On special zeros of p-adic L-functions of Hilbert modular forms

Spiess, Michael

doi:10.1007/s00222-013-0465-0

Cited by 56 publications

(142 citation statements)

References 23 publications

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“…holds, where rec p is the local reciprocity map at a prime of E lying above p. This generalizes Molina's work [17] on exceptional zeros of anticyclotomic p-adic Lfunctions in the CM case and is heavily inspired by Spieß' article [19]. Crucial in the definition of these automorphic periods are extension classes of the Steinberg representation, which were first studied by Breuil in [5].…”

Section: Introductionmentioning

confidence: 52%

“…In Section 3.7 of [19] Spieß constructs extensions of the Steinberg representation associated to characters of the multiplicative group of a p-adic field. Such extensions were already constructed by Breuil in [5] in case the character under consideration is a branch of the p-adic logarithm.…”

Section: L-invariantsmentioning

confidence: 99%

“…Proof. Parts (i) and (iii) are essentially proven in Lemma 3.11 of [19]. For the proof of (ii) let f * (E(l p )) be the pushout of the following diagram:…”

mentioning

confidence: 97%

“…Note that we get rid of the factor 2 showing up in Lemma 3.11 of [19], i.e. the extension class constructed above is "one half" of the extension class constructed in loc.cit.…”

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confidence: 99%

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Leading terms of anticyclotomic Stickelberger elements and 𝑝-adic periods

Bergunde

Gehrmann

2018

Trans. Amer. Math. Soc.

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Abstract. Let E be a quadratic extension of a totally real number field. We construct Stickelberger elements for Hilbert modular forms of parallel weight 2 in anticyclotomic extensions of E. Extending methods developed by Dasgupta and Spieß from the multiplicative group to an arbitrary one-dimensional torus we bound the order of vanishing of these Stickelberger elements from below and, in the analytic rank zero situation, we give a description of their leading terms via automorphic L-invariants. If the field E is totally imaginary, we use the p-adic uniformization of Shimura curves to show the equality between automorphic and arithmetic L-invariants. This generalizes a result of Bertolini and Darmon from the case that the ground field is the field of rationals to arbitrary totally real number fields.

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Section: Introductionmentioning

confidence: 52%

Section: L-invariantsmentioning

confidence: 99%

“…Proof. Parts (i) and (iii) are essentially proven in Lemma 3.11 of [19]. For the proof of (ii) let f * (E(l p )) be the pushout of the following diagram:…”

mentioning

confidence: 97%

“…Note that we get rid of the factor 2 showing up in Lemma 3.11 of [19], i.e. the extension class constructed above is "one half" of the extension class constructed in loc.cit.…”

mentioning

confidence: 99%

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Leading terms of anticyclotomic Stickelberger elements and 𝑝-adic periods

Bergunde

Gehrmann

2018

Trans. Amer. Math. Soc.

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“…In 2009, Mok [15,Theorem 1.1] extended the method to include totally real fields F, provided E has split multiplicative reduction at only a single place of F above p. Recently, Spieß [19,Theorem 5.10] has further developed work on the totally real case, and can now remove Mok's restriction that the elliptic curve be split multiplicative at only a single place.…”

Section: Introductionmentioning

confidence: 97%

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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On Extra Zeros of p-Adic L-Functions: The Crystalline Case

Benois

2014

Contributions in Mathematical and Computational Sciences

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Abstract. We formulate a conjecture about extra zeros of p-adic L-functions at near central points which generalizes the conjecture formulated in [Ben2]. We prove that this conjecture is compatible with Perrin-Riou's theory of p-adic L-functions. Namely, using Nekovář's machinery of Selmer complexes we prove that our L -invariant appears as an additional factor in the Bloch-Kato type formula for special values of Perrin-Riou's module of L-functions.

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On special zeros of p-adic L-functions of Hilbert modular forms

Cited by 56 publications

References 23 publications

Leading terms of anticyclotomic Stickelberger elements and 𝑝-adic periods

Leading terms of anticyclotomic Stickelberger elements and 𝑝-adic periods

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

On Extra Zeros of p-Adic L-Functions: The Crystalline Case

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