Journal ArticleDuke Mathematical Journal · January 1, 2024
Let F be a totally real field of degree n, and let p be an odd prime. We prove the p-part of the integral Gross–Stark conjecture for the Brumer–Stark p-units living in CM abelian extensions of F . In previous work, the first author showed that such a resul ...
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Journal ArticleAnnals of Mathematics · January 1, 2023
Let H=F be a finite abelian extension of number fields with F totally real and H a CM field. Let S and T be disjoint finite sets of places of F satisfying the standard conditions. The Brumer-Stark conjecture states that the Stickelberger element ΦH/FS,T an ...
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Journal ArticleAdvanced Studies in Pure Mathematics · January 1, 2020
In this note, we provide a new proof of the rank 1 Gross–Stark conjecture for a quadratic extension under the assumption that there is only one prime in the base field above a rational prime p. The full Gross–Stark conjecture was proven by the authors in j ...
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Journal ArticleTransactions of the American Mathematical Society · January 1, 2019
We present a conjectural formula for the principal minors and the characteristic polynomial of Gross’s regulator matrix associated to a totally odd character of a totally real field. The formula is given in terms of the Eisenstein cocycle, which was define ...
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Journal ArticleAnnals of Mathematics · November 1, 2018
In 1980, Gross conjectured a formula for the expected leading term at s=0 of the Deligne-Ribet p-adic L-function associated to a totally even character ϕ of a totally real field F. The conjecture states that after scaling by L(ϕω-1,0), this value is equal ...
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Journal ArticleProceedings of the American Mathematical Society · January 1, 2018
We prove that if p ≡ 4, 7 (mod 9) is prime and 3 is not a cube modulo p, then both of the equations x3 + y3 = p and x3 + y3 = p2 have a solution with x, y ∈ ℚ. ...
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Journal ArticleJournal of the European Mathematical Society · January 1, 2018
We prove a conjecture of Gross regarding the “order of vanishing” of Stickelberger elements relative to an abelian tower of fields and give a cohomological construction of the conjectural Gross-Stark units. This is achieved by introducing an integral versi ...
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Journal ArticleAnnales Mathematiques du Quebec · August 1, 2016
This paper is an announcement of the following result, whose proof will be forthcoming. Let F be a totally real number field, and let F⊂ K⊂ L be a tower of fields with L / F a finite abelian extension. Let I denote the kernel of the natural projection from ...
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Journal ArticleInventiones Mathematicae · July 1, 2016
We prove that the p-adic L-series of the tensor square of a p-ordinary modular form factors as the product of the symmetric square p-adic L-series of the form and a Kubota–Leopoldt p-adic L-series. This establishes a generalization of a conjecture of Citro ...
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Journal ArticleCompositio Mathematica · June 18, 2015
We compute the p-adic L-functions of evil Eisenstein series, showing that they factor as products of two Kubota-Leopoldt p-adic L-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens. ...
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Journal ArticleCommentarii Mathematici Helvetici · January 1, 2015
We define a cocycle on GLn(Q) using Shintani's method. This construction is closely related to earlier work of Solomon and Hill, but differs in that the cocycle property is achieved through the introduction of an auxiliary perturbation vector Q. As a corol ...
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Chapter · January 1, 2014
This chapter surveys six different special value formulae for p-adic L-functions, stressing their common features and their eventual arithmetic applications via Kolyvagin’s theory of “Euler systems”, in the spirit of Coates-Wiles and Kato-Perrin-Riou. ...
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Journal ArticleAlgebra and Number Theory · January 1, 2012
In earlier work, the second named author described how to extract Darmon-style ℒ-invariants from modular forms on Shimura curves that are special at p. In this paper, we show that these ℒ-invariants are preserved by the Jacquet-Langlands correspondence. As ...
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Journal ArticleAnnals of Mathematics · January 1, 2011
Let F be a totally real field and χ an abelian totally odd character of F. In 1988, Gross stated a p-adic analogue of Stark's conjecture that relates the value of the derivative of the p-adic L-function associated to χ and the p-adic logarithm of a p-unit ...
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Journal ArticleJournal of Mathematical Sciences · December 1, 2009
Let F be a totally real number field of degree n, and let H be a finite abelian extension of F. Let p denote a prime ideal of F that splits completely in H. Following Brumer and Stark, Tate conjectured the existence of a p-unit u in H whose p-adic absolute ...
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Journal ArticleDuke Mathematical Journal · June 1, 2008
Let F be a totally real number field, and let p be a finite prime of F such that p splits completely in the finite abelian extension H of F. Tate has proposed a conjecture [22, Conjecture 5.4] stating the existence of a p-unit u in H with absolute values a ...
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Journal ArticleCanadian Journal of Mathematics · January 1, 2007
Let K be a real quadratic field, and p a rational prime which is inert in K. Let a be a modular unit on Γ0(N). In an earlier joint article with Henri Darmon, we presented the definition of an element u(α, τ) ε Kpx attached to a and each τ ε K. We conjectur ...
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Journal ArticleAnnales Scientifiques de l'Ecole Normale Superieure · May 1, 2005
We present a construction which lifts Darmon's Stark-Heegner points from elliptic curves to certain modular Jacobians. Let N be a positive integer and let p be a prime not dividing N. Our essential idea is to replace the modular symbol attached to an ellip ...
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Journal ArticleJournal of Algebra · March 15, 2001
For a ring R with identity, define Unipn(R) to be the group of upper-triangular matrices over R all of whose diagonal entries are 1. For i = 1,2,...,n - 1, let Si denote the matrix whose only nonzero off-diagonal entry is a 1 in the ith row and (i + 1)st c ...
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Journal ArticleIsrael Journal of Mathematics · January 1, 2001
Let A = {a1,..., ak} and B = {b1,..., bk} be two subsets of an Abelian group G, k ≤ |G|. Snevily conjectured that, when G is of odd order, there is a permutation π ≤ Sk such that the sums ai + bπ(i), 1 ≤ i ≤ k, are pairwise different. Alon showed that the ...
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Journal ArticleSIAM Journal on Discrete Mathematics · January 1, 1999
Arrow's celebrated impossibility theorem states that a sufficiently diverse domain of voter preference profiles cannot be mapped into social orders of the alternatives without violating at least one of three appealing conditions. Following Fishburn and Kel ...
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