AWS 2018, Problem Session (Algebraic aspects of Iwasawa theory)

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1 AWS 2018, Problem Session (Algebraic aspects of Iwasawa theory) Kâzım Büyükboduk March 3-7, 2018 Contents 1 Commutative Algebra 1 2 Classical Iwasawa Theory (of Tate motives) 2 3 Galois cohomology and Selmer groups 4 4 Iwasawa Invariants of Elliptic Curves 7 A Selmer groups: Definitions and Examples 8 A.1 Greenberg local conditions and Selmer groups Commutative Algebra 1.1. For a positive integer m, we let µ m denote the mth roots of unity. Prove that the integral closure O of Z in Q(µ p ) := n Q(µ pn) is not a Dedekind ring Suppose Γ = Z p and let Λ := Z p [[Γ]]. If 0 M N K 0 is a short exact sequence of Λ-modules, prove that there exists an exact sequence 0 M Γ N Γ K Γ M Γ N Γ K Γ Let Λ := Z p [[T ]] and f = i=0 a it i Λ with a

2 Prove that f Q p [[T ]] Write f 1 = i=0 b it i with b i Q p. Prove that ord p (b i ) (i + 1)ord p (a 0 ) Conclude that Frac(Λ) is not contained in the Laurent series ring Q p ((T )) Suppose that R is a complete local Noetherian domain Prove that pseudo-isomorphism is an equivalence relation in the category of finitely generated R-modules Suppose that 0 M N K 0 is a short exact sequence of finitely generated torsion R-modules. If there no height-one primes of R contained in supp(m) supp(k), then N is pseudoisomorphic to M K Let Λ = Z p [[T ]] and M be a finitely generated torsion Λ-module. Suppose that we are given a sequence g n Λ of distinguished polynomial such that lim deg g n =. n If the sequence {dim Fp M/(g n, p)m} n is bounded, prove that µ(m) = Let K be any field that contains 2pth roots of unity and let K /K denote its cyclotomic Z p -extension. Set Γ K := Gal(K /K) = Z p and Λ := Z p [[Γ K ]]. Let γ denote a topological generator of Γ K and identify Λ with Z p [[T ]] via γ 1 + T. Set u := χ cyc (γ) Prove that u 1 + qz p, where q = p if p > 2 or q = 4 otherwise For any f Z p [[T ]], we may consider M := Λ/fΛ as a G K -module. Let M(i) denote its ith Tate-twist. Prove that we have an isomorphism of Λ-modules. M(i) Λ/f(u i (1 + T ) 1)Λ For a finitely generated Λ-module M, prove that M(i) Γ K is finite for all but finitely many i. 2 Classical Iwasawa Theory (of Tate motives) 2.1. Find a number field K and a prime p such that the first layer K 1 /K in the cyclotomic Z p -tower of K is unramified Suppose L/K is a finite extension of number fields If Leopoldt s conjecture is valid for L and p, prove that it also holds true for K and p. 2

3 Suppose L is a CM extension and K := L + its maximal totally real subfield. If the Leopoldt conjecture holds for K and p, prove that it is also true for L and p Prove that Leopoldt s conjecture for abelian extensions of Q follows from Leopoldt s conjecture for totally real abelian extensions Let K be a number field and K /K a Z p -extension of K. For n 0, let K n /K denote the unique subsextension of K /K that has degree p n over K, set Γ n = Gal(K /K n ) and let A n denote the p-sylow subgroup of the ideal class group of K n. Let L denote the maximal abelian unramified pro-p extension of K and let X := Gal(L /K ) denote its Galois group over K Prove that µ(x) = 0 if and only if {dim Fp A n /pa n } is bounded as n tends to infinity Calculate µ(x) and λ(x) for K = Q (and for the obvious choice of K ) Suppose there is only one prime of K that ramifies in K /K and that prime is totally ramified. Prove that X Γn A n and that A 0 = 0 if and only if A n = 0 for all n Find a number field K and a prime p such that the ν-invariant for the cyclotomic Z p -tower is non-zero Suppose Γ Z p and for each non-negative integer n, set Γ n := Γ pn. Let Λ = Z p [[Γ]] and let M be a finitely generated torsion Λ-module with characteristic polynomial f. Prove that the following three conditions are equivalent: (i) M Γn (ii) M Γn is finite. is finite. (iii) f(ζ 1) is non-zero for any ζ µ p n. When this is the case, prove that M Γn M Γn = p µ(m)pn ζ µ p n f(ζ 1) p Let F be an imaginary quadratic field and p be an odd prime. Let h (p) F denote the order of the p-sylow subgroup of the ideal class group of F. Let F /F denote the cyclotomic Z p -extension of K and X the Galois group of the maximal abelian unramified pro-p extension of F over F. Let f X (T ) denote any generator of the characteristic ideal of X. Suppose the prime p is either inert or ramified in F/Q Prove that f X (0) Prove that ord p h (p) F = ord pf X (0). 3

4 2.7. Let F be an imaginary quadratic field and p be an odd prime. Let h (p) F denote the order of the p-sylow subgroup of the ideal class group of F. Let F /F denote the cyclotomic Z p -extension of K and X the Galois group of the maximal abelian unramified pro-p extension of F over F. Let f X (T ) denote any generator of the characteristic ideal of X. Suppose the prime p is split in F/Q Prove that f X (T ) has a simple zero at T = Suppose F = Q(i). Calculate f X (0) More generally, calculate the ratio f X (0)/h(p) F With the notation of Problem 2.5, let N be a finitely generated Λ-module. Prove for sufficiently large n that rank Zp N Γn = p n rank Λ N + c for some constant c that doesn t depend on n Let K be a number field and K /K its cyclotomic Z p -extension. Let X K denote the Galois group of the maximal abelian unramified pro-p extension of K over K. Set µ K := µ(x K ). Let L be a finite extension of K. If µ L = 0, prove that µ K = 0 as well Let K be a number field and let S denote a set of places of K, containing all primes above p and places above. Let K /K be a Z p -extension of K and let X S denote the Galois group of the maximal abelian pro-p extension of K unramified outside S. Prove that the weak Leopoldt conjecture for K /K holds true if and only if rank Λ X S = r 2 (K) Suppose p is odd, K is a totally real number field and K = K(µ p ). Let S and X S be as in the previous problem. If µ K = 0 (with the notation of Problem 2.9), prove then that µ(x S ) = 0 for all choices of the finite set S (containing all primes of K above p and ). 3 Galois cohomology and Selmer groups 3.1. Let n be a positive integer and let µ p n denote the Galois module of p n th roots of unity Let F be any field. Prove that H 1 (F, µ p n) = F /(F ) pn Conclude that H 1 (F, Z p (1)) = F := lim F /(F ) pn, the p-adic completion of F Let l be a prime and suppose F is a finite extension of Q l. Calculate H 0 (F, µ p n) H 1 (F, µ p n) + H 2 (F, µ p n). 4

5 3.2. Suppose F is any field. Let O be the ring of integers of a finite extension of Q p and let χ be an O-valued character of G F that has finite prime-to-p order. Let L denote the finite extension of F, which is the fixed field of ker(χ). We write O(χ) to denote the free O-module of rank one on which G F acts by χ. Set T n := µ p n O(χ 1 ). Observe that T n = O(χ)/p n O(χ) Prove that H 1 (F, T n ) = ( L /(L ) pn ) χ, where for an O[[GF ]]-module M, we write M χ for the χ-isotypic component of its p-adic completion Prove that H 1 (F, T n) = Hom(G F, Z/p n Z) χ Set T := lim T n = Z p (1) O(χ 1 ) and T = lim T n = O(χ) Q p /Z p. Show that H 1 (F, T ) = (L ) χ and H 1 (F, T ) = Hom(G L, Q p /Z p ) χ Let l be a prime and suppose F is a finite extension of Q l. Let L be given as above and let λ denote its maximal ideal. Let I L G L denote the inertia subgroup so that the Galois group Gal(L ur /L) of the maximal unramified extension of L equals G L /I L. Prove that H 1 ur(f, T ) = Hom(G L /I L, Q p /Z p ) χ 1, the module of unramified homomorphisms into Q p /Z p In the situation of previous example, use local class field theory to conclude that H 1 ur(f, T ) = U χ L, where U L F is the subgroup of units Suppose F is a number field and all other notation in this problem follows that we have set in Problem 3.3. Let F can denote the canonical Selmer structure on T, as defined in Appendix A For each prime λ Σ(F can ) of F that does not lie above p, prove that H 1 F can (F λ, T ) = q λ U χ L q ; H 1 F can (F λ, T ) = q λ Hom(G Lq /I Lq, Q p /Z p ) χ 1. Hint: You may wish to figure out what semi-local Shapiro s lemma should look like Deduce that H 1 F can (F, T ) = (O L [1/p] ) χ and H 1 F can (F, T ) = Pic(O L [1/p]) χ Suppose that χ(p) 1 for primes p of F lying above p. Prove that ( OL [1/p] ) χ = ( O L ) χ and Pic(O L [1/p]) χ = Cl(L) χ. 5

6 Conclude that rank O HF 1 can (F, T ) rank O HF 1 (F, T ) = r can 1 (F ) + r 2 (F ) + 1 p p,χ(p)=1 if χ is non-trivial, and otherwise, that rank O HF 1 can (F, T ) rank O HF 1 (F, T ) = r can 1 (F ) + r 2 (F ) + g(f ) 1 where g(f ) is the number of primes of F that lie above p Prove that for T := Z p (1) Λ cyc we have rank Λcyc HF 1 Λ (F, T) rank Λcyc HF 1 (F, Λ T ) = r 1 (F ) + r 2 (F ) ( = rank Zp IndF/Q Z p (1) ) 3.5. Let F str denote the Selmer structure on T := Z p (1) Λ cyc given as in Example A Given a number field F, let Ô, F denote the kernel of Ô F p Ô F. Prove that HF 1 str (K, T) = lim Ô, n K n. In particular, weak Leopoldt conjecture in this set up is equivalent to the requirement that HF 1 str (K, T) = Let S denote the set of places of K above p and and let X S denote the module introduced in Problem 2.10 (with K = K cyc, the cyclotomic Z p - extension). Prove that H 1 F str (K, T ) = XS Prove that rank Λcyc H 1 F str (K, T) rank Λcyc H 1 F str (K, T ) = r 2 (F ). Remark. As you will learn in J. Coates lectures, Iwasawa has proved Leopoldt s conjecture for cyclotomic Z p -extensions. This, together with Problem 2.10, is a refinement of the equality above Suppose T is a free Z p -module of finite rank, which is endowed with a continuous G Q -action such that the residual representation T/pT is absolutely irreducible, non-trivial and unramified outside a finite set of primes. Prove that rank Zp H 1 F can (Q, T ) rank Zp H 1 F can (Q, T ) = rank Zp T + corank Zp (H 0 (Q p, T )) and explain how this generalizes the conclusion of Problem Suppose T is as in Problem 3.6. Set T := T Λ cyc and let F Λ denote the Selmer structure on T we have introduced in Appendix A. 6

7 Prove for l p that H 1 (Q ur l, T) = Prove that rank Λcyc H 1 F Λ (Q, T) rank Λcyc H 1 F Λ (Q, T ) = rank Zp T =: d (T ) We say that weak Leopoldt conjecture for T holds if H 1 F Λ (Q, T ) is torsion (in view of Problem 3.7.2, this is equivalent to the requirement that rank Λcyc H 1 F Λ (Q, T) = d (T )). If H 0 (Q, T ) = 0 and weak Leopoldt conjecture holds for T, prove that H 1 F Λ (Q, T) is in fact a free Λ cyc -module of rank d (T ). 4 Iwasawa Invariants of Elliptic Curves 4.1. Assume that p is an odd prime and E /Q is an elliptic curve with ordinary reduction at p. Assume also that E[p ] contains a G Q -invariant subgroup Φ of order p, which is either ramified at p and even; or unramified at p and odd. Let Σ denote the set of places that consists of all bad primes for E, the prime p and the infinite place Let Ψ := E[p]/Φ denote the one-dimensional F p -vector space. Prove that the G Q -action on Ψ is either unramified at p and odd (in case Φ is ramified at p and even) or else it is ramified at p and even (in case Φ is unramified at p and odd) Suppose that M is a one-dimensional F p -vector space which is endowed with a G Q -action. For each prime v of Q above l p, prove that H 1 (Q,v, M) has finite cardinality Let M be as in the previous problem. Suppose G Q acts on M via the character χ. Set H 1 ur(q Σ/Q, M) denote the collection of everywhere unramified cohomology classes. Prove that H 1 ur(q Σ /Q, M) = Hom(X χ, M) where X = Gal(L /F ), with F = F Q and F = Q ker χ ; L the maximal abelian unramified pro-p extension of F Show that Ferrero-Washington theorem implies Hur(Q 1 Σ /Q, M) for M as in the previous two problems Prove that H 1 (Q Σ /Q, E[p]) H 1 (Q Σ /Q, E[p ])[p] Prove that µ E = Use Mazur s theorem to determine all possible primes p that the conditions of this problem can apply Let p be an odd prime and suppose E /Q is an elliptic curve with good ordinary reduction at p. Suppose E(Q)[p] = 0. 7

8 Prove that E(K)[p] = 0 for every pro-p extension K/Q Prove that all three arrows lim n HF 1 (Q n, E[p]) lim H 1 Gr n F (Q n, E[p ])[p] H 1 Gr F (Q, T p(e) )[p] Gr are isomorphisms. Conclude that the following assertions are equivalent: µ E = 0. lim n H 1 F Gr (Q n, E[p]) has finite cardinality. The sequence of integers {#H 1 F Gr (Q n, E[p])} n is bounded Let Σ denote the set of places that consists of all bad primes for E, the prime p and the infinite place. Choose any subset Σ 0 of Σ which contains all places of bad reduction, but does not contain p and. We define Sel Σ 0 E[p ] (Q ) to denote the Selmer group by relaxing all local requirements at primes l Σ 0. Define similarly Sel Σ 0 E[p] (Q ) Prove that Sel Σ 0 E[p] (Q ) Sel Σ 0 E[p ] (Q )[p] Prove that Sel Σ 0 E[p ] (Q ) is Λ-torsion and its µ-invariant equals µ E Suppose that E 1 and E 2 are two elliptic curves defined over Q, both with ordinary reduction at p. Suppose moreover that E 1 [p] = E 2 [p] as G Q -modules. Let Σ denote the set of places that consists of all bad primes for E 1 and E 2, the prime p and the infinite place. Choose any subset Σ 0 of Σ which contains all places of bad reduction both for E 1 and E 2, but does not contain p and Prove that the module Sel Σ 0 E 1 [p ] (Q )[p] has finite cardinality if and only if Sel Σ 0 E 2 [p ] (Q )[p] does Deduce that µ E1 = 0 if and only if µ E2 = 0. A Selmer groups: Definitions and Examples Let K be a number field and let R be a complete local noetherian Z p -algebra. Let M be a R[[G K ]]-module which is free of finite rank over R. Definition A.1. A Selmer structure F on M is a collection of the following data: A finite set Σ(F) of places of K, including all infinite places and primes above p as well as all primes where M is ramified. For every λ Σ(F) a local condition on M, i.e., a choice of R-submodule H 1 F(K λ, M) H 1 (K λ, M). 8

9 Definition A.2. The semi-local cohomology group at a rational prime l is defined by setting H i (K l, M) := H i (K λ, M), λ l where the direct sum is over all primes λ of K lying above l. Let λ be a prime of K. There is the perfect local Tate pairing, λ : H 1 (K λ, M) H 1 (K λ, M ) H 2 (K λ, µ p ) Q p /Z p, where M := Hom(M, µ p ) stands for the Cartier dual of M. For a Selmer structure F on M, define H 1 F (K λ, M ) := H 1 F (k λ, M) as the orthogonal complement of H 1 F (K λ, M) with respect to the local Tate pairing. The Selmer structure F on M, with Σ(F) = Σ(F ), defined in this way will be called the dual Selmer structure. Definition A.3. If F is a Selmer structure on M, we define the Selmer module HF 1 (K, M) as HF(k, 1 M) := ker H ( 1 Gal(K Σ(F) /K ), M) H 1 (K λ, M) / HF(K 1 λ, M), λ Σ(F) where K Σ(F) is the maximal extension of K which is unramified outside Σ(F). We also define the dual Selmer structure in a similar fashion; just replace M by M and F by F above. Example A.4. (i) Let R = Z p and let M be a free R-module endowed with a continuous action of G K, which is unramified outside a finite set of places of K. We define a Selmer structure F can on M by setting and Σ(F can ) = {λ : M is ramified at λ} { p} {v }, if λ Σ(F can ), λ p, we define the local condition at λ to be H 1 F can (K λ, M) = ker(h 1 (K λ, M) H 1 (K ur λ, M Q p )), where K ur λ is the maximal unramified extension of K λ; if p, we define the local condition at by setting H 1 F can (k, M) = H 1 (k, M). The Selmer structure F can is called the canonical Selmer structure on M. 9

10 (ii) Let now R = Λ be the cyclotomic Iwasawa algebra and let M be as in the previous example. We define a Selmer structure F Λ on M := M Λ (which we endow with diagonal Galois action) by setting Σ(F Λ ) = F can and HF 1 Λ (K λ, M) = H 1 (K λ, M) for every λ Σ(F Λ ). A trivial extension of Problem tells us that for any λ p, HF 1 Λ (K λ, M) = ker ( H 1 (K λ, M) H 1 (Kλ ur, M) ) consists of unramified classes. As above, we then have a dual Selmer structure F Λ on M := Hom(M, µ p ). In explicit terms, it is given by the local conditions H 1 F Λ (K λ, M ) = 0 for λ Σ(F can ). (iii) For a general R (of characteristic zero) and M, we let F str denote the strict Selmer structure, which we define by setting Σ(F str ) = {λ : M is ramified at λ} { p} {v }, and if λ Σ(F str ), λ p, we define the local condition at λ to be H 1 F str (K λ, M) = ker(h 1 (K λ, M) H 1 (K ur λ, M[1/p])), if p, we define the local condition at by setting H 1 F can (k, M) = 0. Definition A.5. Suppose that F is a local condition (at a prime λ of K) on M. If M is a submodule of M (resp. M is a quotient module), then F induces local conditions (which we still denote by F) on M (resp. on M ), by taking H 1 F (k λ, M ) (resp. H 1 F (k λ, M )) to be the inverse image (resp. the image) of H 1 F (k λ, M) under the natural maps induced by M M, M M. Propagation of a local condition F on M to a submodule M (and a quotient M ) of M is the local condition F on M (and on M ) obtained following the above procedure. For example, if I is an ideal of R, then a local condition on M induces local conditions on M/IM and M[I], by propagation (which is customarily denoted by the same symbol). A.1 Greenberg local conditions and Selmer groups For the sake of simplicity (and since we do not offer anything original in these notes anyway and there are certainly better places to learn more about this material we shall present), we shall only concentrate in the case when K = Q. 10

11 Definition A.6. Let O be the ring of integers of a finite extension of Q p. Suppose T is a free O-module of finite rank which is endowed with a continuous action of G Q, that is unramified outside a finite set of places Σ (which we assume to contain p and ). Set V := T Q p and A := V/T. We say that V (or T ) is geometric if V is de Rham as a G Qp -representation. A geometric Galois representation V (or T ) is critical if the number of positive Hodge-Tate weights of V GQp (counted with multiplicities) equals the dimension of the +1-eigenspace acting on V. Suppose h + denotes the number of positive Hodge-Tate weights of V GQp (counted with multiplicities). We say that V satisfies the Panchishkin condition if T contains a G Qp - stable direct summand F p + T which has rank h + over O and whose all Hodge-Tate weights are positive as well as that the Hodge-Tate weights of T/F p + T are all non-negative. A critical V that satisfies Panchishkin condition is called Panchishkin-ordinary. Exercise A.7. Suppose f is a newform of weight k and let K f denote its Hecke field, a fixed prime of K f and finally, M f, its -adic Galois representation (with Hodge-Tate weights 0 and k 1) attached to f by Deligne. If ord a p (f) = 0, then it is known thanks to Wiles that M f, is Panchishkin-ordinary. Determine all integers j such that M f (j) is Panchishkin-ordinary. Exercise A.8. Suppose f and g are two newforms of respective weights k and l. Let F denote a common extension of K f and K g, and let be a prime of F. Let M f, and M g, denote Deligne s Galois representations. Suppose ord a p (f)a p (g) = 0. Determine all integers j such that M f, M g, (j) is Panchishkin-ordinary. Definition A.9 (Compact Greenberg Selmer groups). Suppose V is a Panchishkinordinary G Q -representation and T V is a G Q -stable lattice. Set F p T := T/F + p T. We define the compact Greenberg Selmer group HF 1 Gr (Q, T ) by setting ( HF 1 Gr (Q, T ) := ker H 1 (Q Σ /Q, T ) ) H 1 (Q ur l, V ) H 1 (Q p, Fp T ). l p, l Σ Likewise, set F p T := F p T Λ and define the compact Iwasawa theoretic Greenberg Selmer group by setting Note that H 1 F Gr (Q, T) := ker ( H 1 (Q, T) H 1 (Q p, F p T) ). H 1 (Q ur l, T) = 0, l p according to Problem 3.7.1, so the classes that in H 1 F Gr (Q, T) are unramified everywhere (away from p). 11

12 Definition A.10 (Discrete Greenberg Selmer groups). We retain the notation in the previous definition. We define the discrete Greenberg Selmer group HF 1 (Q, T ) as the Gr dual Selmer group, in the sense of Definition A.3. In more precise form, H 1 F Gr (Q, T ) := ker ( H 1 (Q Σ /Q, T ) l p, l Σ H 1 (Q l, T ) H 1 f (Q l, T ) H1 (Q p, F p T ) ). where H 1 f (Q l, T ) := im ( ker ( H 1 (Q l, V ) H 1 (Q ur l, V ) ) H 1 (Q l, T ) ) and F p T := ( F + p T ). Likewise, set Fp T := ( F p + T ) and define the discrete Iwasawa theoretic Greenberg Selmer group by setting ( HF 1 Gr (Q, T ) := ker H 1 (Q, T ) ) H 1 (Q l, T ) H 1 (Q p, Fp T ). Exercise A.11. Prove that l p, l Σ H 1 f (Q l, T ) H 1 ur(q l, T ) := ker ( H 1 (Q l, T ) H 1 (Q ur l, T ) ) ( ) H 0 and the quotient Hur(Q 1 l, T )/Hf 1(Q l, T (I l, T Frl =1 ) ) is isomorphic to. H 0 (I l, T ) div The cardinality of the latter Z p -module is precisely the definition of the p-part of the Tamagawa number (due to Fontaine and Perrin-Riou). Remark A.12. Greenberg has defined his Iwasawa theoretic Selmer group Sel T (Q ) p by setting Sel T (Q ) p := ker H1 (Q Σ /Q, T ) H 1 (I η, T ) H 1 (I,p, Fp T ) where I η is the inertia subgroup in G Q,η for each prime η of Q as above and likewise, I,p is the inertia group in G Q,p at the unique prime of Q above p (that we also denote by p, by slight abuse). See Exercise A.15 where this is compared to the Selmer groups H 1 F Gr (Q, T ) introduced above in the particular case when V = V p (E) (or more generally, when V = M f, ). η l p l Σ Exercise A.13. For T as above, prove that the natural map Hom(Λ, T ) =: Hom(Λ, Hom(T, µ p )) Hom(T Λ, µ p ) =: T is an isomorphism of Λ[[G Q ]]-modules. 12

13 Exercise A.14. Using the fact that the restriction maps induce an isomorphism lim H 1 (Q Σ /Q n, T ) res H 1 (Q Σ /Q, T ) n is an isomorphism, prove that H 1 (Q Σ /Q, T ) is naturally isomorphic to H 1 (Q Σ /Q, T ). Exercise A.15. Suppose E /Q is an elliptic curve with ordinary reduction at p and set T = T p (E), T = E[p ] (HW: Use Weil pairing to check that this choice of notation is consistent with our previous definitions). In this case, it follows from a theorem of Lutz that Sel T (Q ) p := ker H1 (Q Σ /Q, T ) H 1 (Q,η, T ) H 1 (I,p, Fp T ). Prove that we have an exact sequence η l p l Σ 0 H 1 F Gr (Q, T ) Sel T (Q ) p M p where the module M p depends only on F p T and it has finite cardinality unless E has split multiplicative reduction at p. In this case, deduce that the Iwasawa µ and λ-invariants of H 1 F Gr (Q, T ) equal to µ E and λ E, respectively. Prove analogous assertions for a general newform f that is ordinary at p (this once, relying on the local-global compatibility of Langlands correspondence). 13