Growth of in towers for isogenous curves

Dokchitser, Tim; Dokchitser, Vladimir

doi:10.1112/s0010437x15007423

Cited by 9 publications

(24 citation statements)

References 21 publications

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“…For one application, seeČesnavičius' work on the parity conjecture [2, §5]. [1], [18], [9], [6], [12] and [8]. Table 1 summarises our results for the valuations of minimal discriminants δ, δ of E, E , their Tamagawa numbers c, c , Kodaira types and the leading term φ * ω ω of φ on the formal group (see §1.1 for the notation).…”

Section: Introductionmentioning

confidence: 97%

Local invariants of isogenous elliptic curves

Dokchitser

2014

Trans. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 97%

Local invariants of isogenous elliptic curves

Dokchitser

2014

Trans. Amer. Math. Soc.

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“…Even in the case where A is defined over Q, and A has complex multiplication, we see that there is an example of this phenomenon; see [4,Remark 1.10]. Here it is shown that the curve 49A2, which has complex multiplication by the order Z[ √ −7], has unbounded 2-part of X in the cyclotomic Z q -extension of Q, for all primes q = 2.…”

Section: Introductionmentioning

confidence: 90%

“…Furthermore, when the elliptic curve is not defined over the imaginary quadratic field K over which A has complex multiplication, then other curves with this phenomenon are easy to construct. For instance, in [4,Remark 1.10], the Dokchitsers give a curve A defined over Q(μ 20 ) with complex multiplication by Z + 5 · Z[ √ −1] such that, for all primes q and all Z q -extensions L ∞ /Q(μ 20…”

Section: Introductionmentioning

confidence: 99%

An analogue of the Washington-Sinnott theorem for elliptic curves with complex multiplication I

Lamplugh

2015

Journal of the London Mathematical Society

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In this paper, we generalize Washington's theorem on the stabilization of the p-part of the ideal class groups in the cyclotomic Z q -extension of an abelian number field for distinct primes p and q. We fix an imaginary quadratic field K and a split prime q of K lying above q and let K ∞/K denote the Z q -extension which is unramified outside q. We show that if F/K is a finite abelian extension, and if F ∞ = F K∞, then, for a prime p satisfying certain conditions, the p-part of the class groups stabilize in the Z p-extension F∞/F . We also show that if E/K is an elliptic curve with complex multiplication by the ring of integers of K, then under the same conditions on a prime p, the p-part of the Selmer groups for E/F n stabilize as Fn runs over the finite layers of F ∞/F . Theorem (Washington). If k is any finite abelian extension of Q, then, for all primes p = q, v p (h m ) is constant for m sufficiently large.Here v p is the usual p-adic valuation, so that, for an integer n, v p (n) = e 0, where p e is the largest power of p dividing n. The theorem of Washington has the following generalization, due to Friedman; see [6]. Fix S to be a finite set of rational primes, and let Q S denote the compositum of all the cyclotomic Z q -extensions of Q for all primes q ∈ S. If k is any finite abelian extension of Q, let k S = kQ S . If we denote the class number of a finite extension F of Q by h F , then we have the following theorem.Theorem (Friedman-Washington). If k is any finite abelian extension of Q, then, for allRemark. It is conjectured that this theorem holds for arbitrary finite extensions k of Q.In the case where A is an elliptic curve defined over Q, the following parallel result is known. We assume the notation above, but assume that k is abelian over Q, and define X S to be the set of finite-order characters of Gal(k S /Q). Then Rohrlich [14] has proved the following theorem.Theorem (Rohrlich). There are only finitely many characters χ ∈ X S such that L(A, χ, 1) = 0.It then follows by some deep results of Kato (see [11, Section 14]) that the Mordell-Weil group A(k S ) is finitely generated, or equivalently that there exists an intermediate extensionRemark. It is not, however, true in general that Sel p ∞ (A/L) stabilizes as L runs over the subfields of k S , which have finite degree over Q. In fact, we have the following example from a recent paper of Tim and Vladimir Dokchitser; see [4, Theorem 1.1 and Example 1.5]. Suppose that k = Q, S = {q} for any prime q, and let Q n be the nth layer of the cyclotomic Z q -extension of Q. If we let A be the curve defined by A : y 2 + y = x 3 − x 2 − 10x − 20, then v 5 (#X 0 (A/Q n )[5 ∞ ]) q n − c q for some constant c q ∈ Z. Here the notation X 0 means the Tate-Shafarevich group X modulo its divisible part X div .

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“…Points (ii) and (iii) were motivated by, respectively, Iwasawa theoretic considerations (see [5] Thm. 5.5) and applications to the p-parity conjecture that require knowing the behaviour of Birch-Swinnerton-Dyer quotients up to rational squares in field extensions (see §1.3 below, and §3.4).…”

Section: Introductionmentioning

confidence: 99%

“…However, the semistability assumption is necessary to obtain such a stable growth: for example, the Tamagawa number of the elliptic curve 243a1 fluctuates between 1 and 3 in the layers of the cyclotomic Z 3 -tower of Q 3 , see [5] Remark 5.4.…”

Section: Introductionmentioning

confidence: 99%

Variation of Tamagawa numbers of semistable abelian varieties in field extensions

Betts

Dokchitser

Morgan

2018

Math. Proc. Camb. Phil. Soc.

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Abstract. We investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on the p-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the form y 2 = f (x), under some simplifying hypotheses.

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Growth of in towers for isogenous curves

Cited by 9 publications

References 21 publications

Local invariants of isogenous elliptic curves

Local invariants of isogenous elliptic curves

An analogue of the Washington-Sinnott theorem for elliptic curves with complex multiplication I

Variation of Tamagawa numbers of semistable abelian varieties in field extensions

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