Coefficient rings of Tate formal groups determining Krichever genera

In the memoir En metod att komma i analytisk besittning af de elliptiska funktionernat we have shown that the method given by Abel in his Recherches sur les fonctions elliptiques,+ which is historically the earliest method of deriving the elliptic functions, is also mathematically sound and leads to the desired result with perfect rigor. The method of Abel has the advantage over all other methods that it does not presuppose any theorems from the general theory of functions. It is consequently the most elementary of all mathematical processes which lead to a real command of the analytical properties of the elliptic functions. A few serious objections can be raised against Abel's own presentation, however. The most important of these is that the passage from the multiplication theorems to the infinite double series and double products is not sufficiently justified; in fact, it is based upon an argument which it is scarcely possible to make mathematically rigorous. This has been pointed out by Broch in his paper Om de elliptiske Funktioners Roekkeudvikling.? In this paper, as well as in his learned and suggestive book Traite Elementaire des Fonctions Elliptiques, ff there is developed a new derivation of this limiting passage which is perfectly distinct from that of Abel.

The object of the memoir is to notice certain forms into which the primary elliptic functions admit of being thrown, and to discuss the identical relations to which they give rise. These latter, it is shown, can be obtained directly by the aid of Fourier’s theorem , or in a less straight-forward manner by ordinary algebra.

In this chapter we consider how elliptic function theory and complex variable theory were finally drawn together in the 1830s and 1840s. As the recognition of the importance of the work of Abel and Jacobi grew, mathematicians came to feel that it was unsatisfactory to base the theory of elliptic functions on the inversion of many-valued integrals. One alternative would have been to adopt and develop Cauchy’s theory of complex integrals. By and large this was not done, and it is interesting to examine why. The study of elliptic integrals was felt by many to be fraught with ambiguity because of the square root in the integrand. Moreover, Cauchy’s system of definitions, based on his newly defined concepts of limit, continuity, differentiability, and integrability, was incompatible with talk of many-valued functions—Cauchy did not define continuity for a many-valued function, and indeed a many-valued function cannot be continuous according to Cauchy’s use of the term. Although a doubly periodic function is a meromorphic function defined on the whole of the complex plane, an elliptic integral makes better sense on something like a Riemann surface (a torus in this case). Thus the many-valued nature of an elliptic integral posed a challenge to mathematicians throughout the 1830s and 1840s. So the perceived problem with the foundations did not meet with a ready answer in the newly emerging theory of complex functions. Matters were to be worse with hyperelliptic integrals, because the corresponding inverse functions could not be treated as multiply-periodic functions in the plane.

Hopf algebras of set systems

2 - ARTHUR CAYLEY (1821–1895)

Introduction Arthur Cayley's 1854 paper On the theory of groups, as depending on the symbolic equation n = 1 inaugurated the abstract idea of a group [2]. I have used this very understandable paper several times for reading, discussion, and homework in teaching introductory abstract algebra, where students are rst introduced to group theory. Many facets of this short paper provide wonderful pedagogical bene ts, not least of which is tremendous motivation for the modern theory. More information on teaching with original historical sources is available at our web resource [8], which includes information on the e cacy of teaching with original sources [3, 4, 5, 9, 10], about courses based entirely on original sources [6, 7], and which provides access to books and other resources for teaching with original sources. Cayley introduces groups as a unifying framework for several critical examples current in the early nineteenth century. These include \substitutions (permutations, with reference to Galois on solutions of equations), \quaternion imaginaries, compositions arising in the theory of elliptic functions, and transposition and inversion of matrices. These phenomena are either familiar to or easily understandable for our students. Cayley observes that they all have common features, which he abstracts in a rst attempt at de ning a group axiomatically. He proceeds to develop initial steps of a theory, including stating and using what Lagrange's theorem tells us on orders of elements, which classi es groups of prime order. Then he provides a detailed classi cation of all groups of order up to six. Cayley gives a brief description of how to form what we call a group algebra, and comments that when applied to the six-element symmetric group on three letters, the group

Invariants, torsion indices and oriented cohomology of complete flags

Let h* be a complex (or real) oriented cohomology theory. It has recently been observed that the of h* plays a very important role in such a theory. Above all, D. Quillen [6] showed that the formal group FU of the complex cobordism theory MU* ( ) is isomorphic to the Lazard universal formal group, and its coefficients generate the ground ring MU* (pt). Unfortunately, in the case of quaternionic oriented cohomology theory (see § 2), there exists no such formal group, since the tensor product of quaternionic line bundles does not yield a quaternionic bundle. This situation makes it difficult, for example, to produce enough generators of MSp*(pt). However there are some substitutes for the formal group. In particular, using the total Pontrjagin class of a certain quaternionic vector bundle 2C (see § 3), N. Ja. Gozman [5] defined a subring A of MU* (pt) which is contained in the image of the forgetful homomorphism (p: MSp* (pt}-*MU*(pt). In this paper, we will generalize his approach to arbitrary quaternionic oriented cohomology theory 7i*, and define a subring Ah of h* (pt) which is generated by the coefficients of certain power series. Then it will be shown that

We explain a new relationship between formal group laws and ring spectra in\nstable homotopy theory. We study a ring spectrum denoted DB which depends on a\ncommutative ring B and is closely related to the topological Andre-Quillen\nhomology of B. We present an explicit construction which to every 1-dimensional\nand commutative formal group law F over B associates a morphism of ring spectra\nF_*: HZ --> DB from the Eilenberg-MacLane ring spectrum of the integers. We\nshow that formal group laws account for all such ring spectrum maps, and we\nidentify the space of ring spectrum maps between HZ and DB. That description\ninvolves formal group law data and the homotopy units of the ring spectrum DB.\n

Computing l-isogenies between elliptic curves defined over a finite field Fq of small characteristic is of some importance for the computation of the cardinalities of elliptic curves using Schoof-Atkin-Elkies method. In a previous publication [3] we showed that this could be achieved in O(l3+e) multiplications in \(\mathbb{F}_q \)using formal groups. This method has been implemented by Lercier and Morain who obtained spectacular results in this direction [7, 8]. Nevertheless, the use of formal groups seems to be a serious deterent both for man and machine to perform such a work. More recently Lercier proposed an algorithm specific for characteristic 2 that has the same assymptotic complexity but is faster by some significative constant factor. In this paper we propose a general algorithm which does not use formal groups. Instead we take advantage of the elementary Galois properties of the p-torsion. This algorithm has the same complexity as the previous ones if we don't use fast multiplication techniques. But, contrary to the previous methods, it allows the use of fast multiplication for polynomials and then turns out to run in O(l2+e) multiplications in the field \(\mathbb{F}_q \). Our algorithm has also the advantage that it is made exclusively of very classical routines in polynomial and elliptic curve arithmetic. Also one may expect that the implementation of this method should require less work than the previous ones thus bringing new people to this kind of calculation.

Buhstaber's work on two-valued formal groups

In this article, we derive the quintuple, Hirschhorn and Winquist product identities using the theory of elliptic functions. Our method can be used to establish generalizations of these identities due to the second author.

According to JACOBI, the theory of elliptic functions was born between the twenty-third of December 1751 and the twenty-seventh of January 1752. On the former date, the Berlin Academy of Sciences handed over to EULER the two volumes of Marchese FAGNANO’s Produzioni Matematiche, published in Pesaro in 1750 and just received from the author; Euler was requested to examine the book and draft a suitable letter of thanks. On the latter date, Euler, referring explicitly to Fagnano’s work on the lemniscate, read to the Academy the first of a series of papers, eventually proving in full generality the addition and multiplication theorems for elliptic integrals.KeywordsNumber TheoryArithmetic ProgressionTwelfth CenturyEuclidean AlgorithmTriangular NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Hermite, in the course of his investigations in the theory of elliptic functions, was led to introduce certain related functions which he called doubly periodic functions of the second kind.

A theta function identity of degree 7 is established with the theory of elliptic functions. A multitude of new modular identities are derived from this theta function identity. Some finite trigonometric sums are given.