Weierstrass preparation theorem for noncommutative rings

It is known from the Lubin-Tate theory that a Lubin–Tate formal group can be constructed by its distinguished isogeny, and an arbitrary power series (with a few restrictions) can be taken as such an isogeny. A similar statement is also known for Honda formal groups. In the present article, a similar statement is proved in detail for p-typical formal groups in the so-called small ramification case. It is also proved that as a distinguished homomorphism, in general, one cannot take a polynomial. Bibliography: 6 titles.     

The Hilbert pairing for formal groups over σ-rings

In the paper, formal groups over the rings of integers of σ-fields are studied. These fields were constructed by the first author in a previous paper. They are a generalization of the inertia field of a classical local field to an arbitrary complete discrete valuation field of characteristic zero. An analog of Honda’s theory for such formal groups is constructed. The arithmetic of the group of points in an extension of a σ-field that contains sufficiently many torsion points is studied. Using the classification of formal groups and the arithmetic results obtained, an explicit formula for the Hilbert pairing for formal groups over σ-fields is proved. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 5–58.     

The Lutz filtration as a Galois module in an extension without higher ramification

One considers the structure of the group of the points of a formal group and its Lutz filtration as a Galois module in an extension without higher ramification of a local field. Making use, on one hand, of Honda's theory on the classification of formal groups over complete local rings and, on the other hand, of a generalization to formal groups of the Artin-Hasse function, one constructs effectively an isomorphism between the group of points and some given additive free Galois module. In particular, in the multiplicative case one gives a new effective proof of Krasner's theorem on the normal basis of the group of principal units of a local field in extensions without higher ramification. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 160, pp. 182–192, 1987.     

Differential Modular Forms on Shimura Curves, I

The quotient of a Shimura curve by the isogeny equivalence relation is not an object of algebraic geometry. The paper shows how this quotient space becomes a geometric object in a more general geometry obtained from 'usual algebraic geometry', by adjoining a new operation; this operation looks like a 'Fermat quotient' and should be viewed as an arithmetic analogue of usual derivations.     

The isogeny conjecture for t-motives associated to direct sums of Drinfeld modules

Let K be a finitely generated field of transcendence degree 1 over a finite field. Let M be a t-motive over K of characteristic p₀, which is semisimple up to isogeny. The isogeny conjecture for M says that there are only finitely many isomorphism classes of t-motives M^′ over K, for which there exists a separable isogeny M^′→M of degree not divisible by p₀. For the t-motive associated to a Drinfeld module this was proved by Taguchi. In this article we prove it for the t-motive associated to any direct sum of Drinfeld modules of characteristic p₀≠0.     

Variation of the unit root along the Dwork family of Calabi–Yau varieties

We study the variation of the unit roots of members of the Dwork families of Calabi–Yau varieties over a finite field by the method of Dwork–Katz and also from the point of view of formal group laws. A p-adic analytic formula for the unit roots away from the Hasse locus is obtained. This work was supported in part by Professor N. Yui’s Discovery Grant from NSERC, Canada.     

The discrete logarithm problem from a local duality perspective

The discrete logarithm problem is analyzed from the perspective of Tate local duality. Local duality in the multiplicative case and the case of Jacobians of curves over p-adic local fields are considered. When the local field contains the necessary roots of unity, the case of curves over local fields is polynomial time reducible to the multiplicative case, and the multiplicative case is polynomial time equivalent to computing discrete logarithm in finite fields. When the local field does not contains the necessary roots of unity, similar results can be obtained at the cost of going to an extension that contains these roots of unity. There was evidence in the analysis that suggests that the minimal extension where the local duality can be rationally and algorithmically defined must contain the roots of unity. Therefore, the discrete logarithm problem appears to be well protected against an attack using local duality. These results are also of independent interest for algorithmic study of arithmetic duality as they explicitly relate local duality in the case of curves over local fields to the multiplicative case and Tate-Lichtenbaum pairing (over finite fields).     

The arithmetic of hyperbolic formal modules

This paper considers hyperbolic formal groups, which come from the elliptic curve theory, in the context of the theory of formal modules. In the first part of the paper, the characteristics of hyperbolic formal groups are considered, i.e., the explicit formulas for the formal logarithm and exponent; their convergence is studied. In the second part, the isogeny and its kernel and height are found; a p-typical logarithm is defined. The Artin–Hasse and Vostokov functions are then constructed; their correctness and main properties are evaluated.     

Paramètres algébriques de groupes formels et critères d'isogénie

We generalise G. and D. Chudnovsky's isogeny criterion for elliptic curves over Q to the case of Abelian varieties of dimension g. The proof is bused on an explicit construction of algebraic parameters of their formal groups.     

Isogeny Covariant Differential Modular Forms Modulo p

An interesting theory arises when the classical theory of modular forms is expanded to include differential analogs of modular forms. One of the main motivations for expanding the theory of modular forms is the existence of differential modular forms with a remarkable property, called isogeny covariance, that classical modular forms cannot possess. Among isogeny covariant differential modular forms there exists a particular modular form that plays a central role in the theory. The main result presented in the article will be the explicit computation modulo p of this fundamental isogeny covariant differential modular form.     

Cohen–Macaulay Loci of Modules

The Cohen–Macaulay locus of any finite module over a noetherian local ring A is studied, and it is shown that it is a Zariski-open subset of Spec A in certain cases. In this connection, the rings whose formal fibres over certain prime ideals are Cohen–Macaulay are studied.     

Software to support numerical analysis teaching

This paper describes a Fortran90 library designed to support the teaching of numerical analysis and its applications. As well as covering traditional material it introduces recent and important ideas in numerical computation such as interval arithmetic and automatic differentiation. The library rests on a module realpac which provides real arithmetic in a range of precisions with a choice of rounding strategies. This, in turn, supports the implementation of an interval arithmetic module intpac. Derived data types and overloaded operations help inexperienced users to interface with unfamiliar data types such as intervals. The library also includes more conventional modules such as lepac for solving linear systems and minpac for nonlinear optimization. These, however, can be enhanced by being linked to more sophisticated tools for sparse matrix handling and automatic differentiation. As well as showing the main structure and scope of the software, the paper mentions some exercises that have successfully been performed by students.     

The Yang-Mills measure in the Kauffman bracket skein module

For each closed, orientable surface , we construct a local, diffeomorphism invariant trace on the Kauffman bracket skein module . The trace is defined when |t| is neither 0 nor 1, and at certain roots of unity. At t = − 1, the trace is integration against the symplectic measure on the SU(2) character variety of the fundamental group of . Received: June 2, 2000     

On the existence of self-dual permutation codes of finite groups

Motivated by a research on self-dual extended group codes, we consider permutation codes obtained from submodules of a permutation module of a finite group of odd order over a finite field, and demonstrate that the condition “the extension degree of the finite field extended by n’th roots of unity is odd” is sufficient but not necessary for the existence of self-dual extended transitive permutation codes of length n + 1. It exhibits that the permutation code is a proper generalization of the group code, and has more delicate structure than the group code.     

Indecomposable Modules and Gelfand Rings

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. It is shown that each indecomposable module over a commutative ring R satisfies a finite condition if and only if R _P is an Artinian valuation ring for each maximal prime ideal P. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings. It is shown that each commutative ring R with a Hausdorff and totally disconnected maximal spectrum is local-global. Moreover, if R is arithmetic, then R is an elementary divisor ring.     

Simplified isogeny formulas on twisted Jacobi quartic curves

Isogenies between elliptic curves play a very important role in elliptic curve related cryptosystems and cryptanalysis. It is widely known that different models of elliptic curves would induce different computational costs of elliptic curve arithmetic, and several works have been devoted to accelerate the isogeny computation on various curve models. This paper studies the case of the Jacobi quartic model, which is a classic form of elliptic curves. A new w-coordinate system on extended Jacobi quartic curves is introduced for Montgomery-like group arithmetic. Explicit formulas for 2-isogenies and odd ℓ-isogenies on the specific curves are presented, and based on the w-coordinate system, the computation of such isogenies could be further simplified.     

On the j-Invariants of the Quadratic Q-Curves

The j-invariants of the quadratic Q-curves without complex multiplicationare studied. Some properties of the norms of these invariantsare shown and a relationship between the field Q(j) and thedegree of an isogeny of the Q-curve to its Galois conjugateis found. In the case when the degree of the isogeny is a primep, some properties of the primes of potentially multiplicativereduction for the Q-curve and of the reduction of j modulo aprime P in Q(j) over p when the Q-curve has potentially goodreduction at P are found.     

Euler Characteristics in Relative K-Groups

Suppose that M is a finite module under the Galois group ofa local or global field. Ever since Tate's papers [17, 18],we have had a simple and explicit formula for the Euler–Poincarécharacteristic of the cohomology of M. In this note we are interestedin a refinement of this formula when M also carries an actionof some algebra A, commuting with the Galois action (see Proposition5.2 and Theorem 5.1 below). This refinement naturally takesthe shape of an identity in a relative K-group attached to A(see Section 2). We shall deduce such an identity whenever wehave a formula for the ordinary Euler characteristic, the keystep in the proof being the representability of certain functorsby perfect complexes (see Section 3). This representabilitymay be of independent interest in other contexts. Our formula for the equivariant Euler characteristic over Aimplies the ‘isogeny invariance’ of the equivariantconjectures on special values of the L-function put forwardin [3], and this was our motivation to write this note. Incidentally,isogeny invariance (of the conjectures of Birch and Swinnerton-Dyer)was also a motivation for Tate's original paper [18]. I am verygrateful to J-P. Serre for illuminating discussions on the subjectof this note, in particular for suggesting that I consider representability.I should also like to thank D. Burns for insisting on a mostgeneral version of the results in this paper. 2000 MathematicsSubject Classification 19A99, 18G99, 11R34.