Elliptic curves with a given number of points over finite fields

Given an elliptic curve $E$ and a positive integer $N$, we consider the problem of counting the number of primes $p$ for which the reduction of $E$ modulo $p$ possesses exactly $N$ points over $\mathbb F_p$. On average (over a family of elliptic curves), we show bounds that are significantly better than what is trivially obtained by the Hasse bound. Under some additional hypotheses, including a conjecture concerning the short interval distribution of primes in arithmetic progressions, we obtain an asymptotic formula for the average.Comment: A mistake was discovered in the derivation of the product formula for K(N). The included corrigendum corrects this mistake. All page numbers in the corrigendum refer to the journal version of the manuscrip

Mapping and Displaying Structural Transformations between XML and PDF

Documents are often marked up in XML-based tagsets to delineate major structural components such as headings, paragraphs, figure captions and so on, without much regard to their eventual displayed appearance. And yet these same abstract documents, after many transformations and 'typesetting' processes, often emerge in the popular format of Adobe PDF, either for dissemination or archiving. Until recently PDF has been a totally display-based document representation, relying on the underlying PostScript semantics of PDF. Early versions of PDF had no mechanism for retaining any form of abstract document structure but recent releases have now introduced an internal structure tree to create the so called 'Tagged PDF'. This paper describes the development of a plugin for Adobe Acrobat which creates a two-window display. In one window is shown an XML document original and in the other its Tagged PDF counterpart is seen, with an internal structure tree that, in some sense, matches the one seen in XML. If a component is highlighted in either window then the corresponding structured item, with any attendant text, is also highlighted in the other window. Important applications of correctly Tagged PDF include making PDF documents reflow intelligently on small screen devices and enabling them to be read out in correct reading order, via speech synthesiser software, for the visually impaired. By tracing structure transformation from source document to destination one can implement the repair of damaged PDF structure or the adaptation of an existing structure tree to an incrementally updated document

Creating Structured PDF Files Using XML Templates

This paper describes a tool for recombining the logical structure from an XML document with the typeset appearance of the corresponding PDF document. The tool uses the XML representation as a template for the insertion of the logical structure into the existing PDF document, thereby creating a Structured/Tagged PDF. The addition of logical structure adds value to the PDF in three ways: the accessibility is improved (PDF screen readers for visually impaired users perform better), media options are enhanced (the ability to reflow PDF documents, using structure as a guide, makes PDF viable for use on hand-held devices) and the re-usability of the PDF documents benefits greatly from the presence of an XML-like structure tree to guide the process of text retrieval in reading order (e.g. when interfacing to XML applications and databases)

DLR equations and rigidity for the Sine-beta process

We investigate Sine$_\beta$, the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of one-dimensional log-gases, or $\beta$-ensembles, at inverse temperature $\beta>0$. We adopt a statistical physics perspective, and give a description of Sine$_\beta$ using the Dobrushin-Lanford-Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine$_\beta$ to a compact set, conditionally to the exterior configuration, reads as a Gibbs measure given by a finite log-gas in a potential generated by the exterior configuration. Moreover, we show that Sine$_\beta$ is number-rigid and tolerant in the sense of Ghosh-Peres, i.e. the number, but not the position, of particles lying inside a compact set is a deterministic function of the exterior configuration. Our proof of the rigidity differs from the usual strategy and is robust enough to include more general long range interactions in arbitrary dimension.Comment: 46 pages. To appear in Communications on Pure and Applied Mathematic

The Right to Arms and Standards of Review: A Tale of Three Circuits Symposium Article

In District of Columbia v. Heller and McDonald v. Chicago, the U.S. Supreme Court recognized an individual right to arms. The Court offered limited guidance as to standard of review, only ruling out rational basis. This Article takes a pragmatic approach to the standard of review issue. First, it explores the practical basis for heightened scrutiny. Judicial review is, at its base, anti-majoritarian. It is therefore often appropriate to employ the loose standard of rational basis. In some contexts, however, majoritarianism is less dependable: the majority may be tempted to unfairly burden the interests of the minority. Here, courts properly employ heightened standards of review. I suggest that this rationale is applicable in the context of firearms control. While, on a national basis, firearms owners are a sufficiently large group to legislatively defend their legitimate interests, the story is different at the state level. The strictest forms of gun control originate in states where gun owners comprise only one-eighth to one-fifth of the population, and have little legislative influence. The result is legislation that is unsupported by existing criminological data, and sometimes serves no demonstrable purpose other than burdening or annoying gun owners. This Article closes by suggesting that varying levels of scrutiny, drawn from election-law cases, are appropriate. The least burdensome legislation is weighed under a near-rational basis standard. Other measures are properly weighed under intermediate review—genuine intermediate review, where the legislation must be supported by hard data, or under stricter exacting scrutiny. Finally, measures regulating what the Court has termed the “core right” to arms—possession by law-abiding citizens in the home—should be evaluated under strict scrutiny

The Rise and Demise of the Collective Right Interpretation of the Second Amendment

This article explores the origins of the two competing theories of the Second Amendment -- the individual rights approach which carried the majority in Heller and McDonald, and the variants of a collective right theory which was previously dominant in the lower courts, and one variant of which was endorsed by the Heller dissents. Careful analysis of states\u27 bills of rights of the Framing period suggests that two guarantees were desired, by different political factions. Framers closely adhering to the Classical Republican point of view favored protection for the militia as a system; those favoring the emerging Jeffersonian point of view favored guarantees of individual rights to arms. The Second Amendment has two clauses because it has two purposes. ...This article will examine the history of the two competing theories in terms of the Framing period, the developments in American law prior to the Civil War, and the understandings of courts, commentators, and Congress following that War. We will then review developments of the Twentieth Century, including the origins of collective right interpretation over that period