GROWTH OF X IN TOWERS FOR ISOGENOUS CURVES TIM AND VLADIMIRDOKCHITSER Abstract. WestudythegrowthofX andp∞-Selmergroupsforisoge- nous abelian varieties in towers of number fields, with an emphasis on elliptic curves. Thegrowth typesare usually exponential, as in theset- 3 ting of ‘positive µ-invariant’ in Iwasawa theory of elliptic curves. The 1 towers we consider are p-adic and l-adic Lie extensions for l 6= p, in 0 particular cyclotomic and other Z-extensions. l 2 n a J 1. Introduction 7 1 The algebraic side of the Iwasawa theory of elliptic curves is concerned withthestudyof thestructureof Selmergroupsincyclotomic Zp-extensions ] T of Q, and other towers of number fields. The aim of the present paper is to N systematically study the behaviour of Selmer groups for isogenous elliptic . curves E,E′ or abelian varieties. The isogeny makes it possible to bypass h Iwasawa theory and, in particular, avoid any assumptions on the reduction t a types. Moreover, it allows us to work with p∞-Selmer groups in general m l-adictowers, bothfor l=pandl=p. For instance, we constructelliptic cur- [ 6 ves whose p-primary part of the Tate-Shafarevich group goes off to infinity 1 inalll-cyclotomic extensionsofQ,incontrasttoWashington’s theoremthat v the p-part of the ideal class group is bounded in these extensions for l = p. 7 6 We show that in the nth layer of the p-cyclotomic tower of Q, the quo- 5 2 tient X [p∞]/X [p∞] (if finite) is pµpn+O(1), as though it came from an E E 4 Iwasa|wa modu|le|with′ λ-in|variant 0 and µ-invariant µ, except that µ may be . 1 fractional whenE haspotentially supersingularreduction atp. Ourformula 0 for µ is explicit and surprisingly simple, and there is similar behaviour in 3 other p-adic and l-adic towers. It would be interesting to understand the 1 : structure theory of the associated Selmer groups that gives rise to such v i growth. X Our main results for Z -extensions and general Lie groups are as follows: l r a Theorem 1.1. Let Q(ln) be the l-cyclotomic tower, p a prime and E,E′ n ∪ two isogenous elliptic curves over Q. Then for all large enough n, X◦ [p∞] | E/Q(ln) | = pµln+ν+ǫ(n), X◦ [p∞] E/Q(ln) | ′ | for some ν Z, some ǫ(n) 6 81/2 and µ 1 Z given by ∈ | | ∈ 12 µ = ord ΩE′ + 0 if l 6= p or ordp(jE) < 0, p ΩE ( 112ordp(∆∆EE′) if l = p and ordp(jE)> 0. If l = p or l does not divide the degree of the isogeny E E′, then ǫ(n)= 0. 6 → Date: Christmas Day 2012. 2000 Mathematics Subject Classification. 11G07 (11G05, 11G10, 11G40, 11R23). 1 2 TIMANDVLADIMIRDOKCHITSER Here and throughout the paper Q(ln) denotes the degree ln-extension of Q in the cyclotomic Z -extension. We write ∆ ,∆ for the minimal discrim- l E E ′ inants1 and j ,j for the j-invariants of the two curves, Ω ,Ω for the E E E E ′ ′ Birch–Swinnerton-Dyer periods (see 2), and X◦ for the Tate-Shafarevich § groupXmoduloitsdivisiblepartXdiv. Write alsoSel forthep∞-Selmer p ∞ group, Seldiv for its divisible part, rk for its Z -corank2, c for the Tama- p p p v ∞ gawa number at v, and f for the residue degree of K /Q . The big O Kv/Qp v p notation refers to the parameter n. Theorem 1.2. Let K be a number field, p a prime number, and K = K ∞ n n ∪ a Z -extension of K, with [K : K] = ln. Let φ : E E′ be an isogeny of l n → elliptic curves over K, with dual isogeny φt. Then Seldiv(E/K )[φ] X◦ [p∞] Ω | p∞ n | | E/Kn | = pµln+O(1), µ = ord E′/K + µ , Seldiv(E′/K )[φt] X◦ [p∞] p Ω v | p∞ n || E′/Kn | (cid:0) E/K(cid:1) Xv where the sum is taken over primes v of bad reduction for E/K, and ordp ccvv((EE′//KK)) if v is totally split in K∞/K, µv = fK1v2/Qp ordv(∆∆EE′) if l=p, v|p is ramified in K∞/K and ordvjE>0,  0 otherwise. If rkpE/Kn is bounded, then ||XX◦E◦E′//KKnn[[pp∞∞]]|| = pµln+O(1) as well. Theorem 1.3. Suppose K is a number field and K /K a Galois extension ∞ whose Galois group is a d-dimensional l-adic Lie group; write K /K for its n nth layer in the natural l-adic Lie filtration. Let p be a prime number, and φ: A A′ an isogeny of abelian varieties over K, with dual φt : A′t At. → → (1) If A is an elliptic curve, then there is µ Q such that ∈ Seldiv(A/K )[φ] X◦ [p∞] | p∞ n | | A/Kn | =pµldn+O(l(d−1)n). Seldiv(A′/K )[φt] X◦ [p∞] | p∞ n || A′/Kn | (2) If either A,A′ are semistable abelian varieties or they are elliptic cur- vesthatdonothaveadditivepotentially supersingularreductionatprimesv p | thatareinfinitelyramifiedinK /K, thenthereareconstantsµ ,...,µ Q ∞ 1 d−1 ∈ such that for all sufficiently large n, Seldiv(A/K )[φ] X◦ [p∞] | p∞ n | | A/Kn | = pµldn+µ1l(d−1)n+...+µd 1ln+O(1). Seldiv(A′t/K )[φt] X◦ [p∞] − | p∞ n || A′/Kn | If A(K )[p∞] is bounded, O(1) may be replaced by a constant µ Q. n d ∈ (3) If rk A/K = O(l(d−1)n) in (1), respectively O(1) in (2), then all p n the conclusions of (1), respectively (2), hold for |X◦A/Kn[p∞]| as well. |X◦A′/Kn[p∞]| 1If the base field is not Q, there may be no global minimal model. We then regard ∆E,∆E′ as ideals that haveminimal valuation at every prime. 2Thus, rkpA/K = rkA/K+t if XdAi/vK ∼= (Qp/Zp)t. Of course, conjecturally, t = 0, X =X◦ and Seldp∞iv(A/K)∼=A(K)⊗Qp/Zp. In any case, X◦ is a torsion abelian group all of whose p-primary parts are finite. GROWTH OF X IN TOWERS FOR ISOGENOUS CURVES 3 Remarks 1.4. (1)Ifp ∤ degφ, thenthe p∞-Selmer groupsandthep-partof X cannothave φ-torsion, so the corresponding quotients in Theorems 1.1-1.3 are trivial. The results can be reduced to those for isogenies of p-power degree. Ω (2)InTheorems1.1,1.2,thequotient E′ isarationalnumber(Lemma2.3). For p-isogenous curves over Q it is 1ΩoEr p±1, see [9] Thm 8.2. The term 112ordp(∆∆EE′) is 0 unless E has additive potentially supersingular reduction at p, see [9] Table 1. In this exceptional case, µ does not have to be an integer, see Example 1.6. (3)SupposeGal(K /K)=Γ=Z . Ifthedualp∞-Selmer group of E/K is ∞ ∼ p ∞ atorsionZ [[Γ]]-module,thentheinvariantµofTheorem1.2isµ(E) µ(E′), p − thedifferenceoftheclassicalµ-invariants ofthetwoSelmergroupsover K . ∞ In this setting, Theorem 1.2 is equivalent to a theorem of Schneider (for odd p), see [25, 23]. (4) Suppose E/Q has good ordinary reduction at p. Then the dual p∞-Sel- mer group of E over the p-cyclotomic extension over Q is a torsion Iwasawa module, by Kato’s theorem [16]. A conjecture of Greenberg ([13] Conj 1.11) assertsthattheisogenyclassofE containsacurveofµ-invariant0. Granting the conjecture, Theorem 1.1 implies that (i) In the isogeny class of E/Q the curve E with the largest period m Ω has µ-invariant 0 at all primes of good ordinary reduction. Em (ii) E has µ-invariant ord ΩEm at p. p ΩE Thus, the theorem provides a conjectural formula for the µ-invariant. We note here that Greenberg’s conjecture is known not to hold in general over number fields, see [11]. (5) Greenberg (see [12] Exc. 4.3–4.5) has observed that if φ : E E′ is a → p-isogeny over K whose kernel is µ , then the map p K×/K×p = H1(Gal(K¯/K),µ ) H1(Gal(K¯/K),E[p]) ∼ p −→ induced by the inclusion µ E[p] gives a way to construct classes in the p ⊂ p-Selmer group of E. The units of K contribute to the Selmer group, and the rank of the unit group is [K:Q]. In particular, one can exhibit µ-like ∼ growth of Sel (E) in towers K /K. It would be interesting to similarly p n explain the Selmer growth in Theorems 1.1–1.3 for p-power isogenies with arbitrary kernels. (6) By a theorem of Washington [32], for p = l the p-part of the ideal 6 class group is bounded in the l-cyclotomic tower. Theorem 1.2 provides examples of elliptic curves over Q for which the analogous statement for the Tate-Shafarevich group fails, see e.g. Example 1.5. Intheoppositedirection, Lamplugh[20]hasrecently proven thefollowing analogueofthetheoremofWashingtonforellipticcurvesE/Qwithcomplex multiplication by the ring of integers of an imaginary quadratic field K. Let p > 3, l > 3 be distinct primes of good reduction of E that split in K/Q. 4 TIMANDVLADIMIRDOKCHITSER Lamplugh proves that if K is the nth layer of the unique Z -extension of K n l unramified outside one of the factors of l in K, then the p∞-Selmer group of E over K stabilises as n . n → ∞ (7) The constants µ and µ ,...,µ in Theorem 1.3 can be made explicit, 1 d as in Theorem 1.2. Following the proof of Theorem 8.7, this requires the knowledge of the decomposition and inertia groups at bad primes; the other ingredients are computed in Proposition 8.5. Example1.5(Ordinaryreduction). Letusshowthatthecurves11A1,11A2 have unbounded 5-primary part of X in the cyclotomic Z -extension of Q l for every prime l. There are 5-isogenies 11A2 11A1 11A3, −→ −→ andΩ = 5Ω = 25Ω = 6.34604.... So,byTheorem1.1,foreveryl 11A3 11A1 11A2 there are ν ,ν′ Z such that l l ∈ X11A2/Q(ln))[5∞] > 25ln−νl, X11A1/Q(ln))[5∞] > 5ln−νl′. | | | | A standard computation with cyclotomic Euler characteristics (e.g. as in [6] 3.11) shows that for every ordinary prime l for which a = 1, e.g. l § 6 l = 3,7,13,17,..., the curves have rank 0 over Q(ln), for all n > 1. For such primes ν ,ν′ can be taken to be 1 and the number of primes above 11 in l l Q(ln), respectively3. For l = 5, these bounds are exact, as X n 11A3/Q(5n) is known to have trivial 5-primary part for all n > 1. S Example 1.6 (Potentially supersingularreduction). Let E/Q bean elliptic curvewithgoodsupersingularreductionatp,andK = Q(pn),thenthlayer n in the p-cyclotomic tower. By a theorem of Kurihara [19], under suitable hypothesis, p |XE/Kn[p∞]|= p⌊µpn−12⌋, µ = p2 1. − Note that such curves cannot have a p-isogeny, by a theorem of Serre ([27] Prop. 12). In contrast, elliptic curves over Q with additive potentially supersingular reduction at p can have a p-isogeny, and there are examples for which X [p∞] > pµpn+ν | E/Kn | with µ > 1. For instance, there is a 9-isogeny φ : 54A2 54A3. These → curves have potentially supersingular reduction at p = 3, and Ω = 9Ω , ∆ = 29311, ∆ = 2 33. 54A3 54A2 54A2 54A3 − − · By Theorem 1.1, there is a constant ν such that for all large enough n, X [3∞] 11 3 4 X [3∞] > | 54A2/Kn | = 33nµ+ν, µ = 2 − = . | 54A2/Kn | X [3∞] − 12 3 | 54A3/Kn | 3Soνl′ is almost always 1 as well; for l<107 theonly exception is l=71. GROWTH OF X IN TOWERS FOR ISOGENOUS CURVES 5 Example 1.7 (False Tate curve tower). To illustrate Theorem 1.3 for a higher-dimensional l-adic Lie group, let Kn = Q(ζ3n, 3√n 7), a ‘false Tate curve tower’ in the terminology of [15, 6]. Let E = 11A1 and E′ = 11A3, as in Example 1.5. We find (see Example 3.3) that either X [5∞] is E/Kn infinite or X [5∞] | E/Kn | = 532n−1−3n. X [5∞] | E′/Kn | Example 1.8. Let K be the unique Z2-extension of Q(i) and let K be ∞ 5 n its nth layer; thus Gal(Kn/Q) ∼= C5n ×D2·5n. If we take the 5-isogenous curvesE = 75A1,E′ = 75A2 over Q,withadditivepotentially supersingular reduction at 5, we find that (see Example 3.4) Seldiv(E/K )[φ] X◦ [5∞] | 5∞ n | | E/Kn | = 5µ52n+µ1(n)5n+µ2(n) Seldiv(E′/K )[φt] X◦ [5∞] | 5∞ n || E′/Kn | with 1 2 µ = , µ (n) = 1 ( 1)n, µ (n) =0. 1 2 −3 − 3 − So the assumption that E does not have potentially supersingularreduction in Theorem 1.3(2) cannot be removed, as the µ may fluctuate with n. i Example 1.9. As opposed to the cyclotomic extensions, for general Z -extensionsofnumberfieldsthereisanextraterminµcomingfromTama- l gawa numbers(compareTheorem1.2 withTheorem1.1). For example, con- sider the 5-isogeny 11A1 11A3 as in Example 1.5 in the 5-anticyclotomic → tower K of K = Q(i). Because 11 is inert in Q(i), and so totally split in ∞ K /K, there is a µ-contribution from the Tamagawa numbers (5 and 1) in ∞ this Z -extension, but not in the cyclotomic one. 5 Remark 1.10 (CM curves with µ > 0). If K /K is a Z -extension and ∞ p Sel (E/K ) is cotorsion over the Iwasawa algebra of Gal(K /K), then it p ∞ ∞ ∞ has a well-defined µ-invariant as in classical Iwasawa theory. Theorem 1.2 gives a formula for its change under isogenies in terms of elementary invari- ants, and allows us to generate examples with positive µ-invariant. Consider, for instance, elliptic curves with complex multiplication and good ordinary reduction at p. Such examples over Q with a p-isogeny are almost non-existent: there are 13 CM j-invariants over Q, and there is only one with a p-isogeny that admits good reduction at p. It is j = 3353 (CM − by Z[1+√ 7]), 2-isogenous to j = 3353173 (CM by Z[√ 7]). (This is easy to 2− − check from the table of CM j-invariants [30] Appendix A and by computing the isogenous curves, e.g. in Magma [2].) The simplest example with these j-invariants is φ :49A1 49A2. −→ Here Ω /Ω = 2, and so 49A2 does have positive µ-invariant for p = 2 49A1 49A2 6 TIMANDVLADIMIRDOKCHITSER as well as unbounded 2-part of X in every cyclotomic Z -extension of Q, by l Theorem 1.1. Assuming Greenberg’s conjecture (Remark 1.4(4)), the curve 49A2 and p=2 is the unique example (up to quadratic twists) of a good ordinary CM curve over Q with positive µ-invariant. Over larger number fields, other examples are easy to construct. For instance, the curve √5 1 E : y2 = x3 24z7√z+3x2+zx, z = − − 2 is defined over K = Q(ζ )+ = Q(√z+3) and has CM by Z + 5iZ. It 20 has good ordinary reduction at the prime above 5, and is 5-isogenous to y2 = x3+zx. ComputingtheperiodsandapplyingTheorem1.2,wefindthat it should have positive µ-invariant both over the Z -cyclotomic extension 5 of K, and over every Z -extension of K(i) = Q(ζ ). 5 20 Let F be the composite of all Z -extensions of K(i), so that G = ∞ 5 Gal(F /K) = Z5. The 5∞-Selmer group of E over F is conjectured to ∞ ∼ 5 ∞ satisfy the (G)-conjecture of non-commutative Iwasawa theory [5]. As H M John Coates remarked to us, this example provides evidence for the conjec- ture as follows. Similar arguments to those given in [4] would show that the (G)-conjecture implies that the G-µ-invariant of the Selmer group over H M F wouldhave to beequal totheusualµ-invariant of theSelmer groupover ∞ the cyclotomic Z -extension of K(i), which we have shown to be non-zero. 5 Thus,grantedthe (G)-conjecture, itwouldfollowthattheG-µ-invariant H M of the Selmer group over F would have to be positive, and then an easy ∞ furtherargumentshowsthattheµ-invariant over every Z -extension of K(i) 5 would also be positive, in accord with what we have proven. Brief overview of the paper. To control the change of Selmer groups, we invoke the theorem on the invariance of the Birch–Swinnerton-Dyer conjecture under isogeny by Cassels and Tate [3, 31]. This is recalled in Theorem 3.1 in 3, after we introduce a convenient choice of periods in 2. § § Pretty much the rest of the paper studies how the terms of the Birch– Swinnerton-Dyer formula behave in towers of local fields and number fields: minimal differentials ( 4), Tamagawa numbers ( 5), torsion ( 6) and the § § § divisible part of Selmer ( 7). At the end of 3 we also give some examples § § how this procedure works. Theorems 1.1-1.3 are proved in 8. In 4, 5 we § § § rely on the results of [9] that describe how local invariants of elliptic curves change under isogeny. The appendix ( 9) concerns the behaviour of conductors of elliptic curves § and Galois representations in field extensions. There is no assumption on the existence of an isogeny here, and the results may be of independent interest. GROWTH OF X IN TOWERS FOR ISOGENOUS CURVES 7 Notation. We write E,E′ for elliptic curves and A,A′ for abelian varieties. We usually have an isogeny E E′ or A A′, denoted by φ. Its dual → → E′ E or (A′)t At is denoted by φt. Number fields are denoted by → → K,F,..., and l-adic fields (finite extensions of Q ) by , ,.... We also use l K F the following notation: normalised absolute value at a prime v v |·| v(), ord () normalised valuation in a local field/at v v · · j j-invariant of an elliptic curve E E ∆ ,∆ minimal discriminant of an elliptic curve over K E E/K δ, δ′ v(∆ ),v(∆ ) when is local E/K E/K ′ K f conductor exponent of E/ when is local E/K K K Om ( ) see Definition 4.1 φ F ωmin = ωmin N´eron minimal exterior form of an abelian variety at a v A,v prime v (minimal differential for an elliptic curve) c ,c (A/K) Tamagawa number over a local field/global field at v A/K v Ω,Ω∗ infinite periods, see Definition 2.1 X,Xdiv,X◦ Tate-Shafarevich group, its divisible part, X◦=X/Xdiv Sel (A/K) limSel (A/K), the p-infinity Selmer group p pn ∞ rk (A/K) Z−→-corank of Sel (A/K) p p p ∞ Q(ln) the nth layer of the cyclotomic Z -extension of Q, i.e. l the unique totally real degree ln subfield of Q(ζ ) ln+2 X[φ],X[p∞] φ-torsion, p-primary component of an abelian group X , integer part (floor) and fractional part x =x x ⌊·⌋ {·} { } −⌊ ⌋ Any two non-zero invariant exterior forms ω ,ω on an abelian variety 1 2 A/K are multiples of one another, ω = αω with α K. We will write 1 2 ∈ ω /ω for the scaling factor α. 1 2 When is an l-adic field, recall that an elliptic curve E/ has additive K K reduction if and only if it has conductor exponent f > 2, and that E/K f = 2 if and only if the ℓ-adic Tate module of E is tamely ramified for E/K some (any) ℓ = l. We will call this tame reduction, and wild if f > 2. E/K 6 If l > 5, the reduction is always tame. We remind the reader that E/ K has potentially good reduction if v(j ) > 0 and potentially multiplicative E reduction if v(j ) < 0. E Finally, an l-adic Lie group G is a closed subgroup GL (Z ) for some k; k l it has a natural filtration by open subgroups, the kernels of the reduction maps mod ln. We use Cremona’s notation (such as 11A1) for elliptic curves over Q. Acknowledgements. Thisinvestigationwasinspiredbyconversationswith John Coates. We would like to thank him and also Masato Kurihara, Ralph GreenbergandKarlRubinfortheircomments. Thefirstauthorissupported by a Royal Society University Research Fellowship. 8 TIMANDVLADIMIRDOKCHITSER 2. Periods Weintroduceaconvenientformofperiodsofabelianvarietiesovernumber fields,thataremodel-independentandwell-suitedfortheBirch–Swinnerton- Dyer conjecture. Definition 2.1. An abelian variety A/K has a non-zero invariant exterior form ω, unique up to K-multiples. If K = C, we define the local period Ω = 2dimAω ω. A/C,ω ∧ ZA(C) If K = R, define Ω Ω = ω and Ω∗ = A/C,ω. A/R,ω | | A/R Ω2 ZA(R) A/R,ω If K is a number field, define the global period Ω = ω/ωmin Ω . A/K | v |v A/Kv,ω v∤∞ v|∞ Y Y Here v runs through places of K, and the term at v in the first product is the normalised v-adic absolute value of the quotient of ω by the N´eron minimal form at v. Remark 2.2. Note that both Ω∗ and Ω are independent of the A/Kv A/K choice of ω, by the product formula for the second one. An elliptic curve E/Q can be put in minimal Weierstrass form, with the global minimal differential ω = dx . Then Ω = Ω , which is 2y+a1x+a3 E/Q E/R,ω the traditional real period Ω or 2Ω , depending on whether or not E(R) + + is connected. If E(C) = C/Zτ+Z under the usual complex uniformisation, then Ω∗ = Imτ. E/R Lemma 2.3. Let φ : A A′ be an isogeny of abelian varieties over a → number field K. Then both ΩA′/K and, for real places v, Ω∗A′/Kv are positive Ω Ω A/K ∗A/Kv rational numbers. They have trivial p-adic valuation for all p ∤ degφ. Proof. Fix a non-zero invariant exterior form ω′ on A′, and set ω = φ∗ω′. For v , |∞ Ω cokerφ :A(K ) A′(K ) A′/Kv,ω′ = | v → v |. Ω kerφ :A(K ) A′(K ) A/Kv,ω | v → v | Thisisapositiverational, andconsideringtheconjugateisogenyφ′ :A′ A → (so that φ′ φ,φ φ′ are the multiplication-by-degφ maps), we see that the ◦ ◦ only prime factors of ker and coker are those dividing degφ. The claim | | | | for Ω∗ now follows. As for the global periods, Ω ω′/ωmin Ω A′/K = | A′,v|v A′/Kv,ω′ Ω ω/ωmin Ω A/K v∤∞ | A,v|v v|∞ A/Kv,ω Y Y GROWTH OF X IN TOWERS FOR ISOGENOUS CURVES 9 isapositiverational. Ifv ∤ degφ,thenφ∗(ωmin)isaunitmultipleofωmin,so A,v A,v ′ |ω′/ωAm′i,nv|v = 1. So ΩA′/K has trivial p-adic valuation at primes p ∤ degφ. (cid:3) |ω/ωAm,ivn|v ΩA/K Lemma 2.4. Let φ : A A′ be an isogeny of abelian varieties over a → number field K, and F/K a finite extension. Then ωmin Ω = Ω[F:K] Ω∗ #{w|v complex} v , A/F A/K A/Kv ωmin v real v,w|v (cid:12) w (cid:12)w Y (cid:0) (cid:1) Y (cid:12) (cid:12) (cid:12) (cid:12) where v runs over places of K and w over places of F above v. (cid:12) (cid:12) Proof. Chooseaninvariant exterior formω forA/K. We computetheterms in Ω using ω. A/F [F:K] Let v be a place of K. If v is complex, then Ω = Ω . If w|v A/Fw,ω A/Kv,ω v is real, then, writing Σ ,Σ for the set of real and complex places w v + − Q | in F, we have Ω = Ω Ω2 Ω∗ = Ω[F:K] (Ω∗ )|Σ |. A/Fw,ω A/Kv,ω A/Kv,ω A/Kv A/Kv,ω A/Kv − Yw|v wY∈Σ+ wY∈Σ − If v ∤ , then ∞ ω ω ωmin ω [F:K] ωmin = v = v . ωwmin w ωvmin w ωwmin w ωvmin v ωwmin w Yw|v(cid:12) (cid:12) Yw|v(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Yw|v(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Multiplying(cid:12) the t(cid:12)erms ove(cid:12)r all p(cid:12)la(cid:12)ces v(cid:12)of K(cid:12)gives(cid:12)the claim(cid:12). (cid:12) (cid:3) Remark 2.5. For elliptic curves, the term ωmin/ωmin relates to the be- v w haviour of the minimal discriminant of E in F /K (cf. [29] Table III.1.2), w v ωmin 1 ∆ ord w = ord E/K . w ωmin 12 w ∆ v E/F (cid:16) (cid:17) (cid:16) (cid:17) 3. BSD invariance under isogeny We now state a version of the invariance of the Birch–Swinnerton-Dyer conjecture under isogeny for Selmer groups (see page 7 for the notation). Theorem 3.1. Let φ : A A′ be a isogeny of abelian varieties over a number field K, and φt : A′t→ At the dual isogeny. If the degree of φ is a → power of p, then |Seldp∞iv(A/K)[φ]| |X◦A/K[p∞]| = |A(K)[p∞]||At(K)[p∞]| ΩA′/K cv(A′/K). |Seldp∞iv(A′t/K)[φt]||X◦A′/K[p∞]| |A′(K)[p∞]||A′t(K)[p∞]| ΩA/K vY∤∞ cv(A/K) Otherwise, the left-hand side and the right-hand side have the same p-part. Proof. This is essentially [7] Thm 4.3, that says |Seldp∞iv(A/K)[φ]| |X◦A/K[p∞]| = |A(K)tors||At(K)tors| ΩA′/K cv(A′/K). |Seldp∞iv(A′t/K)[φt]|p|Ydegφ|X◦A′/K[p∞]| |A′(K)tors||A′t(K)tors| ΩA/K vY∤∞ cv(A/K) The term Q(φ) in [7] in exactly Seldiv(A/K)[φ]. (cid:3) p ∞ 10 TIMANDVLADIMIRDOKCHITSER Corollary 3.2. Let φ : A A′ be an isogeny of abelian varieties over a → number field K with dual φt : A′t At, and F/K a finite extension. If the → degree of φ is a power of p, then Seldiv(A/F)[φ] X◦ [p∞] A(F)[p∞] At(F)[p∞] Ω [F:K] | p∞ | | A/F | = | || | A′/K Seldiv(A′t/F)[φt] X◦ [p∞] A′(F)[p∞] A′t(F)[p∞] Ω × | p∞ || A′/F | | || |(cid:16) A/K(cid:17) Ω∗A′/Kv #{w|v complex} cv(A′/F) ωAm′i,nv/ωAm′i,nw , × Ω∗ c (A/F) ωmin/ωmin v real A/Kv v∤∞ v v,w|v (cid:12) A,v A,w (cid:12)w Y (cid:0) (cid:1) Y Y (cid:12) (cid:12) (cid:12) (cid:12) where v ranges over places of K, and w v are places of F. If φ has arbitrary (cid:12) (cid:12) | degree, then the left-hand side and the right-hand side have the same p-part. Proof. Combine Theorem 3.1 with Lemma 2.4. (cid:3) Corollary 3.2 is our main tool for studying the Selmer growth in towers in 8. As we now illustrate, it already enables us to construct explicit § examples of interesting growth of Selmer and X. The general behaviour of the Tamagawa number quotient will be discussed in 5, torsion quotient § in 6, and the contribution from exterior forms in 4, under the name of § § Om (F ). φ w Example 3.3. Let Kn = Q(ζ3n, 3√n 7), a ‘false Tate curve tower’ in the terminology of [15, 6], and let φ : E = 11A1 E′ = 11A3 be the 5-isogeny → as in Example 1.5. A result of Hachimori and Matsuno [14] Thm 3.1 and a cyclotomic Euler characteristic computation as in [6] 3.11 show that § rkE/K =rk E/K =0 for all n > 1. Therefore n 3 n Sel (E/K ) = X [5∞], 5∞ n E/Kn and similarly for E′. The periods of the two curves are 1 Ω = 1.2692... = Ω , Ω∗ = 1.1493... = 5Ω∗ , E/Q 5 E′/Q E/R E′/R and both curves have torsion of size 5 over all K . Applying Corollary 3.2, n we find that either X [5∞] is infinite for some n, or E/Kn |XE/Kn[5∞]| = 52 ΩE′/Q 2·32n−1 Ω∗E′/R 32n−1 cv(E′/Kn) 1 X [5∞] 52 · Ω · Ω∗ · c (E/K ) · | E′/Kn | E/Q E/R v|11 v n (cid:0) (cid:1) (cid:0) (cid:1) Y = 52·32n−1 = 532n−1−3n. 532n−1 53n · Example 3.4 (Fluctuation inSelmergrowth). Inthepreviousexample, the quotient of the Tate-Shafarevich groups grew like 532n−1−3n. Theorem 1.3 shows that such growth of the form ppolynomialinln is a general phenomenon. However, the assumption on primes of additive potentially supersingular reduction is essential, as we now illustrate.