Non-commutative Iwasawa theory for elliptic curves with multiplicative reduction

Lei

2016

Ramanujan J

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Section: Estimations Of #X(e/k N )[ P ∞ ]mentioning

confidence: 80%

Estimating the growth in Mordell–Weil ranks and Shafarevich–Tate groups over Lie extensions

Lei

2016

Ramanujan J

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“…)ޑ‬ (b) While throughout one has assumed that q = p, the situation where q = p has been addressed by Lei and the author in [4]. The p-adic L-functions constructed in [4, Theorems 1 and 4] also exhibit exceptional zeroes, and moreover satisfy various p-power congruences predicted by a non-commutative Iwasawa Main Conjecture.…”

Section: Remarksmentioning

confidence: 99%

EXCEPTIONAL ZEROES OF P-ADIC L-FUNCTIONS OVER NON-ABELIAN FIELD EXTENSIONS

2015

Glasgow Math. J.

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Suppose E is an elliptic curve over ‫,ޑ‬ and p > 3 is a split multiplicative prime for E. Let q = p be an auxiliary prime, and fix an integer m coprime to pq. We prove the generalised Mazur-Tate-Teitelbaum conjecture for E at the prime p, overThe proof makes use of an improved p-adic L-function, which can be associated to the Rankin convolution of two Hilbert modular forms of unequal parallel weight.2000 Mathematics Subject Classification. 11F33, 11F41, 11F67, 11G40. Introduction.Let E denote a modular elliptic curve defined over the rationals of conductor N E . The behaviour of its Hasse-Weil L-function L(E, s) at s = 1 is a fundamental topic in modern number theory. Thanks to the efforts of Birch and Swinnerton-Dyer, there are some deep conjectures describing both the order of vanishing at s = 1 for these L-functions, and also a detailed formula predicting their leading terms. Despite much strong progress over the last thirty years, the original conjectures themselves remain unproven (except for curves whose analytic rank is ≤ 1).Assume p is a prime number. We fix once and for all embeddings τ p : ‫ޑ‬ → ‫ޑ‬ p and τ ∞ : ‫ޑ‬ → ‫,ރ‬ which enable us to view L-values both p-adically and over ‫.ރ‬ In an attempt to understand these questions from a non-Archimedean standpoint, Mazur et al. [13,21] constructed p-adic avatars of the classical complex L-series. For almost all primes, the order of vanishing of the p-adic L seems to agree with that of its complex cousin. However, in 1986, Mazur, Tate and Teitelbaum [14] discovered if p is a prime of split multiplicative reduction, the p-adic avatar vanishes at s = 1 regardless of how the classical L-function behaves there. Based on extensive calculation, they conjectured a derivative formula at s = 1, involving a mysterious L-invariant term defined via Iwasawa's logarithm (normalised so that log p (p) = 0).Throughout we suppose E has split multiplicative reduction at a prime p = 2. As a local G ‫ޑ‬ p -module, the elliptic curve admits the rigid-analytic parametrisation

“…• the elliptic curve E = 11A3, the prime p = 3 and (∆ 1 , ∆ 2 ) in the list (2,5), (2,7), (2,13), (2,17), (2,19), (2,23), (2,31), (2,37), (2,41), (5,7), (5,13), (5,17), (5,19), (5,23), (7,13), (7,17);…”

Section: Conjecture 12 the Family Of Congruencesmentioning

confidence: 99%

“…(i) As no Main Conjecture can be formulated over Q (d) ∞,∆ for Tate motives, the next obvious place to look for examples is from the theory of elliptic curves. If U (m) = Gal(Q(µ p ∞ )/Q(µ p m )), then sequences of p-adic L-functions belonging to the algebras Z p U (m) [p −1 ] have already been constructed in [1,[5][6][7]. (ii) Some weak congruences were established under technical hypotheses in [1,[5][6][7], inspired by the numerical evidence of the Dokchitser brothers [8].…”

Section: Introductionmentioning

confidence: 99%

Higher Order Congruences Amongst Hasse–weil -Values

Peters²

2014

J. Aust. Math. Soc.

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For the (d + 1)-dimensional Lie group G = Z × p Z ⊕d p , we determine through the use of p-power congruences a necessary and sufficient set of conditions whereby a collection of abelian L-functions arises from an element in K 1 (Z p G ). If E is a semistable elliptic curve over Q, these abelian L-functions already exist; therefore, one can obtain many new families of higher order p-adic congruences. The first layer congruences are then verified computationally in a variety of cases.2010 Mathematics subject classification: primary 11R23; secondary 11G40, 19B28.