Algebraic Correspondence Research Articles

On Brumer's family of {RM}-curves of genus two

We reconstruct Brumer's family with 3-parameters of curves of genus two whose jacobian varieties admit a real multiplication of discriminant 5. Our method is based on the descent theory in geometric Galois theory which can be compared with a classical problem of Noether. Namely, we first construct a 3-parameter family of polynomials $f(X)$ of degree 6 whose Galois group is isomorphic to the alternating group $A_5$. Then we study the family of curves defined by $Y^2=f(X)$, showing that they are equivalent to Brumer's family. The real multiplication will be described in three distinct ways, i.e., by Humbert's modular equation, by Poncelet's pentagon, and by algebraic correspondences.

Explicit construction of a global uniformization for an algebraic correspondence

Explicit construction of a global uniformization for an algebraic correspondence

The mixed volume optimization problem

The mixed volume optimization problem is to determine the point of duality Q for a given convex set K that minimizes the “mixed volume” of the associated polar set (K ∗;Q) . In the plane, the mixed volumes translate as the area and length; in space, the mixed volumes include the volume, surface area, and mean width. In this paper, the geometric optimization problems associated with minimizing mixed volumes are examined from two perspectives: enumerative search and symbolic computation. The problem of minimizing the polar area through an enumerative search is first considered. The dual polygon (P m ∗;Q) is constructed for an arbitrary point of duality Q∈ P m ° by using an algebraic correspondence between the edges of P m and the vertices of (P m ∗;Q) , and the area of (P m ∗;Q) , A(P ∗ m;Q) , is calculated and minimized using naive search techniques. A result due to Santaló is applied to verify the minimizing solution, and computational tests are described for various classes of randomly generated polygons. Statistical evidence indicates that a “good” approximation to the minimum area polar polygon occurs when the duality point is located at the center-of-gravity of P m . The polar area problem is then investigated using symbolic procedures. Explicit symbolic expressions for the polar area and length functionals are computed and solved directly using the differential optimality conditions and Newton's iterative method of solution. The mixed volume and surface area functionals are formulated and solved using numerical products, and the mean width functional is described. Examples are used throughout to illustrate the methodology.

The local monodromy as a generalized algebraic correspondence (with an appendix by Spencer Bloch)

For an algebraic, normal-crossings degeneration over a local field the local monodromy operator and its powers naturally define Galois equivariant classes in the \ell -adic (middle dimensional) cohomology groups of some precise strata of the special fiber of a normal-crossings model associated to the fiber product degeneration. The paper addresses the question whether these classes are algebraic. It is shown that the answer is positive for any degeneration whose special fiber has (locally) at worst triple points singularities. These algebraic cycles are responsible for and they explain geometrically the presence of poles of local Euler L-factors at integers on the left of the left-central point.

ALGORITHM OF GLOBAL UNIFORMIZATION OF ALGEBRAIC CORRESPONDENCE

The problem of global uniformization of algebraic correspondence is investigated. The weaker assumptions are used in the analysis.

Numerical calculation of the multiplicity of a solution to algebraic equations

A method to calculate numerically the multiplicity of a solution to a system of algebraic equations is presented. The method is an application of Zeuthen’s rule which gives the multiplicity of a solution as the multiplicity of a united point of an algebraic correspondence defined naturally by the system. The numerical calculation is applicable to a large scale system of algebraic equations which may have a solution that we cannot calculate the multiplicity by a symbolic computation.

Commuting algebraic correspondences and groups

In this paper, the naturalness of the appearance of the group approach in the study of problems of algebraic commutation is considered. New, previously unknown solutions are obtained. Special attention is paid to an example of commutation obtained from the icosahedral equation.

A Gallery of Iterated Correspondences

Algebraic correspondences on the Riemann sphere generalise both Kleinian groups and rational maps. We classify quadratic correspondences satisfying certain “diagram conditions” and derive canonical forms for them. We exhibit examples for which the regular setand limit setare both nontrivial, and show that such examples can simultaneously exhibit Kleinian-grouplike and rational-map-like behaviour. We also state some general theorems and conjectures.

Computer-aided kinematic analysis of basic spatial mechanisms

Using the spherical coordinate system in conjunction with the principle of algebraic correspondence between the displacement equations of the plane, spherical and spatial four-bar mechanisms and the plane and spatial slider-crank mechanisms, an efficacious procedure has been developed to carry out kinematic analysis for the coupler and output link of the mechanisms. The iteration-free nature of the procedure precludes any non-convergence problem and the inherent complexity of the 3-dimensional analysis is mitigated substantially.

On algebraic equivalence of tariffing systems

Based on a concept of insurance form we show that the following three tariffing systems are mathematically equivalent: the model of experience rating, the over-loss risk covering and the bonus system. The equivalence is described by explicit one-to-one algebraic correspondences. As by-product a known result on self-financed insurance forms is derived very simply.

On a computationally efficient microcomputer kinematic analysis of the basic linkage mechanisms

Based on the algebraic correspondence between the displacement equations of the plane, spherical and skew four-bar linkages and the plane and skew slider-crank mechanisms, a highly compact, computationally efficient procedure has been developed for a microcomputer kinetmatic analysis of these linkages.

Relative equilibria of the four-body problem

AbstractBy employing a regularizing transformation, the problem of bifurcation of relative equilibria in the Newtonian 4-body problem is reduced to a study of an algebraic correspondence between real algebraic varieties. The finiteness theorems of algebraic geometry are used to find an upper bound for the number of affine equivalence classes of relative equilibria which holds for all masses in the complement of a proper, algebraic subset of the space of all masses.

Umlauffähigkeit räumlicher mechanismen

For four-membered plane, spherical and spatial mechanisms whose transfer functions are of the same structure, a common theory for the conditions of complete rotation is given. The transfer equation has to allow for the interpretation as the algebraic correspondence of type (1,2) between two conic sections. For this, the theory of conic pencils is required establishing a complete classification of these drives. The reality of the cluster points and the kind of the conic sections supply information on the conditions for complete rotation and the kind of output motion. The conditions for complete rotation are represented independent of the transmission type.

The ring of algebraic correspondences on a generic curve of genus g

0. Let p be a prime number or zero and let g be a non-negative integer. Then there is a coarse moduli space Mg for complete non-singular irreducible curves of genus g defined over fields of characteristic p, which is an irreducible variety over the algebraic closure F̅p of the prime field Fp. (Especially, F0 is also denoted by Q as usual.) ([8], [2]). The curve corresponding to a generic point of Mg over F̅p is called a generic curve of genus g.

An algebraic correspondence with applications to projective bundles and blowing up Chern classes

New techniques are developed, based on the consideration of the projective bundle associated with a direct sum of two vector bundles, to give a simpler solution of the problem of blowing up Chern classes which was previously solved by Porteous[12] using the Grothendieck Riemann-Roch theorem.

Automorphic forms and automorphy preserving differential operators on products of halfplanes

The theory of automorphic functions of one complex variable is now nearly a century old, and has antecedents -the theories of elliptic and elliptic modular functions -which can be t raced through ABEL'S far-reaching paper of 1827 [1] to the addition theorems for certain elliptic integrals discovered by FAO~A_~O in 1714 [18]. I t can be said to have at ta ined a state of matur i ty and to be, today, well understood in general al though a number of impor tan t specific problems still remain unsolved. Despite its origin as a branch of Analysis, and its reliance on essentially analyt ical techniques for the establishment of its major results, the theory maintains to this day a curiously algebraic outlook: its main results concern the algebraic structure of certain graded rings of functions, and the algebraic correspondences that persist amongst

On Dirichlet Series and Abelian Varieties Attached to Automorphic Forms

Since Hecke [13] had given a general theory of constructing Dirichlet series with Euler-product and functional equation out of elliptic modular forms of any level, several authors considered its generalization for other types of automorphic forms. In the case of the Hilbert modular group of level one, Herrmann [14] succeeded in this problem; he has shown the necessity of considering not only the product of the upper half-planes but also the domain r consisting of the points (zl, * * , Zr) of the r-dimensional complex vector space Cr such that Im(z,) # 0, * ., Im(zr) # 0, and h distinct discontinuous groups commensurable to each other, h being the class number of the totally real number field in the problem. On the other hand, the unit-group of an order in an indefinite quaternion algebra over the rational number field Q yields a fuchsian group. In this case, Eichler [6] defined Hecke's operators as representations of algebraic correspondences, called modular correspondences, and proved a formula for the trace of the operators. The trace-formula was proved also by Selberg [22] in a more general formulation. Recently, Godement [9] has given a theory of zeta-functions attached to division algebras; namely, he has shown the possibility of applying the adele-idele method of Iwasawa-Tate [15, 28] to automorphic functions and forms with respect to the unit-group of a division algebra over Q. The case of non-holomorphic automorphic functions of this type has been investigated by Tamagawa [27]. In [25] I have treated cusp-forms with respect to the unit-group of an indefinite quaternion algebra over Q. Now, the purpose of Part I of the present paper is to develop an analogous theory for automorphic forms on the domain ; mentioned above with respect to the unit-groups of an indefinite quaternion algebra. Let F be a totally real algebraic number field of degree t, and D be a quaternion algebra over F. Denote by r the number of infinite prime spots of F unramified in D, and suppose that r > 0. Let D,1 and D-2 denote the product of r copies of the total matric algebra M2(R) over the real number field R and the product of t r copies of the division ring K of real quaternions, respectively. Then, D( Q R is isomorphic to D,1 x D12. Let o be a maximal order in D and I' be the group of units in o. Let y, and 72 237

Cohomology theory and algebraic correspondences

Cohomology theory and algebraic correspondences

Some enumerative formulae for algebraic curves

This paper is in two parts. In Part I we are concerned with one or more linear series on an algebraic curve; we consider a set of points on the curve which are contained with assigned multiplicities in a set of each of the linear series and, by persistent use of Severi's equivalence relation for the united points of an algebraic correspondence with valency, we derive formulae for the number of such sets of points when the constants involved are such as to make this number finite. All this is essentially a generalization of the formula for the number of points in the Jacobian set of a linear series of freedom 1, and the main result is Theorem 3.

The moduli of hyperelliptic curves

Introduction. We can state the problem of the moduli of algebraic curves in the following way. Consider all irreducible algebraic curves, defined over a ground field k, of a fixed genus g. Can we, in some natural fashion, associate to each birational class of these curves a point of an algebraic variety N, defined over k, in such a way that for distinct birational classes we obtain distinct points? We impose the further condition that those points of N which do not represent any of the above birational classes form a proper algebraic subvariety of N, defined over k. We shall consider this problem only for the special case of hyperelliptic curves. van der Waerden [3] has attacked the general problem. There is, however, an important gap in the proof of Theorem 7 [p. 698, loc. cit.]. We shall use the notation of van der Waerden in pointing this out. I is an irreducible algebraic system of regular plane curves of genus g [?4, p. 698, loc. cit.]. The algebraic correspondence Q3 transforms I, onto itself. The difficulty arises from the fact that, as far as we know, an irreducible regular plane curve r of genus g may be fundamental for the correspondence