William G. McCallum

On the method of Coleman and Chabauty

Let C be a curve of genus g >_ 2, defined over a number field K , and let J be the Jacobian of C. Coleman [C2], following Chabauty, has shown how to obtain good bounds on the cardinality of C(K) if the rank r of the Mordell-Weil group J(K) is less than g. The key to the method is to construct a logarithm on J(Kv), for some valuation v of K , whose kernel contains J(K), and whose restriction to C(Kv) is represented explicitly as the integral of a differential. This paper is an attempt to make the case, through a detailed examination of the case of Fermat curves, that this method can be fashioned into a quite precise tool for bounding rational points on curves. We show how to transform an element of the Selmer group of the Jacobian of a Fermat curve of degree 19 into a p-adic analytic function on the curve itself, whose zero set contains all the rational points. As a consequence, we prove the second case of Fermats Last Theorem for regular primes, The method depends on the existence of a suitable element in the Selmer group; for the lack of a satisfactory theory of descent for Jacobians of Fermat curves, we can only show that this element exists in the case that p is regular. Of course, in that case, Kummer had already proved the whole of Fermats Last Theorem. However, we believe the interest of this paper is in the method, not the theorem, and as such is independent of Kumaner, and also of the recent work of Wiles. Our method is different, and offers a development of the method of Coleman and Chabauty, many aspects of which are generalizable to arbitrary curves, although we do not attempt to make that generalization here. We now describe the contents of the paper in more detail. Let p be an odd prime, and let F be the pm Fermat curve, with projective equation

A cup product in the Galois cohomology of number fields

This paper is devoted to the consideration of a certain cup product in the Galois cohomology of an algebraic number field with restricted ramification. Let n be a positive integer, let K be a number field containing the group μn of nth roots of unity, and let S be a finite set of primes including those above n and all real archimedean places. Let GK,S denote the Galois group of the maximal extension of K unramified outside S (inside a fixed algebraic closure of K). We consider the cup product H(GK,S, μn)⊗H(GK,S, μn)→ H(GK,S, μ⊗2 n ). ()

The Goals of the Calculus Course

It has been more than a decade since the beginning of efforts to renew calculus teaching in this country. Those efforts, successful beyond the wildest dreams of their initiators, have reached a plateau. The reaction has been at times intemperate and at times thoughtful, and has led to serious efforts to reinvigorate the traditional curriculum (see How to Teach Mathematics 1 and its appendixes).

Common U.S. math standards.

The United States has never had national standards or curricula for mathematics. Instead, each state decides, for example, what version of high-school algebra should be taught and how to assess whether students have learned it. In March 2010, the National Governors Association and the Council of Chief State School Officers released Common Core State Standards (CCSS) for English language arts and mathematics (www.corestandards.org). The goal is to have all states adopt these as their standards, making more uniform the knowledge and skills expectations for students in preparation for success in college and careers.

The Common Core State Standards in Mathematics

The US Common Core State Standards in Mathematics were released in 2009 and have been adopted by 45 states. We describe the background, process, and design principles of the standards.

Assessing the Strands of Student Proficiency in Elementary Algebra

In order to assess something, you have to have some idea of what that something is. Algebra means many different things in today’s schools (for a discussion, see [RAN03], Chapter 4). In particular, the study of algebra is often blended with the study of functions. Although it is true that the notion of a function is lurking behind much of beginning algebra, and that there is algebraic aspect to many of the tasks we want our students to carry out with functions, the conglomeration of algebra and functions has led to considerably muddy waters in the teaching of algebra. Therefore, I’d like to spend some time clarifying my own stance before talking about how to assess proficiency. In the process, while acknowledging other possible uses, I will use the word algebra to mean elementary algebra, that is, the study of algebraic expressions and equations in which the letters stand for numbers. In the progression of ideas from arithmetic to algebra to functions, there is an increase in abstraction at each step, and the increase at the second step is at least as large as that at the first. In the step from arithmetic to algebra, we learn to represent numbers by letters, and calculations with numbers by algebraic expressions. In the step from algebra to functions we learn of a new sort of object, function, and represent functions themselves by letters. There are two complementary dangers in the teaching of algebra, each of which can cause students to miss the magnitude of this step. The first danger is that functions and function notation appear to the students to be mostly a matter of using a sort of auxiliary notation, so that if a function f is defined by f(x) = x − 2x − 3, say, then f(x) is nothing more than a short-hand notation for the expression x−2x−3. This results in a confusion between functions and the expressions representing them, which in turn leads to a broad area of confusion around the ideas of equivalent expressions and transforming expressions. For example, we want students to understand that (x + 1)(x − 3) and (x − 1) − 4 are equivalent expressions, each of which reveals different aspects of the same function. But without a strong notion of function as an object distinct from the expression defining it, the significance of

The Double Continuity of Algebra

We consider Klein’s double discontinuity between high school and university mathematics in relation to algebra as it is studied in both settings. We give examples of two kinds of continuities that might mend the break: (1) examples of how undergraduate courses in algebra and number theory can provide useful tools for prospective teachers in their professional work, as they design and sequence mathematical tasks, and (2) examples of how questions that arise in secondary pre-college mathematics can be extended and analyzed with methods from algebra and algebraic geometry, using both a careful analysis of algebraic calculations and the application of algebraic methods to geometric problems. We discuss useful sensibilities, for high school teachers and university faculty, that are suggested by these examples. We conclude with some recommendations about the content and structure of abstract algebra courses in university.

Curricular Coherence in Mathematics

Building on the work of Schmidt et al., we propose a definition of curricular coherence for K–12 mathematics that encompasses both the arrangement of topics, which we call coherence of content, and the habits of mind the curriculum fosters in students, which we call coherence of practice. We give examples to illustrate each.

What Is and What Might Be the Legacy of Felix Klein

Felix Klein always emphasised the great importance of teaching at the university, and he strongly promoted the modernisation of mathematics in the classrooms. The three books “Elementary Mathematics from a higher (advanced) standpoint” from the beginning of the last century gave and still give a model for university lectures especially for student teachers. The “Merano Syllabus” (1905), essentially initiated and influenced by Felix Klein, pleaded for an orientation of mathematics education at the concept of function, an increased emphasis on spatial geometry and an introduction of calculus in high schools. The Thematic Afternoon “The legacy of Felix Klein” will give an overview about the ideas of Felix Klein, it will highlight some developments in university teaching and school mathematics related to Felix Klein in the last century, and it will especially be asked for the meaning, the importance and the legacy of Klein’s ideas nowadays and in the future in an international, worldwide context.

Brauer points on Fermat curves

The Hasse principle is said to hold for a class of varieties over a number field K if for any variety X in the class, the set of rational points X(K) is non-empty whenever the set of adelic points X(AK) is non-empty. Manin [Man] observed that the failure of the Hasse principle can often be explained in terms of the Brauer group of X, Br(X). The product rule implies that X(K) must be contained in the set of Brauer points