aa r X i v : . [ m a t h . N T ] J a n SELMER GROUPS OF ELLIPTIC CURVES IN DEGREE p EXTENSIONS
JULIO BRAU
Abstract.
We study the growth of the Galois invariants of the p -Selmer group of anelliptic curve in a degree p Galois extension. We show that this growth is determinedby certain local cohomology groups and determine necessary and sufficient conditionsfor these groups to be trivial. Under certain hypotheses this allows us to give necessaryand sufficient conditions for there to be growth in the full p -Selmer group in a degree p Galois extension. Introduction
Let K be a number field and E/K an elliptic curve defined over K . For a positive integer n , the n -Selmer group of E over K is defined by Sel n ( E/K ) = ker (cid:16) H ( K, E [ n ]) −→ Y v H ( K v , E ( K v )) (cid:17) where v runs over all places of K , and E [ n ] denotes E ( K )[ n ] . For p an odd prime, let L/K be a Galois extension of degree p with Galois group G . Then there is a natural actionof G on the p -Selmer group of E/L , denoted by
Sel p ( E/L ) . In this paper we discuss thesize of Sel p ( E/L ) G . Growth of the p -Selmer group in Galois extensions has been studiedby several authors, often with the aim of obtaining unboundedness results for the Tate-Shafarevich group (see [Bar10], [Bar13], [Č12], [Klo05], [KS03]). Indeed, the unboundednessof dim F p X ( E/ Q )[ p ] has been shown for primes p or p = 13 (see [Cas64], [Fis01], [Kra83]and [Mat07]). In [Mat09], Matsuno showed that as E varies over elliptic curves over Q , the F p -dimension of the Tate-Shafarevich group, and in particular the p -Selmer group of E overa fixed cyclic degree p extension of Q can be arbitrarily large.In this paper we will primarily be concerned with studying the growth of the p -Selmergroup of an elliptic curve in a degree p Galois extension, and in particular determiningthe exact F p -dimension of the Galois invariants of the p -Selmer group. As an example in[Dok07], Tim Dokchitser analyzes the growth of the -Selmer group of X (11) in cubicextensions of the form K ( √ m ) , where K = Q ( ζ ) . He shows that the -Selmer group growsif and only if | m or there exists a prime v of K such that v | m and ˜ E ( k v )[3] = 0 . Thisis done by applying a formula of Hachimori and Matsuno for the λ -invariant in p -powerGalois extensions (Theorem 3.1 of [HM99] and Corollaries 3.20, 3.24 of [DD05]). If we let K = Q ( ζ p ) , this same method can be applied to give necessary and sufficient conditions forhaving growth of the p -Selmer group in extensions of the form K ( p √ m ) for elliptic curveswith ordinary reduction at primes above p satisfying certain additional conditions, amongthem having trivial p -Selmer group over K and trivial cyclotomic Euler characteristic.As a corollary of our main result, we give necessary and sufficient conditions for havinggrowth of the p -Selmer group in the same setting as above, however without any assumptionon the Euler characteristic and without requiring ordinary reduction above p . Indeed, ourresults are mainly concerned with computing the exact F p -dimension of Sel p ( E/L ) G as best s we can, as opposed to questions of unboundedness. This is illustrated by the followingsimpler version of our main result, for which we establish the following notation.Let E/ Q be an elliptic curve over the rationals with j -invariant j := j ( E/ Q ) . Let ℓ be aprime at which E has semistable reduction and m a positive integer. Let s > be such that ℓ s k m , that is, such that m = ℓ s d with d coprime to ℓ . Since E is semistable at ℓ it followsthat we may write j = ℓ − n ab with n a positive integer and a and b coprime to ℓ . Then define(1) u ℓ,m := d n (cid:18) ab (cid:19) s . Theorem 1.1.
Let p be an odd prime and E a semistable elliptic curve over Q with goodreduction at p . Let K = Q ( ζ p ) and suppose that Sel p ( E/K ) is trivial. Consider the familyof p -extensions L m = K ( p √ m ) . Then dim F p Sel p ( E/L m ) Gal( L m /K ) = X v δ v with v running over all primes of K and the δ v are given by the following table:Reduction type of E at v v ramified in L m /K v inert in L m /K v split in L m /K good ordinary, v | p δ v = ( or if ˜ E ( k v )[ p ] = 00 otherwise δ v = 0 good supersingular, v | p δ v = ( p − if p | m otherwise δ v = 0 good, v ∤ p δ v = dim F p ˜ E ( k v )[ p ] δ v = 0 split multiplicative, v ∤ p δ v = ( if u qv − p ℓ v ,m ≡ ℓ v otherwise δ v = ( if p | c v otherwise δ v = 0 nonsplit multiplicative, v ∤ p δ v = 0 Here c v is the Tamagawa number of E/K v , ℓ v is the prime in Q below v and q v is the size of the residue field of K v . We remark that the sum given in the theorem is finite, as there are only finitely many v such that v is ramified in L m /K or p | c v , and δ v = 0 for all other primes. Note also thatthe contribution of the δ v coming from primes v | p of good supersingular reduction growsarbitrarily large with p , and this is the only type of reduction for which this phenomenonoccurs. This theorem essentially follows from the more general scenario of consideringarbitrary Galois extensions of degree p . However in the more general setting, we are onlyable to obtain upper and lower bounds for the contribution of supersingular primes to the F p -dimension of Sel p ( L/K ) G . In order to state the more general case we use the followingnotation. Suppose L/K is a Galois extension of degree p with Galois group G , and let w bea prime of L . Then denote by π w a uniformiser for the completion L w . Let σ be a generatorfor G , and set t w = ord w ( σ ( π w ) − π w ) − . Recall then that t w is the largest index forwhich the higher ramification group of L w /K v is non-trivial, where v is a prime of K below w . When v is (totally) ramified in L , then t w is related to the discriminant ∆ L/K via theformula ( t w + 1)( p −
1) = ord v (∆ L/K ) . or more details see section 3.3. Now suppose v ∤ p and E has semistable reduction at v .Then we may write L w = K v ( p √ π v ) where π v is a uniformiser in K v (see section 5.1). Writing O v for the ring of integers of K v , we let u v be the unit in O v such that j ( E/K v ) = π − mv u v ,where − m = ord v ( j ( E )) . Theorem 1.2.
Let p be an odd prime and E a semistable elliptic curve over a number field K with good reduction at all primes above p . Let L/K be a Galois extension of degree p .Suppose also that Sel p ( E/K ) is trivial. Then dim F p Sel p ( E/L ) Gal(
L/K ) = X v δ v where the δ v are as in Theorem 1.1, except for the following two cases: (i) If v ∤ p is a prime of split multiplicative reduction, we have that δ v = ( if u qv − p v ≡ π v otherwise . (ii) If v | p is a prime of good supersingular reduction, we have that δ v = 0 ⇐⇒ t w = 1 where w is a place of L above v . Further, for t w > we have f v δ v min { f v ( t w − , [ K v : Q p ] + 2 } where f v is the inertia degree of v in K/ Q . Corollary 1.3.
Under the assumptions of Theorem 1.2, we have that
Sel p ( E/L ) is trivialif and only if δ v = 0 for all v . Example 1.4.
Let E/ Q be the elliptic curve 17a1 in Cremona’s database. It is given bythe Weierstrass equation Y + XY + Y = X − X − X − . Denote K = Q ( ζ ) . Thiscurve satisfies E ( K ) ≃ Z / Z and X ( E/K )[3] = 1 , hence
Sel ( E/K ) is trivial. Let us useTheorem 1.1 and Corollary 1.3 to determine necessary and sufficient conditions for thereto be change in 3-Selmer in extensions of the form L m = K ( √ m ) . First note E has goodsupersingular reduction at and split multiplicative reduction at . Since q v = 17 and u ,m is already defined over Q , it follows that u qv − ,m ≡ π v ) , hence there is -Selmer growth in every extension K ( √ m ) for all cube-free m divisible by . It follows from Theorem 1.1 and Corollary 1.3 that Sel ( E/L m ) is trivial if and only ifthe following conditions are satisfied:(i) ∤ m .(ii) ∤ m .(iii) There does not exist a prime v of K such that v | m and ˜ E ( k v )[3] = 0 .Note that there are infinitely many cube-free m satisfying the above three conditions andhence there are infinitely many extensions of the form L m such that rk E/L m = 0 . Weremark that there are also infinitely many v such that ˜ E ( k v )[3] = 0 . This can be seen forexample by considering primes which split completely in K ( E [3]) . Incidentally this showsthat dim F Sel ( E/L m ) is unbounded as m varies over cube-free integers. The first fewprimes v for which ˜ E ( k v )[3] = 0 are the primes above , and . emark 1.5. Note that the conditions obtained in the previous example are compatiblewith what is predicted by the global root number. Indeed, for m divisible by paritypredicts that rk E/ Q ( √ m ) is odd and hence positive. It follows that the rank and hence -Selmer should grow in every extension of the form K ( √ m ) for cube-free m divisible by , which is what we had previously obtained.The idea of the proofs is to relate the F p -dimension of the Galois invariants of Selmer tothe sum of the F p -dimensions of certain local cohomology groups which will be introducedin section 4. This is accomplished using a result of Mazur and Rubin which uses Cassels-Poitou-Tate global duality. Indeed, using this we obtain that dim F p Sel p ( E/L ) G = X v δ v where the δ v are the F p -dimensions of the above mentioned local cohomology groups. Section5 is almost entirely devoted to computing the F p -dimensions of these local cohomologygroups which in many cases is equivalent to computing local norm indices. Of course the δ v depend heavily on the type of reduction of E at v as well as the splitting behavious of v in L . In section 3 we study the local norm map of formal groups in cyclic Galois extensions, asthis is used in section 5 to bound the δ v coming from supersingular primes. Theorem 1.2 isproved in section 5, along with Corollary 1.3. As mentioned earlier, Theorem 1.1 essentiallyfollows from Theorem 1.2, except that because of the nature of the extensions consideredthere we are able to obtain more precise information. In section 6 we finish the proof ofTheorem 1.1. Acknowledgements.
I would like to thank Tim Dokchitser and Peter Stevenhagen forthe constant guidance and support, as well as many helpful discussions and suggestions.I am also grateful to Vladimir Dokchitser for many discussions regarding the material ofthe paper. I also thank Alex Bartel, Kestutis Česnavičius and Adam Morgan for helpfulconversations. 2.
Background and notation
In this section we recall some properties of Selmer groups of elliptic curves (see e.g. [Sil09],§X), as well as establish some notation.If K is a number field we will write G K for the absolute Galois group Gal(
K/K ) . For G a profinite group and A a discrete G -module, we write H n ( G, A ) to denote the coho-mology groups of G formed with continuous cochains. As usual, if A is a G K -module, wewrite H n ( K, A ) to denote the cohomology group H n ( G K , A ) , and sometimes we will write H n ( L/K, A ) to denote H n (Gal( L/K ) , A ) . We will use inf , res and cor to denote the in-flation, restriction and corestriction maps of cohomology, respectively. For M an abeliangroup and n a positive integer, we write M [ n ] to denote the n -torsion subgroup of M , thatis, the subgroup of elements of M annihilated by n . If M is a torsion abelian group and p is a prime, we denote by M [ p ∞ ] = ∪ n M [ p n ] the p -primary component of M .As mentioned in the introduction, the n -Selmer group of E over K is defined by Sel n ( E/K ) = ker (cid:16) H ( K, E [ n ]) −→ Y v H ( K v , E ( K v )) (cid:17) . The n -Selmer group fits into the exact sequence −→ E ( K ) /nE ( K ) −→ Sel n ( E/K ) −→ X ( E/K )[ n ] −→ . Proposition 2.1.
Let p be a prime let S be a finite set of primes of K containing allarchimidean primes, all primes dividing p and all primes where E has bad reduction. Then Sel p ( E/K ) = ker (cid:16) H ( K S /K, E [ p ]) −→ Y v ∈ S H ( K v , E ( K v ))[ p ] (cid:17) . where K S denotes the maximal extension of K unramified outside S .Proof. This is [Mil06] §I, Corollary 6.6. (cid:3)
Proposition 2.2 (Cassels-Poitou-Tate) . Let E , S and K be as above. Then there is anexact sequence Sel p ( E/K ) −−−−→ H ( K S /K, E [ p ]) −−−−→ Q v ∈ S H ( K v , E ( K v ))[ p ] yQ v ∈ S H ( K v , E ( K v ))[ p ] ←−−−− H ( K S /K, E [ p ]) ←−−−− \ Sel p ( E/K ) y \ E ( K )[ p ] −−−−→ . Proof.
See for instance [CS00] §1.7. (cid:3)
Proposition 2.3.
Let G be a pro- p -group and A a G -module which is uniquely divisible by p . Then H i ( G, A ) = 0 for all i ≥ .Proof. The exact sequence −→ A [ p ] −→ A −→ A −→ yields an exact sequence of cohomology groups H n ( G, A [ p ]) −→ H n ( G, A ) p −→ H n ( G, A ) −→ H n +1 ( G, A [ p ]) . Since A is uniquely divisible by p we have that A [ p ] is trivial and so we obtain an isomorphism [ p ] : H n ( G, A ) ∼ −→ H n ( G, A ) , hence H n ( G, A ) is uniquely divisible by p for all n . But G is a pro- p -group and so everyelement of H n ( G, A ) is annihilated by some power of p for n ≥ so the result follows. (cid:3) The norm map on formal groups
In this section we will analyse the cokernel of the norm map of formal groups in cyclicdegree p extensions. .1. Generalities on formal groups.
For this section we let K be a finite extension of Q p with residue field k and normalized valuation v K . We denote its ring of integers by O K and its maximal ideal by m K . Let F be a one-dimensional commutative formal group lawdefined over O K . Recall this is given by a power series F ∈ O K [[ X, Y ]] which satisfies thefollowing properties:(i) F ( X, Y ) = F ( Y, X ) (ii) F ( X,
0) = F (0 , X ) = X (iii) F ( X, F ( Y, Z )) = F ( F ( X, Y ) , Z ) As usual, we define a group operation on the set m K by x ⊕ F y = F ( x, y ) and we denote the corresponding abelian group by F ( m K ) . When there is no risk of confu-sion we will omit the subscript F from ⊕ F .Define F n ( X , . . . X n ) ∈ O K [[ X , . . . , X n ]] inductively by F ( X , X ) = F ( X , X ) , F n +1 ( X , . . . , X n +1 ) = F ( F ( X , . . . , X n ) , X n +1 ) . Then given a positive integer n , the multiplication by n map on F ( m K ) is simply given bythe power series [ n ] F ( X ) = F n ( X, . . . , X ) . Consider the multiplication-by- p map [ p ] F ( X ) = F p ( X, . . . , X ) mod m K . Recall that (see [Sil09]) the height of F , denoted ht ( F ) is the largestinteger h such that [ p ] F ( X ) ≡ g ( X p h ) mod m K for some power series g ( X ) ∈ O K [[ X ]] ,where we set h = ∞ if [ p ] F ( X ) ≡ m K .3.2. The norm map of a formal group.
Let L be a finite Galois extension of K withGalois group G = { σ , . . . , σ n } . Then we also have an abelian group structure on m L definedusing F as above, and we denote this group by F ( m L ) . Recall that there is natural actionof G on F ( m L ) and that the norm map is defined by N F L/K : F ( m L ) −→ F ( m K ) x σ ( x ) ⊕ · · · ⊕ σ n ( x ) . The following lemma, which appeared in [Haz74], will be quite useful when studying thenorm map on formal groups. In order to state it we need some notation. Given a monomial M = X r . . . X r n n in the variables X , . . . , X n , define Tr( M ) = X r · · · X r n n + X r · · · X r n − n X r n + · · · + X r n · · · X r n n − . Lemma 3.1.
Let ( F , F ) be a formal group as above with height h < ∞ . Then (2) F n ( X , . . . , X n ) = Tr( X ) + ∞ X i =1 a i ( X X · · · X n ) i + X M a M Tr( M ) where M runs over all monomials of degree at least which are not of the form ( X · · · X n ) i .If in addition n = p , then v K ( a i ) > unless i = kp h − with k = 1 , . . . , and v K ( a i ) = 0 when i = p h − .Proof. Equation (2) follows from the fact that F n ( X , . . . , X n ) = F n ( X σ (1) , . . . , X σ ( n ) ) forevery permutation σ of { , . . . , n } . The second statement follows directly from the definitionof height, since reducing mod m K gives something of the form g ( X p h ) . (cid:3) orollary 3.2. Let
L/K be a finite Galois extension of degree n , and let Tr L/K and N L/K denote the usual trace and field norm. Then for all x ∈ F ( m L ) we have N F ( x ) ≡ Tr L/K ( x ) + ∞ X i =1 a i N L/K ( x ) i mod Tr L/K ( x O L ) . If n = p then the a i satisfy the same statements as in Lemma 3.1.Proof. By definition we have N F L/K ( x ) = F n ( σ ( x ) , . . . , σ n ( x )) , so the result follows fromLemma 3.1. (cid:3) Norm map of formal groups in cyclic extensions.
As before K is a finite extensionof Q p and ( F , F ) is a one-dimensional commutative formal group law over O K . Denote by f K the inertia degree of K over Q p , that is, the dimension of k as an F p -vector space. Let L/K be a cyclic totally ramified Galois extension with G ≃ Z /p Z . Let σ be a generator of G and put t = v L ( σ ( π L ) − π L ) − . Then the ramification groups of G are G = G = · · · = G t { } = G t +1 = · · · where G is the inertia group of L/K . Since
L/K is totally ramified of degree p , it is wildlyramified and hence t > . Denote by D = D L/K the different, and let m be defined by theequation D = m mL . Then it is known (see for example [Ser79], §IV.3) that m = ( t + 1)( p − . Note since L/K is totally ramified, m also equals ord K (∆ L/K ) and we have the following lemma. Lemma 3.3.
Let n > be an integer, and set r = ⌊ ( m + n ) /p ⌋ . Then Tr L/K ( m nL ) = m rK . Proof.
Since the trace is K -linear, it follows that Tr L/K ( m nL ) is an ideal of O K . Also, bydefinition of the different we have Tr L/K ( m nL ) ⊂ m rK ⇐⇒ m nL ⊂ D − m rK = m pr − mL ⇐⇒ r ( m + n ) /p. (cid:3) We now give a necessary and sufficient condition for the norm map to be surjective in thecase where the height of the formal group is at least . For this we will need the followinglemma, which is essentially the same as Lemma 2 in [Ser79] §V.1. Lemma 3.4.
Let
A, B be abelian groups filtered by subgroups A = A ⊃ A ⊃ · · · , B = B ⊃ B ⊃ · · · such that A is complete with respect to the topology defined by the A n and T B n = { } . Let u : A → B be a homomorphism and suppose there exist indices t < t < · · · such that u ( A t i ) ⊂ B i and the induced maps A t i → B i /B i +1 are surjective forall i > . Then u is surjective.Proof. Let b ∈ B . Then there exists a ∈ A t and b ∈ B such that u ( a ) = b − b . By thesame argument there exists a ∈ A t and b ∈ B such that u ( a ) = b − b . Continuing inthis manner we obtain a sequence a + a + · · · which converges to a ∈ A since A is complete.For each n > we have that u ( a + a + · · · + a n − ) − b = − b n ∈ B n and a ′ = P ∞ i = n a i ∈ A t n ,so it follows that u ( a ) − b = u ( a + · · · + a n − + a ′ ) − b = u ( a + · · · + a n − ) − b + u ( a ′ ) s in B n . We conclude u ( a ) − b ∈ T B n = 0 , hence u ( a ) = b . (cid:3) Proposition 3.5.
Suppose F has height h > , and let t = v L ( σ ( π L ) − π L ) − . Then N F L/K ( F ( m L )) = F ( m K ) ⇐⇒ t = 1 . Moreover, for t > we have f K dim F p F ( m K ) /N F L/K ( F ( m L )) min { f K ( t − , [ K : Q p ] + h } Proof.
We show first that F ( m tL ) surjects via the norm onto F ( m tK ) . Note that by Lemma3.3 we have Tr L/K ( m tL ) = m tK and Tr L/K ( m t +1 L ) = m t +1 K . Hence we may set t i = ( i − p + t for i > . Then again by the lemma the t i satisfy(i) Tr L/K ( m t i L ) = m t + i − K (ii) Tr L/K ( m t i +1 L ) = m t + iK .Using Corollary 3.2 and the fact that h > we obtain N F ( x ) ≡ Tr L/K ( x ) mod m t + iK for all x in F ( m t i L ) . It follows that the induced maps F ( m t i L ) → F ( m t + i − K ) / F ( m t + iK ) aresurjective for all i > , and the claim follows from Lemma 3.4.From this we immediately obtain that the norm map is surjective for t = 1 . Now supposethat t > . Then keeping the above notation we have m > p − and hence by Lemma 3.3we have Tr L/K ( m L ) ⊂ m K . Let x ∈ F ( m L ) . Then by Corollary 3.2 and using that h > we conclude N F ( x ) ∈ F ( m K ) , hence N F L/K ( F ( m L )) ⊂ F ( m K ) . It follows then that dim F p F ( m K ) / F ( m K ) dim F p F ( m K ) /N F L/K ( F ( m L )) . But F ( m K ) / F ( m K ) ≃ m K / m K , which is a vector space over the residue field k of degree , hence the left hand side of theinequality. For the right hand side, note that from the claim it follows that N F L/K ( F ( m L )) ⊃F ( m tK ) and hence dim F p F ( m K ) /N F L/K ( F ( m L )) dim F p F ( m K ) / F ( m tK ) which has F p -dimension f K ( t − by the same argument as before. Finally, recall that for K a finite extension of Q p , we have an isomorphism of abelian groups F ( m K ) ≃ Z [ K : Q p ] p × T where T is a finite group such that T /pT has F p -dimension at most h . The result thenfollows from the fact that p F ( m K ) ⊂ N F L/K ( F ( m L )) . (cid:3) Note that since G is a cyclic, F ( m K ) /N F L/K ( F ( m L )) is equal to H ( G, F ( m L )) . We nowend this section with a result which will be useful when computing Selmer ranks of ellipticcurves. Proposition 3.6.
Suppose F and L/K are as in this section. Then H ( G, F ( m L )) ≃ H ( G, F ( m L )) . roof. Because both groups are F p -vector spaces, it suffices to show that the Herbrandquotient h G ( F ( m L )) = H ( G, F ( m L )) H ( G, F ( m L )) is equal to 1. In order to show this, recall that for large enough n the formal logarithminduces an isomorphism F ( m nL ) ∼ −→ b G a ( m nL ) with the additive formal group on m L . Thisisomorphism is G -equivariant since F is defined over K . Thus we have a short exact sequenceof G -modules −→ b G a ( m nL ) −→ F ( m L ) −→ C −→ where C is finite. By a well-known property (see for example [Ser79], §VIII.4) of the Her-brand quotient one has h G (cid:0) F ( m L ) (cid:1) = h G (cid:0) b G a ( m nL ) (cid:1) h G ( C ) . Finally, we know that for n divisible by p we have b G a ( m nL ) ≃ O L as G -modules and O L isan induced G -module, hence h G (cid:0) b G a ( m nL ) (cid:1) = 1 . Also h G ( C ) = 1 as C is finite. We concludethen that h G ( F ( m L )) = 1 , and this completes the proof. (cid:3) Twisted Selmer groups of elliptic curves
Let K be a number field and p an odd prime. For E an elliptic curve defined over K ,recall that E [ p ] is a 2-dimensional F p -vector space with a continuous G K -action and with aperfect, skew-symmetric, G K -equivariant self-duality E [ p ] × E [ p ] −→ µ p . A Selmer structure F on E [ p ] is a collection of subspaces H F ( K v , E [ p ]) for every prime v of K such that for all but finitely many v we have H F ( K v , E [ p ]) = H ( K ur v /K, E [ p ] I v ) where K ur v denotes the maximal unramified extension of K v and I v is the inertia group. GivenSelmer structures F and G on E [ p ] , define Selmer structures F + G and F ∩ G by H F + G ( K v , E [ p ]) = H F ( K v , E [ p ]) + H G ( K v , E [ p ]) H F∩G ( K v , E [ p ]) = H F ( K v , E [ p ]) ∩ H G ( K v , E [ p ]) . Given a Selmer structure F on E [ p ] define the Selmer group by H F ( K, E [ p ]) = ker (cid:16) H ( K, E [ p ]) −→ Y v H ( K v , E [ p ]) /H F ( K v , E [ p ]) (cid:17) . Example 4.1.
For each v in K let H F ( K v , E [ p ]) = im( λ E,v ) , where λ E,v : E ( K v ) /pE ( K v ) −→ H ( K v , E [ p ]) is the Kummer map. Then by Lemma 19.3 of [Cas65], H F ( K v , E [ p ]) = H ( K ur v /K, E [ p ]) for all v such that v ∤ p and E has good reduction at v . With this Selmer structure givenon E [ p ] we have that H F ( K, E [ p ]) = Sel p ( E/K ) is the usual p -Selmer group as defined insection 1.Let L/K be a Galois extension of number fields of degree p with Galois group G suchthat G is generated by σ . Denote by Res LK E the Weil restriction of scalars of E from L to K . Then for every K -algebra X there is an isomorphism, functorial in X ,(3) (Res LK E )( X ) ≃ E ( X ⊗ K L ) . The action of G on E ( X ⊗ K L ) induces a canonical inclusion Z [ G ] ֒ → End K (Res LK E ) (seefor instance section 4 of [MRS07]). Let O be the ring of integers of the cyclotomic field of th roots of unity, and p the maximal ideal of O . Let A be the twist of E by the irreduciblerational representation of G corresponding to L . Hence A is the abelian variety denoted E L in Definition 5.1 of [MRS07]. If we let N L/K = X g ∈ G g ∈ Z [ G ] then it follows from Proposition 4.2 in [MRS07] that E = ker( σ − ⊂ Res LK E, A = ker( N L/K ) ⊂ Res LK E. Hence, A is an abelian variety of dimension p − over K , and it is the kernel of the map N L/K : Res LK E → E . We also have from Theorem 3.4 of [MR07] that the following holds:(i) The inclusion Z [ G ] ֒ → End K (Res LK E ) induces a ring homomorphism Z [ G ] → End K ( A ) that factors Z [ G ] ։ O ֒ → End K ( A ) where the first map is induced by the projection in (3.2) of [MR07].(ii) For every commutative K -algebra X , the isomorphism of (3) restricts to an isomor-phism, functorial in X , A ( X ) ≃ { x ∈ E ( X ⊗ K L ) : X g ∈ G (1 ⊗ g )( x ) = 0 } . Proposition 4.2.
There is a canonical G K -isomorphism A [ p ] ∼ −→ E [ p ] .Proof. This is Proposition 4.1 of [MR07]. (cid:3)
Using Proposition 4.2 we may define another Selmer structure A on E [ p ] as follows. Let π be a generator for p . For every place v of K let λ A,v : A ( K v ) /πA ( K v ) ֒ → H ( K v , A [ p ]) denote the Kummer map and set H A ( K v , E [ p ]) to be the image of λ A,v composed with theisomorphism of Proposition 4.2. For each place w of L , let H F ′ ( L w , E [ p ]) be the image ofthe Kummer map E ( L w ) /pE ( L w ) ֒ → H ( L w , E [ p ]) . Lemma 4.3.
Let v be a prime of K , and w a prime of L above v . If res denotes therestriction map H ( K v , E [ p ]) −→ H ( L w , E [ p ]) then we have that ker(res) ⊂ H F + A ( K v , E [ p ]) .Proof. Note that there is nothing to prove if v splits completely in L , so suppose there isone prime w of L above v . It is equivalent to show that the image of the inflation map H ( L w /K v , E ( L w )[ p ]) −→ H ( K v , E [ p ]) is contained in H F + A ( K v , E [ p ]) . The short exact sequence −→ E [ p ] −→ (Res LK E )[ p ] σ − −−−→ A [ p ] −→ induces the exact sequence in cohomology(4) −→ A ( K v )[ p ]( σ − (cid:0) (Res LK E )( K v )[ p ] (cid:1) f −−→ H ( K v , E [ p ]) g −−→ H ( K v , (Res LK E )[ p ]) . ote that (Res LK E )( K v ) ≃ E ( K v ⊗ L ) = E ( L w ) , hence we may identify A ( K v ) with theelements x ∈ E ( L w ) such that N L w /K v ( x ) = 0 } . We have then isomorphisms A ( K v )[ p ]( σ − (cid:0) (Res LK E )( K v )[ p ] (cid:1) ≃ H ( L w /K v , E ( L w )[ p ]) ,H ( K v , (Res LK E )[ p ]) ≃ H ( L w , E [ p ]) where the first isomorphism follows from G being cyclic and the second one is by Shapiro’slemma (see for instance the proof of Proposition 3.1 in [MR07]). Using these identificationsone sees that indeed the maps f and g in (4) are the usual inflation and restriction maps.We now show that the image of f is contained in H F + A ( K v , E [ p ]) . To this end, let α ∈ O × be such that απ p − = p . Let P ∈ A ( K v )[ p ] , and let Q ∈ A ( K v ) be such that π ( Q ) = P .Set P ′ = απ p − P . Then P ′ ∈ A ( K v ) hence ( σ − acts as π on it, so we obtain ( σ − P ′ = π ( P ′ ) = pP = 0 and it follows that P ′ ∈ E ( K v ) . Now let Q ∈ E ( K v ) such that pQ = P ′ . Finally, set Q = Q − Q ∈ (Res LK E )( K v )[ p ] . Then as Q ∈ E we have that ( σ − Q ) = 0 andso ( σ − Q ) = P . It follows that the image of P under f is represented by the cocycle { Q g − Q } = { ( Q g − Q ) − ( Q g − Q ) } . This cocycle also represents λ A,v ( P ) − λ E,v ( απ p − P ) which is contained in H F + A ( K v , E [ p ]) and this completes the proof. (cid:3) Proposition 4.4.
Let v be a prime of K and w a prime of L above v . Then res − (cid:0) H F ′ ( L w , E [ p ]) (cid:1) = H F + A ( K v , E [ p ]) Proof.
The inclusion res − (cid:0) H F ′ ( L w , E [ p ]) (cid:1) ⊃ H F + A ( K v , E [ p ]) follows from considering thefollowing commutative diagram with exact rows −−−−→ E ( K v ) /pE ( K v ) λ E,v −−−−→ H ( K v , E [ p ]) y y res −−−−→ E ( L w ) /pE ( L w ) λ E,w −−−−→ H ( L w , E [ p ]) and the corresponding one for A , −−−−→ A ( K v ) /πA ( K v ) λ A,v −−−−→ H ( K v , E [ p ]) y π p − y res −−−−→ E ( L w ) /pE ( L w ) λ E,w −−−−→ H ( L w , E [ p ]) . For the reverse inclusion, we again suppose there is only one prime w of L above v , asotherwise there is nothing to prove. Take x ∈ res − (cid:0) H F ′ ( L w , E [ p ]) (cid:1) , so that res( x ) ∈ H F ′ ( L w , E [ p ]) . Then res( x ) = λ E,w ( P ) for some P ∈ E ( L w ) . From the commutativediagram −−−−→ E ( K v ) /pE ( K v ) λ E,v −−−−→ H ( K v , E [ p ]) x N Lw/Kv x cor −−−−→ E ( L w ) /pE ( L w ) λ E,w −−−−→ H ( L w , E [ p ]) e obtain that λ E,v (cid:0) N L w /K v ( P ) (cid:1) = cor (cid:0) λ E,w ( P ) (cid:1) = cor(res( x ))= 0 . It follows that N L w /K v ( P ) = pQ for some Q ∈ E ( K v ) . Finally, set R = P − Q . Then N L w /K v ( R ) = 0 hence we may view R as an element of A ( K v ) . It follows that λ A,v ( R ) + λ E,v ( Q ) − x is in contained in the kernel of res , and the result now follows from Lemma4.3. (cid:3) Growth in Selmer groups
In this section we use the results of previous sections to study the growth of p -Selmergroups in cyclic extensions. Assume for now that L/K is a finite Galois extension of numberfields and
E/K is an elliptic curve over K . For v a (archimidean or non-archimidean) primeof K we define W v,L by W v,L = ker (cid:16) H ( K v , E ( K v ) −→ H ( L w , E ( L w )) (cid:17) , where w is any place of L lying above v . As L/K is Galois, this definition is independentof the choice of w . Also, if we set G v = Gal( L w /K v ) then by inflation-restriction we have W v,L ≃ H ( G v , E ( L w )) . It follows from this that W v,L is finite.Suppose now that L/K has odd prime degree p , and let G = Gal( L/K ) . Let S be afinite set of primes of K containing all archimidean primes, all primes above p , primes ofbad reduction of E and all primes ramified in L/K . Lemma 5.1.
For every prime v in K , we have W v,L ≃ H F + A ( K v , E [ p ]) /H F ( K v , E [ p ]) .Proof. Consider the commutative diagram with exact rows −−−−→ E ( K v ) /pE ( K v ) λ E,v −−−−→ H ( K v , E [ p ]) β −−−−→ H ( K v , E ) y res y res ′ y −−−−→ E ( L w ) /pE ( L w ) λ E,w −−−−→ H ( L w , E [ p ]) −−−−→ H ( L w , E ) . It is not hard to see β induces a map from H F + A ( K v , E [ p ]) to W v,L . Indeed, if x isin H F + A ( K v , E [ p ]) then by Proposition 4.4 we have that res ′ ( β ( x )) = 0 , hence β ( x ) ∈ ker(res ′ ) = W v,L . It is easy to see that this map is onto and its kernel is H F ( K v , E [ p ]) . (cid:3) Proposition 5.2. If Sel p ( E/K ) is trivial, then dim F p Sel p ( E/L ) Gal(
L/K ) = X v ∈ S dim F p W v,L roof. Using Propositions 1.3, 2.1 and 4.4 of [MR07] we obtain dim F p H F + A ( K, E [ p ]) /H F∩A ( K, E [ p ]) = X v ∈ S dim F p H A ( K v , E [ p ]) /H F∩A ( K v , E [ p ])= X v ∈ S dim F p H F + A ( K v , E [ p ]) /H F ( K v , E [ p ])= X v ∈ S dim F p W v,L where the third equality follows from Lemma 5.1. Now if Sel p ( E/K ) is trivial, then so is H F∩A ( K, E [ p ]) . Also, by Proposition 4.4 we have that H F + A ( K, E [ p ]) = res − (Sel p ( E/L )) and res induces an isomorphism H F + A ( K v , E [ p ]) ≃ Sel p ( E/L ) Gal(
L/K ) , and the result fol-lows. (cid:3) Remark 5.3.
We can also deduce Proposition 5.2 as follows. Let X ′ the image of therestriction map H ( K, E [ p ]) → H ( L, E [ p ]) . Then using inflation-restriction we have acommutative diagram −−−−−→ H ( G, E ( L )[ p ]) −−−−−→ H ( K, E [ p ]) −−−−−→ X ′ −−−−−→ y ψ y ϕ y ϕ L −−−−−→ Q v ∈ S W v,L [ p ] −−−−−→ Q v ∈ S H ( K v , E )[ p ] −−−−−→ Q v ∈ S Q w | v H ( L w , E )[ p ] and so by the snake lemma we obtain an exact sequence(5) −→ ker ψ −→ Sel p ( E/K ) −→ X −→ (cid:16) Y v ∈ S W v,L [ p ] (cid:17) / Im ψ −→ coker ϕ, where X = ker ϕ L . Note that here X is contained in Sel p ( E/L ) G , and that X = Sel p ( E/L ) G if E ( K )[ p ] = 0 .Suppose now that E has trivial p -Selmer group over K . Then ψ becomes the zero mapand by the Cassels-Poitou-Tate (Proposition 2.2) exact sequence we have that coker ϕ isalso trivial since it is isomorphic to a subgroup of \ Sel p ( E/K ) , hence (5) becomes Sel p ( E/L ) G ≃ Y v ∈ S W v,L [ p ] . Hence in this setting to analyse the F p -dimension of the G -invariants of p -Selmer it sufficesto know the F p -dimensions of the local cohomology groups W v,L for v ∈ S . The behaviourof these groups will depend on the reduction type of the places as well as whether or notthey lie above p .5.1. F p -dimensions of W v,L . For the remainder of the section assume that E is a semistableelliptic curve defined over K . In this section we study the behaviour of W v,L for the different v ∈ S . First note that when v is archimidean or is split in L then W v,L is trivial, hencewe only consider inert and ramified primes. These cases are the content of the propositionsto follow. Continue to assume that L/K is a finite cyclic extension of number fields of oddprime degree p . Let w be a prime of L above v . We will write k w and k v for the residuefields of L w and K v , respectively, and we denote their sizes by q w and q v . Hence q w = q f v v where f v is the inertial degree of v in L . Also, N L w /K v will denote the norm map from L w to K v . First we recall the following well-known result. We let E ( L w ) denote the subgroupof E ( L w ) consisting of points with non-singular reduction and E ( L w ) denote the kernel ofthe reduction map E ( L w ) → ˜ E ns ( k w ) . emma 5.4. Let v be a place K and w a place of L above v , and set G = Gal( L w /K v ) .Then there is an exact sequence of G -modules −→ E ( L w ) −→ E ( L w ) −→ ˜ E ns ( k w ) −→ . Proof.
The exact sequence follows from [Sil09] §VII.2, Proposition 2.1. Since E has semistablereduction at v it follows that any minimal model for E over K v stays minimal over L w , henceall the groups in the sequence are G -modules. The result follows since the maps in the se-quence are G -equivariant. (cid:3) Recall that if b E is the formal group associated to the elliptic curve E , then there is anisomorphism b E ( m K v ) ≃ E ( K v ) , and in what follows me will sometimes write b E in place of E . Using this isomoprhism and Proposition 3.6 gives that H ( G, E ( L w )) = H ( G, E ( L w )) . In the next lemma we show the same thing holds for W v,L . Lemma 5.5.
Keeping the notation of this section, we have W v,L ≃ H ( G, E ( L w )) . Proof.
The argument is the same as that of Proposition 3.6. Lemma 5.4 gives the exactsequence −→ E ( L w ) −→ E ( L w ) −→ ˜ E ns ( k w ) −→ . By what was just shown E ( L w ) has trivial Herbrand quotient, as does ˜ E ns ( k w ) since it isfinite, hence the same is true of E ( L w ) . Finally there is an exact sequence −→ E ( L w ) −→ E ( L w ) −→ E ( L w ) /E ( L w ) −→ where E ( L w ) /E ( L w ) is finite, and the result follows by the same argument. (cid:3) Case when v ∤ p . Keeping the notation of this section, let v be a prime in S whichdoes not divide p . Proposition 5.6.
Suppose that E has good reduction at v . Then (i) If v is ramified in L , then dim F p W v,L = dim F p ˜ E ( k v )[ p ] . (ii) If v is inert in L then W v,L is trivial.Proof. Let w be the prime of L above v , and set G = Gal( L w /K v ) , so that G is a cyclicgroup of order p . Recall that W v,L is isomorphic to H ( G, E ( L w )) . By Lemma 5.4 we havean exact sequence of G -modules −→ E ( L w ) −→ E ( L w ) −→ ˜ E ( k w ) −→ which induces an exact sequence of cohomology groups H ( G, E ( L w )) −→ H ( G, E ( L w )) −→ H ( G, ˜ E ( k w )) −→ H ( G, E ( L w )) . Let ℓ denote the characteristic of k v . By [Sil09] §IV, Proposition 2.3 and §VII, Proposition2.2 we have that E ( L w ) is uniquely divisible by p , as ℓ = p . It follows then from Proposition2.3 that H i ( G, E ( L w )) = 0 for i = 1 , , and therefore we obtain H ( G, E ( L w )) ∼ −→ H ( G, ˜ E ( k w )) . For (i), as v is totally ramified in L , we have k w = k v and G acts trivially on it. Hence H ( G, ˜ E ( k w )) ≃ Hom( G, ˜ E ( k v )) nd the result follows since G ≃ Z /p Z .Part (ii) follows from Theorem 2 of [Lan56], which gives that H ( G, ˜ E ( k w )) is trivial,hence so is W v,L . (cid:3) To deal with primes of split multiplicative reduction we introduce the following notation.Let w be a prime of L above v . Then L w /K v is tamely ramified with Gal( L w /K v ) ≃ Z /p Z ,hence by Lemma 1 of [Gre74], K v contains a primitive p th root of unity so we may write L w = K v ( p √ π v ) where π v is a uniformiser in K v . We then write u v for a unit in O v suchthat j ( E/K v ) = π − mv u v , where − m = ord v ( j ( E )) . Proposition 5.7.
Suppose E has split multiplicative reduction at v . Then (i) If v is ramified in L , then dim F p W v,L = ( if u qv − p v ≡ π v otherwise (ii) If v is inert in L then dim F p W v,L = ( if p | c v otherwiseProof. By Lemma 5.5 we have that W v,L ≃ H ( G, E ( L w )) and this in turn is simply E ( K v ) /N L w /K v E ( L w ) since G is cyclic. As E has split multi-plicative reduction at v , there is a unique q ∈ K v with q ∈ m K v such that E is isomorphicover K v to the Tate curve E q ( K v ) ≃ K × v /q Z . As these isomorphisms are compatible withthe action of Galois we obtain W v,L ≃ (cid:0) K × v /q Z (cid:1). N L w /K v (cid:0) L × w /q Z (cid:1) . Consider the commutative diagram −−−−→ q Z −−−−→ L × w −−−−→ L × w /q Z −−−−→ y p y N Lw/Kv y N Lw/Kv −−−−→ q Z −−−−→ K × v −−−−→ K × v /q Z −−−−→ where the rows are exact. From local reciprocity we know that K × v /N L w /K v ( L × w ) is Z /p Z hence by the snake lemma we obtain dim F p W v,L = ( if q ∈ N L w /K v ( L × w )0 otherwise . For part (i) let j = j ( E/K v ) , and recall ([Sil94], §V.5) that there is a series g ( T ) = T + 744 T + · · · ∈ Z [[ T ]] such that q = g (cid:18) j (cid:19) . Now j = π − mv u v where m is a positive integer, and so q = π mv ( u − v + · · · )= π mv s v here s v ≡ u − v mod π v . Also, since L w = K v ( p √ π v ) , it follows that π v ∈ N L w /K v ( L × w ) ,hence q is a norm from L w if and only if s v is, and the result follows from the computationof the tame Hilbert symbol, see the corollary to Proposition 8 in ([Ser79], §XIV).For (ii), let π be any uniformiser for K v . Then as v is inert, under this choice of uniformiserwe may identify K × v with Z × O × v and L × w with Z × O × w , these identifications being compatiblewith the action of Galois. It follows that under this identification, the field norm maps ( m, u ) to ( pm, N L w /K v ( u )) for m ∈ Z , u ∈ O × w . Finally, recall that in unramified extensions thenorm map is surjective on the units (see for instance [Ser79], §V.2, Proposition 3) hence weobtain q ∈ N L w /K v ( L × w ) ⇐⇒ p | ord v ( q ) . The result now follows from the fact that ord v ( q ) = c v (see proof of [Sil94], §V.4, Proposition4.1). (cid:3) Proposition 5.8.
Suppose E has non-split multiplicative reduction at v . Then W v,L istrivial.Proof. The exact sequence −→ E ( L w ) −→ E ( L w ) −→ ˜ E ns ( k w ) −→ induces an exact sequence of cohomology groups H ( G, E ( L w )) −→ H ( G, E ( L w )) −→ H ( G, ˜ E ns ( k w )) −→ H ( G, E ( L w )) and by the same argument as in Proposition 5.6 we obtain H ( G, E ( L w )) ≃ H ( G, ˜ E ns ( k w )) . First suppose v is inert. Then again using Theorem 2 of [Lan56] we obtain that H ( G, E ( L w )) is trivial. Also, we have an exact sequence −→ E ( L w ) −→ E ( L w ) −→ E ( L w ) /E ( L w ) −→ which induces the exact sequence H ( G, E ( L w )) −→ H ( G, E ( L w )) −→ H ( G, E ( L w ) /E ( L w )) . As E has non-split multplicative reduction at v , it follows that the group E ( L w ) /E ( L w ) isof order at most , hence all the terms in the above sequence are trivial.Now suppose v is ramified in L . Recall q v denotes the size of the residue field k v , where q v is coprime to p . Then as v is ramified, it follows that p divides q v − . However ˜ E ns ( k w ) is a group if size q v + 1 , so H ( G, ˜ E ns ( k w )) must be trivial, hence by the same argument asabove so is W v,L . (cid:3) Case when v | p . We now wish to study the groups W v,L for primes v which divide p . In this case the formal group E ( L w ) is not uniquely divisible by p , so its cohomologygroups are not necessarily trivial. The situation depends on the reduction type of E at v , and results from section 2 will be useful particularly when E has good supersingularreduction at v . The case of ordinary reduction was studied by Mazur and Rubin in [MR07].We say that E has anomalous reduction at v if ˜ E ( k v )[ p ] is non-trivial. Proposition 5.9.
Let E have good ordinary reduction at v . Then i) If v is ramified in L , then dim F p W v,L = ( or if E has anomalous reduction at v otherwise . (ii) If v is inert in L , then W v,L is trivial.Proof. Part (i) follows immediately from Proposition B.2 in [MR07]. For part (ii), note thatby the same argument as above we have an exact sequence of cohomology groups H ( G, E ( L w )) −→ H ( G, E ( L w )) −→ H ( G, ˜ E ( k w )) −→ H ( G, E ( L w )) . where w is a prime of L above v . Denote by b E the formal group associated to the ellipticcurve E . Now since v is unramified in L , the norm map on b E is surjective and hence itscohomology is trivial. Finally, again using Theorem 2 of [Lan56] we have that H ( G, ˜ E ( k w )) is trivial and therefore the same thing is true of W v,L . (cid:3) Proposition 5.10.
Suppose E has good supersingular reduction at v . (i) Suppose v is ramified in L . Let w be the place of L above v , and let π L be auniformizer for the completion L w . Set t = ord w ( σ ( π L ) − π L ) − . Then dim F p W v,L = 0 ⇐⇒ t = 1 Moreover, for t > we have f K dim F p W v,L min { f K ( t − , [ K : Q p ] + 2 } . (ii) If v is inert in L , then W v,L is trivial.Proof. Consider again the exact sequence of cohomology groups E ( K v ) −→ ˜ E ( k v ) −→ H ( G, E ( L w )) −→ H ( G, E ( L w )) −→ H ( G, ˜ E ( k w )) . As E has supersingular reduction at v we have p ∤ | E ( k w ) | and so H ( G, ˜ E ( k w )) = 0 . Also,the map E ( K v ) → ˜ E ( k v ) is simply the reduction map, hence it is surjective. These togetherimply that we have an isomorphism H ( G, E ( L w )) ≃ H ( G, E ( L w )) = W v,L . For part (i) b E has height as E has supersingular reduction at v , hence the result followsfrom propositions 3.5 and 3.6. Part (ii) follows from the same reasoning as Proposition5.9. (cid:3) Proof of Theorem 1.2.
By Proposition 5.2, dim F p Sel p ( E/L ) Gal(
L/K ) = X v ∈ S dim F p W v,L . If we let δ v = dim F p W v,L then the theorem then follows from propositions 5.6, 5.7, 5.8, 5.9and 5.10 which compute the F p -dimensions of W v,L for the different primes in S . (cid:3) Proof of Corollary 1.3.
By Lemma 1 in ([Ser79], §IX) we have that
Sel p ( E/L ) is trivial ifand only if Sel p ( E/L ) Gal(
L/K ) is trivial, and the result follows. (cid:3) . Extensions of the form K ( p √ m ) In this section we will finish the proof of Theorem 1.1. Let then E be an elliptic curveover Q . Consider the extension of E to K = Q ( ζ p ) and let L m = K ( p √ m ) . For a fixed m , if v is a prime of K then again we let δ v = dim F p W v,L m . We also denote by ℓ v the prime of Q below v , and set q v to be the size of the residue field of K v . That is, if f v is the order of ℓ v (mod p ) , then q v = ℓ f v v .Note that most of the statement follows directly from Theorem 1.2, and the only state-ments which need to be modified are those for primes v ∤ p of split multiplicative reductionramified in L m , and primes v | p of good supersingular reduction ramified in L m . This is donein the two propositions below. To deal with the split multiplicative reduction case recall thenotation introduced in section 1. Let j denote the j -invariant of E and let v ∤ p be a primeof K at which E has split multiplicative reduction. Note that since ℓ v is unramified in K , itfollows that E has semistable reduction at ℓ v . For each positive integer m let s > be suchthat m = ℓ s d where d is coprime to ℓ . Since E is semistable at ℓ v we may write j = ℓ − nv ab with n a positive integer and a and b coprime to ℓ v . Then we set(6) u ℓ v ,m := d n (cid:18) ab (cid:19) s . Proposition 6.1.
Let v be a prime of K such that E has split multiplicative reduction at v . Then δ v = ( if u qv − p ℓ v ,m ≡ ℓ v otherwiseProof. We have that δ v = 1 if and only if the tame Hilbert symbol ( m, q ) v is equal to 1. Weargue as in the proof of part (i) of Proposition 5.7. We have that q = g (cid:18) j (cid:19) = g (cid:18) ℓ nv ba (cid:19) = ℓ nv s v where s v ≡ ba (mod ℓ v ) . Now again a calculation using the corollary to Proposition 8 in([Ser79], §XIV) gives the desired result. (cid:3) Now consider v a prime of K dividing p , and again we let δ v = dim F p W v,L m which as wehave seen is equal to the F p -dimension of H ( G, E ( L w )) , where E can be identified withthe formal group of E . The only thing left to show is that when considering extensions ofthe form K ( p √ m ) , then δ v = p − if p | m , and equals otherwise. This is shown in thefollowing proposition, which is done for general formal groups of height h > . Proposition 6.2.
Let F be a formal group over Q p of height h > . Let K = Q p ( ζ p ) and L m = K ( p √ m ) . Then dim F p F ( m K ) /N F L m /K ( F ( m L m )) = ( p − if p | m otherwiseProof. Let t m be the largest for which the higher ramification group of L m /K is non-trivial.Then one can show that t m = p − when p | m and t m otherwise. The rest of the proofis a very slight modification of that of Proposition 3.5. Indeed, when p | m then t m = p − nd so it follows by Proposition 3.5 that F ( m p − L m ) surjects via the norm onto F ( m p − K ) .Recall from Corollary 3.2 that for x ∈ F ( m L m ) we have N F L m /K ( x ) ≡ Tr L m /K ( x ) + ∞ X i =1 a i N L/K ( x ) i mod Tr L m /K ( x O L m ) . Also, since h > then v K ( a i ) > v K ( p ) = p − unless i = kp h − , since F is definedover Q p . But by Lemma 3.3 we have that Tr L m /K ( m L m ) = m p − K and so it follows that N F L m /K ( F ( m L m )) ⊂ F ( m p − K ) . As we have already seen that the reverse inclusion holds, weconclude that N F L m /K ( F ( m L m )) = F ( m p − K ) , as desired. Finally, when p does not divide m we have that t m and by Proposition 3.5we have that N F L m /K ( F ( m L m )) = F ( m K ) . This concludes the proof. (cid:3)
Proof of Theorem 1.1.
This follows immediately from Theorem 1.2 as well as propositions6.1 and 6.2. (cid:3)
References [Bar10] A. Bartel. Large Selmer groups over number fields.
Math. Proc. Cam. Philos. Soc. , 148(01):73–86,2010.[Bar13] A. Bartel. Elliptic curves with p -Selmer growth for all p . Q. J. Math , 64(4):947–954, 2013.[BCP97] W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system. I: The user language.
J.Symb. Comput. , 24(3-4):235–265, 1997.[Cas64] J.W.S. Cassels. Arithmetic on curves of genus 1, VI. The Tate-Shafarevich group can be arbitrarilylarge.
J. reine angew. Math. , 214/215:65–70, 1964.[Cas65] J.W.S. Cassels. Arithmetic of curves of genus 1, VIII. On conjectures of Birch and Swinnerton-Dyer.
J. Reine Angew. Math. , 217:180–199, 1965.[CG96] J. Coates and R. Greenberg. Kummer theory for abelian varieties over local fields.
Invent. math. ,124:129–174, 1996.[CS00] J. Coates and R. Sujatha.
Galois Cohomology of Elliptic Curves . Tata Institute of FundamentalResearch, Narosa Publ. House, 2000.[DD05] T. Dokchitser and V. Dokchitser. Computations in non-commutative Iwasawa theory.
Proc. LondonMath. Soc. , 91:300–324, 2005.[Dok07] T. Dokchitser. Ranks of elliptic curves in cubic extensions.
Acta Arith. , 126:357–360, 2007.[Fis01] T. Fisher. Some examples of 5 and 7 descent for elliptic curves over Q . J. Eur. Math. Soc. , 3:169–201,2001.[Gre74] M. J. Greenberg. An elementary proof of the Kronecker-Weber Theorem.
Amer. Math. Monthly ,81(601-607), 1974.[Gre99] R. Greenberg. Iwasawa theory for elliptic curves. In
Arithmetic Theory of Elliptic Curves . Springer,1999.[Haz74] M. Hazewinkel. On norm maps for one dimensional formal groups I. The cyclotomic γ -extension. J. Algebra , 32:89–108, 1974.[HM99] Y. Hachimori and K. Matsuno. An analougue of Kida’s formula for Selmer groups of elliptic curves.
J. Algebraic Geom. , 8:581–601, 1999.[Klo05] R. Kloosterman. The p -part of Shafarevich-Tate groups of elliptic curves can be arbitrarily large. J. Theorie Nombres Bordeaux , 17:787–800, 2005.[Kra81] K. Kramer. Arithmetic of elliptic curves upon quadratic extension.
Trans. Am. Math. Soc. , 264:121–135, 1981.[Kra83] K. Kramer. A family of semistable elliptic curves with large Tate-Shafarevich groups.
Proc. Amer.Math. Soc. , 89:379–386, 1983.[KS03] R. Kloosterman and E. F. Schaefer. Selmer groups of elliptic curves that can be arbitrarily large.
J.Number Theory , 99:148–163, 2003. Lan56] S. Lang. Algebraic Groups over Finite Fields.
Amer. J. Math. , 78(3):555–563, 1956.[Mat07] K. Matsuno. Construction of elliptic curves with large Iwasawa λ -invariants and large Tate-Shafarevich groups. manuscr. math. , 122:289–304, 2007.[Mat09] K. Matsuno. Elliptic curves with large Tate-Shafarevich groups over a number field. Math. Res.Lett. , 16(3):449–461, 2009.[Maz72] B. Mazur. Rational points of abelian varieties with values in towers of number fields.
Invent. math. ,18:183–266, 1972.[Mil06] J. S. Milne.
Arithmetic Duality Theorems . BookSurge, 2nd edition, 2006.[MR07] B. Mazur and K. Rubin. Finding large Selmer rank via an arithmetic theory of local constants.
Annals of Mathematics , 166:579–612, 2007.[MR10] B. Mazur and K. Rubin. Ranks of twists of elliptic curves and Hilbert’s tenth problem.
InventionesMath. , 181:541–575, 2010.[MRS07] B. Mazur, K. Rubin, and A. Silverberg. Twisting commutative algebraic groups.
J. Algebra ,314:419–438, 2007.[New12] R. Newton. Explicit local reciprocity for tame extensions.
Math. Proc. Camb. Philos. Soc. ,152(3):425–454, 2012.[Ser79] J.-P. Serre.
Local Fields . Springer, 1979.[Sil94] J. Silverman.
Advanced Topics in the Arithmetic of Elliptic Curves . Springer, 1994.[Sil09] J. Silverman.
The Arithmetic of Elliptic Curves . Springer, 2nd edition, 2009.[Č12] K. Česnavičius. Selmer groups and class groups. arxiv: 1307.4261, 2012.. Springer, 2nd edition, 2009.[Č12] K. Česnavičius. Selmer groups and class groups. arxiv: 1307.4261, 2012.