T by T ∗ = HomZp (T; µp1 ) with the adjoint Galois action. Let be a Selmer structure on T . Recall that consists of choices of local Selmer structuresF F 1 1 H (Kv;T ) H (Kv;T ) F ⊆ for each place v of K; these are assumed to coincide with the unramified choice at almost all places v. Tate local duality yields a Cartier dual Selmer structure ∗ on 1 F T ∗, and the Selmer group S(K; T ∗) is defined to be the subgroup of H (K; T ∗) of 1 cocycles whose local restrictions lie in H (Kv;T ) for every place v. ∗ ∗ Kolyvagin has shown that in some situationsF one can use duality theorems in Galois cohomology to bound the order of such a Selmer group. In this section we will explain Kolyvagin's general strategy; the key step is the production of what we will call a Kolyvagin system. Date: May 4, 2001. Notes from Arizona Winter School, March 10{15, 2000. 1 2 TOM WESTON, UNIVERSITY OF MICHIGAN Fix a Selmer structure on T . If a is a fractional ideal of K, we define modified F Selmer structures on T and T ∗ by H1 (K ;T ) v - a; ( v a = F1 F H (Kv;T ) v a; j and 1 H (Kv;T ∗) v - a; ∗ = ( F ∗ Fa 0 v a: j (Here for simplicity of notation we say that v divides the fractional ideal a if v occurs with non-zero exponent in the prime factorization of a.) We will just write the Selmer groups S (K; T ) (resp. S (K; T ), resp. S (K; T ), resp. S (K; T )) ∗ ∗ a a∗ ∗ F F a F F as S(K; T ) (resp. S(K; T ∗), resp. Sa(K; T ), resp. S (K; T ∗)). For any place v we 1 1 1 will write Hs (Kv;T ) for the singular quotient H (Kv;T )=H (Kv;T ). F If M is a Zp-module, we write M _ for its dual Hom(M; Qp=Zp). Unwinding the discussion of local and global duality in Mazur's lecture, one finds that for any and a as above there is an exact sequence F 1 a (1) 0 S(K; T ) Sa(K; T ) Hs (Kv;T ) S(K; T ∗)_ S (K; T ∗)_ 0: ! ! !v ⊕a ! ! ! j This exact sequence suggests the following approaching to bounding the order of a the Selmer group S(K; T ∗): first, one bounds the order of S (K; T ∗) for “sufficiently divisible" ideals a. In practice it is often not difficult to show that one can choose a an ideal a so that S (K; T ∗) vanishes. For such an a, (1) yields an isomorphism 1 S(K; T ∗)_ = Hs (Kv;T ) = img Sa(K; T ): ∼ v⊕a j 1 In particular, if one can exhibit elements of Sa(K; T ) which fill up v aHs (Kv;T ) ⊕ j with finite index η, then one concludes that #S(K; T ∗) η. (More generally, if a j a S (K; T ∗) is non-zero but computable, one has #S(K; T ∗) η #S (K; T ∗).) An j · 1 explicitly given submodule of Sa(K; T ) which has finite index in v aHs (Kv;T ) is called a Kolyvagin system. ⊕ j a In practice, one can bound S (K; T ∗) (for sufficiently divisible a) for a very general class of Galois representations T ; it involves only the coarse structure of T . In contrast, constructing a Kolyvagin system for T is quite difficult and has only been done for certain special Galois representations. In the remainder of this paper we will study the situation in the simplest case of the Galois representation T = Zp(1). 3. Example: Ideal class groups We now set T = Zp(1), so that T ∗ = Qp=Zp. We will define the Selmer structure on T momentarily. F To exhibit a Kolyvagin system, we need some method of constructing elements of H1(K; T ) with controlled ramification. We will use the Kummer map 1 κ : K× H (K; µn) ! σ(α1=n) α σ : 7! 7! α1=n L-FUNCTIONS AND CYCLOTOMIC UNITS 3 Here α1=n is a fixed choice of nth root of α in K¯ ; changing the choice of α1=n changes κ by a coboundary. The Kummer map is compatible with changing n; in particular, 1 if we take the inverse limit over powers of p we obtain a map κ : K× H (K; T ). The same construction yields local Kummer maps ! 1 κv : K× H (Kv;T ) v ! for each v. In fact, κv induces an isomorphism 1 κv : Kv× H (Kv;T ) d ! where Kv× is the p-adic completion of Kv×. Considerd the exact sequence ordv 0 × K× Z 0 !OKv ! v −! ! with p-adic completion ordv (2) 0 × Kv× Zp 0: ! OKv ! −! ! d d1 We define the local Selmer structure H (Kv;T ) by requiring that the exact se- quence F 1 1 1 0 H (Kv;T ) H (Kv;T ) Hs (Kv;T ) 0 ! F ! ! ! corresponds to (2) under the isomorphism κv of the middle terms. (It is straight- forward to check that this choice agrees with the natural local Selmer structure if v - p.) As always we take the Cartier dual Selmer structure on T ∗. We see from the definition that for each v there is a canonical isomorphism 1 ∼= H (Kv;T ) Zp s −! fitting into a commutative diagram κ 1 1 (3) K× / H (K; T ) / Hs (Kv;T ) ord v ∼= Z / Zp 1 In particular, it follows from (3) that for any α K×, κ(α) H (K; T ) satisfies the local condition given by at every place which2 is relatively2 prime to α. In the notation introduced earlier, thisF means that κ(α) S (K; T ): 2 (α) We now use κ to produce a Kolyvagin system for T . To motivate our construc- tion, we recall that class field theory yields a canonical isomorphism (4) S(K; T ∗) ∼= A_ where A is the p-part of the ideal class group of K and A_ = Hom(A; Qp=Zp). 1 (To see this, note that H (K; T ∗) = Hom(GK ; Qp=Zp). The local conditions ∗ correspond to allowing only those homomorphisms which factor through Gal(L=KF ) for L=K unramified. Since the Galois group of the maximal unramified abelian extension of K is canonically isomorphic to the ideal class group of A, the iso- morphism (4) follows.) Furthermore, for any ideal a, the restricted Selmer group a S (K; T ∗) corresponds to the dual of the quotient of A by the subgroup generated a by the prime factors of a. In particular, S (K; T ∗) will vanish if the prime factors 4 TOM WESTON, UNIVERSITY OF MICHIGAN of a generate A. (One can also easily prove this vanishing by using the Tcheba- torev density theorem instead of (4); this approach has the advantage of making the results below somewhat less vacuous.) Write A = Z=pn1 Z Z=pnr Z ⊕ · · · ⊕ with generators l1;:::; lr (which we can take to be prime). Set a = l1 lr; as we a ··· observed above, we have S (K; T ∗) = 0. In particular, by (1) we have 1 (5) S(K; T ∗)_ = Hs (Kv;T )= img Sa(K; T ): ∼ v⊕a j We will now compute this directly, without any reference to (4). pni By definition, li is principal; choose a generator αi and set ci = κ(αi) ni 2 Sli (K; T ). Since ordli (αi) = p , by (3) we have 1 ni (6) H (Kl ;T )=(Zp img ci) = Z=p Z: s i · ∼ These elements c1; : : : ; cr form a Kolyvagin system. Indeed, let C be the Zp-span of c1; : : : ; cr in Sa(K; T ). By (6) we have 1 n1 nr Hs (Kv;T )= img C = Z=p Z Z=p Z = A: v⊕a ∼ ⊕ · · · ⊕ ∼ j Combining this with (5) yields a surjection (7) A S(K; T ∗)_: In fact, it is not difficult to see that img C = img Sa(K; T ), so that the surjection (7) is an isomorphism. Of course, all we have done is reproduced the result (4).
Recommended publications
  • Part III Essay on Serre's Conjecture

    Part III Essay on Serre's Conjecture

    Serre’s conjecture Alex J. Best June 2015 Contents 1 Introduction 2 2 Background 2 2.1 Modular forms . 2 2.2 Galois representations . 6 3 Obtaining Galois representations from modular forms 13 3.1 Congruences for Ramanujan’s t function . 13 3.2 Attaching Galois representations to general eigenforms . 15 4 Serre’s conjecture 17 4.1 The qualitative form . 17 4.2 The refined form . 18 4.3 Results on Galois representations associated to modular forms 19 4.4 The level . 21 4.5 The character and the weight mod p − 1 . 22 4.6 The weight . 24 4.6.1 The level 2 case . 25 4.6.2 The level 1 tame case . 27 4.6.3 The level 1 non-tame case . 28 4.7 A counterexample . 30 4.8 The proof . 31 5 Examples 32 5.1 A Galois representation arising from D . 32 5.2 A Galois representation arising from a D4 extension . 33 6 Consequences 35 6.1 Finiteness of classes of Galois representations . 35 6.2 Unramified mod p Galois representations for small p . 35 6.3 Modularity of abelian varieties . 36 7 References 37 1 1 Introduction In 1987 Jean-Pierre Serre published a paper [Ser87], “Sur les representations´ modulaires de degre´ 2 de Gal(Q/Q)”, in the Duke Mathematical Journal. In this paper Serre outlined a conjecture detailing a precise relationship between certain mod p Galois representations and specific mod p modular forms. This conjecture and its variants have become known as Serre’s conjecture, or sometimes Serre’s modularity conjecture in order to distinguish it from the many other conjectures Serre has made.
  • Arxiv:0906.3146V1 [Math.NT] 17 Jun 2009

    Arxiv:0906.3146V1 [Math.NT] 17 Jun 2009

    Λ-RINGS AND THE FIELD WITH ONE ELEMENT JAMES BORGER Abstract. The theory of Λ-rings, in the sense of Grothendieck’s Riemann– Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Λ-algebraic geometry. We show that Λ-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry. Introduction Many writers have mused about algebraic geometry over deeper bases than the ring Z of integers. Although there are several, possibly unrelated reasons for this, here I will mention just two. The first is that the combinatorial nature of enumer- ation formulas in linear algebra over finite fields Fq as q tends to 1 suggests that, just as one can work over all finite fields simultaneously by using algebraic geome- try over Z, perhaps one could bring in the combinatorics of finite sets by working over an even deeper base, one which somehow allows q = 1. It is common, follow- ing Tits [60], to call this mythical base F1, the field with one element. (See also Steinberg [58], p. 279.) The second purpose is to prove the Riemann hypothesis. With the analogy between integers and polynomials in mind, we might hope that Spec Z would be a kind of curve over Spec F1, that Spec Z ⊗F1 Z would not only make sense but be a surface bearing some kind of intersection theory, and that we could then mimic over Z Weil’s proof [64] of the Riemann hypothesis over function fields.1 Of course, since Z is the initial object in the category of rings, any theory of algebraic geometry over a deeper base would have to leave the usual world of rings and schemes.
  • Full Text (PDF Format)

    Full Text (PDF Format)

    ASIAN J. MATH. © 2017 International Press Vol. 21, No. 3, pp. 397–428, June 2017 001 FUNCTIONAL EQUATION FOR THE SELMER GROUP OF NEARLY ORDINARY HIDA DEFORMATION OF HILBERT MODULAR FORMS∗ † ‡ SOMNATH JHA AND DIPRAMIT MAJUMDAR Abstract. We establish a duality result proving the ‘functional equation’ of the characteristic ideal of the Selmer group associated to a nearly ordinary Hilbert modular form over the cyclotomic Zp extension of a totally real number field. Further, we use this result to establish an ‘algebraic functional equation’ for the ‘big’ Selmer groups associated to the corresponding nearly ordinary Hida deformation. The multivariable cyclotomic Iwasawa main conjecture for nearly ordinary Hida family of Hilbert modular forms is not established yet and this can be thought of as a modest evidence to the validity of this Iwasawa main conjecture. We also prove a functional equation for the ‘big’ Z2 Selmer group associated to an ordinary Hida family of elliptic modular forms over the p extension of an imaginary quadratic field. Key words. Selmer Group, Iwasawa theory of Hida family, Hilbert modular forms. AMS subject classifications. 11R23, 11F33, 11F80. Introduction. We fix an odd rational prime p and N a natural number prime to p. Throughout, we fix an embedding ι∞ of a fixed algebraic closure Q¯ of Q into C and also an embedding ιl of Q¯ into a fixed algebraic closure Q¯ of the field Q of the -adic numbers, for every prime .LetF denote a totally real number field and K denote an imaginary quadratic field. For any number field L, SL will denote a finite set of places of L containing the primes dividing Np.
  • Background on Local Fields and Kummer Theory

    Background on Local Fields and Kummer Theory

    MATH 776 BACKGROUND ON LOCAL FIELDS AND KUMMER THEORY ANDREW SNOWDEN Our goal at the moment is to prove the Kronecker{Weber theorem. Before getting to this, we review some of the basic theory of local fields and Kummer theory, both of which will be used constantly throughout this course. 1. Structure of local fields Let K=Qp be a finite extension. We denote the ring of integers by OK . It is a DVR. There is a unique maximal ideal m, which is principal; any generator is called a uniformizer. We often write π for a uniformizer. The quotient OK =m is a finite field, called the residue field; it is often denoted k and its cardinality is often denoted q. Fix a uniformizer π. Every non-zero element x of K can be written uniquely in the form n uπ where u is a unit of OK and n 2 Z; we call n the valuation of x, and often denote it S −n v(x). We thus have K = n≥0 π OK . This shows that K is a direct union of the fractional −n ideals π OK , each of which is a free OK -module of rank one. The additive group OK is d isomorphic to Zp, where d = [K : Qp]. n × ∼ × The decomposition x = uπ shows that K = Z × U, where U = OK is the unit group. This decomposition is non-canonical, as it depends on the choice of π. The exact sequence 0 ! U ! K× !v Z ! 0 is canonical. Choosing a uniformizer is equivalent to choosing a splitting of this exact sequence.
  • On Functional Equations of Euler Systems

    On Functional Equations of Euler Systems

    ON FUNCTIONAL EQUATIONS OF EULER SYSTEMS DAVID BURNS AND TAKAMICHI SANO Abstract. We establish precise relations between Euler systems that are respectively associated to a p-adic representation T and to its Kummer dual T ∗(1). Upon appropriate specialization of this general result, we are able to deduce the existence of an Euler system of rank [K : Q] over a totally real field K that both interpolates the values of the Dedekind zeta function of K at all positive even integers and also determines all higher Fitting ideals of the Selmer groups of Gm over abelian extensions of K. This construction in turn motivates the formulation of a precise conjectural generalization of the Coleman-Ihara formula and we provide supporting evidence for this conjecture. Contents 1. Introduction 1 1.1. Background and results 1 1.2. General notation and convention 4 2. Coleman maps and local Tamagawa numbers 6 2.1. The local Tamagawa number conjecture 6 2.2. Coleman maps 8 2.3. The proof of Theorem 2.3 10 3. Generalized Stark elements and Tamagawa numbers 13 3.1. Period-regulator isomorphisms 13 3.2. Generalized Stark elements 16 3.3. Deligne-Ribet p-adic L-functions 17 3.4. Evidence for the Tamagawa number conjecture 18 arXiv:2003.02153v1 [math.NT] 4 Mar 2020 4. The functional equation of vertical determinantal systems 18 4.1. Local vertical determinantal sytems 19 4.2. The functional equation 20 5. Construction of a higher rank Euler system 24 5.1. The Euler system 25 5.2. The proof of Theorem 5.2 26 6.
  • Multiplicative Reduction and the Cyclotomic Main Conjecture for Gl2

    Multiplicative Reduction and the Cyclotomic Main Conjecture for Gl2

    Pacific Journal of Mathematics MULTIPLICATIVE REDUCTION AND THE CYCLOTOMIC MAIN CONJECTURE FOR GL2 CHRISTOPHER SKINNER Volume 283 No. 1 July 2016 PACIFIC JOURNAL OF MATHEMATICS Vol. 283, No. 1, 2016 dx.doi.org/10.2140/pjm.2016.283.171 MULTIPLICATIVE REDUCTION AND THE CYCLOTOMIC MAIN CONJECTURE FOR GL2 CHRISTOPHER SKINNER We show that the cyclotomic Iwasawa–Greenberg main conjecture holds for a large class of modular forms with multiplicative reduction at p, extending previous results for the good ordinary case. In fact, the multiplicative case is deduced from the good case through the use of Hida families and a simple Fitting ideal argument. 1. Introduction The cyclotomic Iwasawa–Greenberg main conjecture was established in[Skinner and Urban 2014], in combination with work of Kato[2004], for a large class of newforms f 2 Sk.00.N// that are ordinary at an odd prime p - N, subject to k ≡ 2 .mod p − 1/ and certain conditions on the mod p Galois representation associated with f . The purpose of this note is to extend this result to the case where p j N (in which case k is necessarily equal to 2). P1 n Recall that the coefficients an of the q-expansion f D nD1 anq of f at the cusp at infinity (equivalently, the Hecke eigenvalues of f ) are algebraic integers that generate a finite extension Q. f / ⊂ C of Q. Let p be an odd prime and let L be a finite extension of the completion of Q. f / at a chosen prime above p (equivalently, let L be a finite extension of Qp in a fixed algebraic closure Qp of Qp that contains the image of a chosen embedding Q.
  • Viewed As a Vector Space Over Itself) Equipped with the Gfv -Action Induced by Ψ

    Viewed As a Vector Space Over Itself) Equipped with the Gfv -Action Induced by Ψ

    New York Journal of Mathematics New York J. Math. 27 (2021) 437{467. On Bloch{Kato Selmer groups and Iwasawa theory of p-adic Galois representations Matteo Longo and Stefano Vigni Abstract. A result due to R. Greenberg gives a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic Zp-extensions at good ordinary primes p. We extend Green- berg's result to more general p-adic Galois representations, including a large subclass of those attached to p-ordinary modular forms of weight at least 4 and level Γ0(N) with p - N. Contents 1. Introduction 437 2. Galois representations 440 3. Selmer groups 446 4. Characteristic power series 449 Γ 5. Relating SelBK(A=F ) and S 450 6. Main result 464 References 465 1. Introduction A classical result of R. Greenberg ([9, Theorem 4.1]) establishes a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic Zp-extensions at good ordinary primes p. Our goal in this paper is to extend Greenberg's result to more general p-adic Galois representa- tions, including a large subclass of those coming from p-ordinary modular forms of weight at least 4 and level Γ0(N) with p a prime number such that p - N. This generalization of Greenberg's theorem will play a role in our Received November 11, 2020. 2010 Mathematics Subject Classification. 11R23, 11F80.
  • BOUNDING SELMER GROUPS for the RANKIN–SELBERG CONVOLUTION of COLEMAN FAMILIES Contents 1. Introduction 1 2. Modular Curves

    BOUNDING SELMER GROUPS for the RANKIN–SELBERG CONVOLUTION of COLEMAN FAMILIES Contents 1. Introduction 1 2. Modular Curves

    BOUNDING SELMER GROUPS FOR THE RANKIN{SELBERG CONVOLUTION OF COLEMAN FAMILIES ANDREW GRAHAM, DANIEL R. GULOTTA, AND YUJIE XU Abstract. Let f and g be two cuspidal modular forms and let F be a Coleman family passing through f, defined over an open affinoid subdomain V of weight space W. Using ideas of Pottharst, under certain hypotheses on f and g we construct a coherent sheaf over V × W which interpolates the Bloch-Kato Selmer group of the Rankin-Selberg convolution of two modular forms in the critical range (i.e the range where the p-adic L-function Lp interpolates critical values of the global L-function). We show that the support of this sheaf is contained in the vanishing locus of Lp. Contents 1. Introduction 1 2. Modular curves 4 3. Families of modular forms and Galois representations 5 4. Beilinson{Flach classes 9 5. Preliminaries on ('; Γ)-modules 14 6. Some p-adic Hodge theory 17 7. Bounding the Selmer group 21 8. The Selmer sheaf 25 Appendix A. Justification of hypotheses 31 References 33 1. Introduction In [LLZ14] (and more generally [KLZ15]) Kings, Lei, Loeffler and Zerbes construct an Euler system for the Galois representation attached to the convolution of two modular forms. This Euler system is constructed from Beilinson{Flach classes, which are norm-compatible classes in the (absolute) ´etalecohomology of the fibre product of two modular curves. It turns out that these Euler system classes exist in families, in the sense that there exist classes [F;G] 1 la ∗ ^ ∗ cBF m;1 2 H Q(µm);D (Γ;M(F) ⊗M(G) ) which specialise to the Beilinson{Flach Euler system at classical points.
  • An Introduction to Motives I: Classical Motives and Motivic L-Functions

    An Introduction to Motives I: Classical Motives and Motivic L-Functions

    An introduction to motives I: classical motives and motivic L-functions Minhyong Kim February 3, 2010 IHES summer school on motives, 2006 The exposition here follows the lecture delivered at the summer school, and hence, contains neither precision, breadth of comprehension, nor depth of insight. The goal rather is the curious one of providing a loose introduction to the excellent introductions that already exist, together with scattered parenthetical commentary. The inadequate nature of the exposition is certainly worst in the third section. As a remedy, the article of Schneider [40] is recommended as a good starting point for the complete novice, and that of Nekovar [37] might be consulted for more streamlined formalism. For the Bloch-Kato conjectures, the paper of Fontaine and Perrin-Riou [20] contains a very systematic treatment, while Kato [27] is certainly hard to surpass for inspiration. Kings [30], on the other hand, gives a nice summary of results (up to 2003). 1 Motivation Given a variety X over Q, it is hoped that a suitable analytic function ζ(X,s), a ζ-function of X, encodes important arithmetic invariants of X. The terminology of course stems from the fundamental function ∞ s ζ(Q,s)= n− nX=1 named by Riemann, which is interpreted in this general context as the zeta function of Spec(Q). A general zeta function should generalize Riemann’s function in a manner similar to Dedekind’s extension to number fields. Recall that the latter can be defined by replacing the sum over positive integers by a sum over ideals: s ζ(F,s)= N(I)− XI where I runs over the non-zero ideals of the ring of integers F and N(I)= F /I , and that ζ(F,s) has a simple pole at s =1 (corresponding to the trivial motiveO factor of Spec(|OF ), as| it turns out) with r1 r2 2 (2π) hF RF (s 1)ζ(F,s) s=1 = − | wF DF p| | By the unique factorization of ideals, ζ(F,s) can also be written as an Euler product s 1 (1 N( )− )− Y − P P 1 as runs over the maximal ideals of F , that is, the closed points of Spec( F ).
  • Iwasawa Theory of the Fine Selmer Group

    Iwasawa Theory of the Fine Selmer Group

    J. ALGEBRAIC GEOMETRY 00 (XXXX) 000{000 S 1056-3911(XX)0000-0 IWASAWA THEORY OF THE FINE SELMER GROUP CHRISTIAN WUTHRICH Abstract The fine Selmer group of an elliptic curve E over a number field K is obtained as a subgroup of the usual Selmer group by imposing stronger conditions at places above p. We prove a formula for the Euler-cha- racteristic of the fine Selmer group over a Zp-extension and use it to compute explicit examples. 1. Introduction Let E be an elliptic curve defined over a number field K and let p be an odd prime. We choose a finite set of places Σ in K containing all places above p · 1 and such that E has good reduction outside Σ. The Galois group of the maximal extension of K which is unramified outside Σ is denoted by GΣ(K). In everything that follows ⊕Σ always stands for the product over all finite places υ in Σ. Let Efpg be the GΣ(K)-module of all torsion points on E whose order is a power of p. For a finite extension L : K, the fine Selmer group is defined to be the kernel 1 1 0 R(E=L) H (GΣ(L); Efpg) ⊕Σ H (Lυ; Efpg) i i where H (Lυ; ·) is a shorthand for the product ⊕wjυH (Lw; ·) over all places w in L above υ. If L is an infinite extension, we define R(E=L) to be the inductive limit of R(E=L0) for all finite subextensions L : L0 : K.
  • An Introductory Lecture on Euler Systems

    An Introductory Lecture on Euler Systems

    AN INTRODUCTORY LECTURE ON EULER SYSTEMS BARRY MAZUR (these are just some unedited notes I wrote for myself to prepare for my lecture at the Arizona Winter School, 03/02/01) Preview Our group will be giving four hour lectures, as the schedule indicates, as follows: 1. Introduction to Euler Systems and Kolyvagin Systems. (B.M.) 2. L-functions and applications of Euler systems to ideal class groups (ascend- ing cyclotomic towers over Q). (T.W.) 3. Student presentation: The “Heegner point” Euler System and applications to the Selmer groups of elliptic curves (ascending anti-cyclotomic towers over quadratic imaginary fields). 4. Student presentation: “Kato’s Euler System” and applications to the Selmer groups of elliptic curves (ascending cyclotomic towers over Q). Here is an “anchor problem” towards which much of the work we are to describe is directed. Fix E an elliptic curve over Q. So, modular. Let K be a number field. We wish to study the “basic arithmetic” of E over K. That is, we want to understand the structure of these objects: ² The Mordell-Weil group E(K) of K-rational points on E, and ² The Shafarevich-Tate group Sha(K; E) of isomorphism classes of locally trivial E-curves over K. [ By an E-curve over K we mean a pair (C; ¶) where C is a proper smooth curve defined over K and ¶ is an isomorphism between the jacobian of C and E, the isomorphism being over K.] Now experience has led us to realize 1. (that cohomological methods apply:) We can use cohomological methods if we study both E(K) and Sha(K; E) at the same time.
  • THE UNIVERSAL P-ADIC GROSS–ZAGIER FORMULA by Daniel

    THE UNIVERSAL P-ADIC GROSS–ZAGIER FORMULA by Daniel

    THE UNIVERSAL p-ADIC GROSS–ZAGIER FORMULA by Daniel Disegni Abstract.— Let G be the group (GL2 GU(1))=GL1 over a totally real field F , and let be a Hida family for G. Revisiting a construction of Howard× and Fouquet, we construct an explicit section Xof a sheaf of Selmer groups over . We show, answering a question of Howard, that is a universal HeegnerP class, in the sense that it interpolatesX geometrically defined Heegner classes at all the relevantP classical points of . We also propose a ‘Bertolini–Darmon’ conjecture for the leading term of at classical points. X We then prove that the p-adic height of is givenP by the cyclotomic derivative of a p-adic L-function. This formula over (which is an identity of functionalsP on some universal ordinary automorphic representations) specialises at classicalX points to all the Gross–Zagier formulas for G that may be expected from representation- theoretic considerations. Combined with a result of Fouquet, the formula implies the p-adic analogue of the Beilinson–Bloch–Kato conjecture in analytic rank one, for the selfdual motives attached to Hilbert modular forms and their twists by CM Hecke characters. It also implies one half of the first example of a non-abelian Iwasawa main conjecture for derivatives, in 2[F : Q] variables. Other applications include two different generic non-vanishing results for Heegner classes and p-adic heights. Contents 1. Introduction and statements of the main results................................... 2 1.1. The p-adic Be˘ılinson–Bloch–Kato conjecture in analytic rank 1............... 3 1.2.