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Shintani's Method: zeta values and Stark units
Full text linkWe prove a formula relating Dedekind zeta functions associated to a number field to certain Shintani zeta functions, whose analytic properties and values at non-positive integers have been well studied by Takuro Shintani. This allows us to compute explicit formulas for Dedekind zeta functions, partial zeta functions and certain -series and their derivatives evaluated at non-positive integers. We relate the explicitly given value of the derivative of partial zeta functions at to those predicted by abelian Stark's conjecture. Though this conjecture remains open, we are able to write down explicit formulas for the absolute values of the conjectured Stark units.The main ingredient in these formulas is an explicit proof of Shintani's unit theorem for number fields of arbitrary signature. This says that the totally positive units of a number field has a fundamental domain given by a signed union of polyhedral cones in the Minkowski space of the field. Existence of such domains was known to Shintani. In the case is a totally real field, Colmez, Diaz y Diaz--Friedman and Charollois-Dasgupta-Greenberg were able to construct such domains and give their generators explicitly. We give an explicit construction of such domains for number fields of arbitrary signature with an exact formula for the domain. Moreover, our construction is cohomological, allowing for future cohomological applications of Shintani's method as in the work of Charollois--Dasgupta--Greenberg.This construction allows us to write Dedekind zeta functions and partial zeta functions in terms of certain analytic zeta functions defined over polyhedral cones (Shintani zeta functions). Thus we are able to translate questions about special values of Dedekind zeta functions to those about special values of Shintani zeta, whose values at non-positive integers are given by closed finite expressions due to work of Shintani
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Families of half-integer weight Eisenstein series
Full text linkWe use explicit formulas for the Fourier coefficients of a certain set of half-integer weight Eisenstein series to determine the appropriate analogue of -stabilization for those forms. We discover that the series does not live in a -adic family in a traditional sense, but that it can be recognized as a linear combination, with slightly different non-analytic coefficients, of two -adic families
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On the Compatibility of Two Conjectures Concerning -adic Gross-Stark Units
Full text linkWe give a proof of the consistency of Dasgupta's conjectural -adic formula for Gross-Stark units with Dasgupta and Spiess's alternative conjectural formula for these units. We give details of the proof when is a totally real number field of degree 2, which had been previously proven by Dasgupta and Spiess. We present work towards proving the case for a general totally real number field. Finally, we give a proof when is a totally real number field of degree 3
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Stark's Conjectures for p-adic L-functions
Full text linkWe give a new definition of a p-adic L-function for a mixed signature character of a real quadratic field and for a nontrivial ray class character of an imaginary quadratic field. We then state a p-adic Stark conjecture for this p-adic L-function. We prove our conjecture in the case when p is split in the imaginary quadratic field by relating our construction to Katz's p-adic L-function. We also prove our conjecture in the real quadratic setting for one special case and give numerical evidence in one specific example
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
A Shintani-type formula for Gross--Stark units over function fields
No full textLet 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 values are related in a precise way to the partial zeta-functions of the extension H/F. Gross later refined this conjecture by proposing a formula for the p-adic norm of the element u. Recently, using methods of Shintani, the first author refined the conjecture further by proposing an exact formula for u in the p-adic completion of H. In this article we state and prove a function field analogue of this Shintani-type formula. The role of the totally real field F is played by the function field of a curve over a finite field in which n places have been removed. These places represent the “real places” of F. Our method of proof follows that of Hayes, who proved Gross’s conjecture for function fields using the theory of Drinfeld modules and their associated exponential functions
Brumer-Stark Units and Explicit Class Field Theory
Full text linkLet be a totally real field of degree and an odd prime. We prove
the -part of the integral Gross--Stark conjecture for the Brumer--Stark
-units living in CM abelian extensions of . In previous work, the first
author showed that such a result implies an exact -adic analytic formula for
these Brumer--Stark units up to a bounded root of unity error, including a
``real multiplication'' analogue of Shimura's celebrated reciprocity law from
the theory of Complex Multiplication. In this paper we show that the
Brumer--Stark units, along with other easily described elements (these
are simply square roots of certain elements of ) generate the maximal
abelian extension of . We therefore obtain an unconditional construction of
the maximal abelian extension of any totally real field, albeit one that
involves -adic integration for infinitely many primes .
Our method of proof of the integral Gross--Stark conjecture is a
generalization of our previous work on the Brumer--Stark conjecture. We apply
Ribet's method in the context of group ring valued Hilbert modular forms. A key
new construction here is the definition of a Galois module \nabla_{\!\sL}
that incorporates an integral version of the Greenberg--Stevens \sL-invariant
into the theory of Ritter--Weiss modules. This allows for the reinterpretation
of Gross's conjecture as the vanishing of the Fitting ideal of
\nabla_{\!\sL}. This vanishing is obtained by constructing a quotient of
\nabla_{\!\sL} whose Fitting ideal vanishes using the Galois representations
associated to cuspidal Hilbert modular forms..Comment: 70 pages (to appear in Duke Math. J.
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