Non-abelian congruences between L-values of elliptic curves

Delbourgo, Daniel; Ward, Thomas

doi:10.5802/aif.2377

Cited by 11 publications

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“…Delbourgo and the author also proved a congruence of this form (using Hilbert modular forms as described above) but modulo a smaller power of p. We did this for semistable elliptic curves in [5], and extended our results to the case of CM curves in [6].…”

Section: Introductionmentioning

confidence: 85%

“…This approach was used by Bouganis and V. Dokchitser in [3] to prove algebraicity properties for these L-values. It was then used in [5] by Delbourgo and the author to construct an abelian p-adic L-function L p (E, ρ n ) ∈ Z p [[U (n) ]] interpolating the values L(E, ρ n,Q ⊗ ψ, 1) for characters ψ : U (n) → Q × .…”

Section: Introductionmentioning

confidence: 99%

“…for any h in S k (cm 0 , η). Holomorphic projection was not required in [5], where the only weight considered was k = 2; however the Eisenstein series E k−1 (s − k + 1) will be non-holomorphic at some critical values if k > 2. For convenience we will put Φ(g, s) := Hol g m0 E k−1 (s, ηω −1 ) .…”

Section: Introductionmentioning

confidence: 99%

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Congruences for convolutions of Hilbert modular forms

Ward¹

2012

Math. Proc. Camb. Phil. Soc.

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Section: Introductionmentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

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Congruences for convolutions of Hilbert modular forms

Ward¹

2012

Math. Proc. Camb. Phil. Soc.

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“…For elliptic curves there are some evidences for the existence of such non-abelian p-adic L-functions offered in [4,9] and also some computational evidences offered in [10,11]. Also, there is some recent progress, achieved in [6], for elliptic curves with complex multiplication defined over Q with respect to the padic Lie extension obtained by adjoing to Q the p-power torsion points of the elliptic curve.…”

Section: Introductionmentioning

confidence: 99%

Non-abelian p-adic L-functions and Eisenstein series of unitary groups – The CM method

Bouganis¹

2014

Ann. inst. Fourier

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“…We can now define a two-variable p-adic L-function 3 interpolating the standard versions of Panchiskin et al [2,5,16,17] over Spec(‫. )މ‬ (A.1.3) Thirdly, replacing 'g l V p ' with 'g l ' one deduces…”

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

Delbourgo

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

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Non-abelian congruences between L-values of elliptic curves

Cited by 11 publications

References 8 publications

Congruences for convolutions of Hilbert modular forms

Congruences for convolutions of Hilbert modular forms

Non-abelian p-adic L-functions and Eisenstein series of unitary groups – The CM method

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

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