Principal Divisors Research Articles

The Galois module structure of holomorphic poly-differentials and Riemann-Roch spaces

Suppose X is a smooth projective geometrically irreducible curve over a perfect field k of positive characteristic p. Let G be a finite group acting faithfully on X over k such that G has non-trivial, cyclic Sylow p-subgroups. If E is a G-invariant Weil divisor on X with deg(E)>2g(X)−2, we prove that the decomposition of H0(X,OX(E)) into a direct sum of indecomposable kG-modules is uniquely determined by the class of E modulo G-invariant principal divisors, together with the ramification data of the cover X→X/G. The latter is given by the lower ramification groups and the fundamental characters of the closed points of X that are ramified in the cover. As a consequence, we obtain that if m>1 and g(X)≥2, then the kG-module structure of H0(X,ΩX⊗m) is uniquely determined by the class of a canonical divisor on X/G modulo principal divisors, together with the ramification data of X→X/G. This extends to arbitrary m>1 the m=1 case treated by the first author with T. Chinburg and A. Kontogeorgis. We discuss applications to the tangent space of the global deformation functor associated to (X,G) and to congruences between prime level cusp forms in characteristic 0. In particular, we complete the description of the precise kPSL(2,Fℓ)-module structure of all prime level ℓ cusp forms of even weight in characteristic p=3.

On Fermat Last Theorem: The New Efficient Expression of a Hypothetical Solution as a Function of Its Fermat Divisors

Denote by a non-trivial primitive solution of Fermat’s equation (p prime).We introduce, for the first time, what we call Fermat principal divisors of the triple defined as follows. , and . We show that it is possible to express a,b and c as function of the Fermat principal divisors. Denote by the set of possible non-trivial solutions of the Diophantine equation . And, let (p prime). We prove that, in the first case of Fermat’s theorem, one has . In the second case of Fermat’s theorem, we show that , ,. Furthermore, we have implemented a python program to calculate the Fermat divisors of Pythagoreans triples. The results of this program, confirm the model used. We now have an effective tool to directly process Diophantine equations and that of Fermat.

Non-negative divisors and the Grauert metric

Grauert showed that it is possible to construct complete Kähler metrics on the complement of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics on the complement of a principal divisor in \(\mathbb {C}^n\), \(n \ge 1\). In addition, we also study how this metric and its holomorphic sectional curvature behave when the corresponding principal divisors vary continuously.

Geometry of the scaling site

In this paper we construct the scaling site \({\mathscr {S}}\) by implementing the extension of scalars on the arithmetic site \({\mathscr {A}}\), from the smallest Boolean semifield \({\mathbb B}\) to the tropical semifield \({\mathbb R}_+^\mathrm{max}\). The obtained semiringed topos is the Grothendieck topos \({[0,\infty )\rtimes {{\mathbb N}^{\times }}}\), semi-direct product of the Euclidean half-line and the monoid \({\mathbb N}^{\times }\) of positive integers acting by multiplication, endowed with the structure sheaf of piecewise affine, convex functions with integral slopes. We show that pointwise \({[0,\infty )\rtimes {{\mathbb N}^{\times }}}\) coincides with the adele class space of \({\mathbb Q}\) and that this latter space inherits the geometric structure of a tropical curve. We restrict this construction to the periodic orbit of the scaling flow associated to each prime p and obtain a quasi-tropical structure which turns this orbit into a variant \(C_p={\mathbb R}_+^*/p^{\mathbb Z}\) of the classical Jacobi description \({\mathbb C}^*/q^{\mathbb Z}\) of an elliptic curve. On \(C_p\), we develop the theory of Cartier divisors, determine the structure of the quotient \(\mathrm{Div}(C_p)/{\mathcal P}\) of the abelian group of divisors by the subgroup of principal divisors, develop the theory of theta functions, and prove the Riemann–Roch formula which involves real valued dimensions, as in the type II index theory. We show that one would have been led to the same definition of \({\mathscr {S}}\) by analyzing the well known results on the localization of zeros of analytic functions involving Newton polygons in the non-archimedean case and the Jensen formula in the complex case.

Reciprocity laws and K-theory

We associate to a full flag $\mathcal{F}$ in an $n$-dimensional variety $X$ over a field $k$, a "symbol map" $\mu_{\mathcal{F}}:K(F_X) \to \Sigma^n K(k)$. Here, $F_X$ is the field of rational functions on $X$, and $K(\cdot)$ is the $K$-theory spectrum. We prove a "reciprocity law" for these symbols: Given a partial flag, the sum of all symbols of full flags refining it is $0$. Examining this result on the level of $K$-groups, we re-obtain various "reciprocity laws". Namely, when $X$ is a smooth complete curve, we obtain degree of a principal divisor is zero, Weil reciprocity, Residue theorem, Contou-Carr\`{e}re reciprocity. When $X$ is higher-dimensional, we obtain Parshin reciprocity.

Faithful tropicalization of Mumford curves of genus two

In the present paper, we investigate the question if the skeleton of a Mumford curve of genus two can be tropicalized faithfully in dimension three, i.e. if there exists an embedding of the curve in projective three space such that the tropicalization maps the skeleton of the curve isometrically to its image. Baker, Payne and Rabinoff showed that the skeleton of every analytic curve can be tropicalized faithfully. However the dimension of the ambient space in their proof can be quite large. We will define a map from the skeleton to the tropicalization of the Jacobian, which is an isometry on the cycles. It allows us to find principal divisors and simultaneously to determine the retractions of their support on the skeleton which is necessary to calculate their tropicalization. It turns out that a Mumford curve of genus two whose cycles of its skeleton are either disjoint or share an edge of length at most half of the length of the cycles, can be tropicalized faithfully in dimension three.

Breeding on eight strains of Pseudostellaria heterophylla based on phenotypic traits and quality in Guizhou province

To provide new germplasm materials for breeding new varieties of Pseudostellaria heterophylla. The method of single plant selection was adopted, with the comparative experiments being carried out under the same conditions in Shibing county. The 8 plants of Shibing SB-4 were compared respectively with factor analysis for 27 phenotypic traits and 8 yield traits, and single factor variance analysis for the contents of polysaccharides. Using factor analysis, 27 phenotypic traits were classified into 7 principal divisors and 8 yield traits were simplified into 3 principal divisors. The 4 strains of P. heterophylla, ZT-01, ZT-02, ZT-06 and ZT-07, performed better than others in the phenotypic traits, and ZT-01, ZT-02, ZT-03 and ZT-07 in the yield traits. The contents of polysaccharides of ZT-01, ZT-02, ZT-05 and ZT-08 showed significantly higher value. There is significant difference among the 8 strains of P. heterophylla in phenotypic traits, yield traits and quality traits, making it possible to select certain strains for different purposes. ZT-01 and ZT-02 can be breaded further. ZT-06 and ZT-07 were used as ornamental cultivars for its great phenotypic traits. ZT-03 with good resistance and high yield was taken as resistant variety, and ZT-05 would face next selection on the basis of its high content of polysaccharide.

The Skeleton of the Jacobian, the Jacobian of the Skeleton, and Lifting Meromorphic Functions From Tropical to Algebraic Curves: Fig. 1.

Let K be an algebraically closed field which is complete with respect to a nontrivial, non-Archimedean valuation and let \Lambda be its value group. Given a smooth, proper, connected K-curve X and a skeleton \Gamma of the Berkovich analytification X^\an, there are two natural real tori which one can consider: the tropical Jacobian Jac(\Gamma) and the skeleton of the Berkovich analytification Jac(X)^\an. We show that the skeleton of the Jacobian is canonically isomorphic to the Jacobian of the skeleton as principally polarized tropical abelian varieties. In addition, we show that the tropicalization of a classical Abel-Jacobi map is a tropical Abel-Jacobi map. As a consequence of these results, we deduce that \Lambda-rational principal divisors on \Gamma, in the sense of tropical geometry, are exactly the retractions of principal divisors on X. We actually prove a more precise result which says that, although zeros and poles of divisors can cancel under the retraction map, in order to lift a \Lambda-rational principal divisor on \Gamma to a principal divisor on X it is never necessary to add more than g extra zeros and g extra poles. Our results imply that a continuous function F:\Gamma -> R is the restriction to \Gamma of -log|f| for some nonzero meromorphic function f on X if and only if F is a \Lambda-rational tropical meromorphic function, and we use this fact to prove that there is a rational map f : X --> P^3 whose tropicalization, when restricted to \Gamma, is an isometry onto its image.

The $\mathbb{Q}$-rational cuspidal group of $J_{1}(2p)$

Let $p$ be a prime not equal to 2 or 3. In this paper we study the $\mathbb{Q}$-rational cuspidal group $\mathcal{C}_{\mathbb{Q}}$ of the jacobian $J_{1}(2p)$ of the modular curve $X_{1}(2p)$. We prove that the group $\mathcal{C}_{\mathbb{Q}}$ is generated by the $\mathbb{Q}$-rational cusps. We determine the order of $\mathcal{C}_{\mathbb{Q}}$, and give numerical tables for all $p\leq127$. These tables give also other cuspidal class numbers for the modular curves $X_{1}(2p)$ and $X_{1}(p)$. We give a basis of the group of the principal divisors supported on the $\mathbb{Q}$-rational cusps, and using this we determine the explicit structure of $\mathcal{C}_{\mathbb{Q}}$ for all $p\leq127$. We determine the structure of the Sylow $p$-subgroup of $\mathcal{C}_{\mathbb{Q}}$, and the explicit structure for all $p\leq4001$.

The double ramification cycle and the theta divisor

We compute the classes of universal theta divisors of degrees zero and g-1 over the Deligne-Mumford compactification of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of M\uller. We also obtain a formula for the class in the Chow ring of the moduli space of curves of compact type of the double ramification locus, given by the condition that a fixed linear combination of the marked points is a principal divisor, reproving a recent result of Hain. Our approach for computing the theta divisor is more direct, via test curves and the geometry of the theta divisor, and works easily over the entire Deligne-Mumford compactification. We use our extended result in another paper to study the partial compactification of the double ramification cycle.

A generalisation of Miller's algorithm and applications to pairing computations on abelian varieties

In this paper, we use the theory of theta functions to generalise to all abelian varieties the usual Miller's algorithm to compute a function associated with a principal divisor. We also explain how to use the Frobenius morphism on abelian varieties defined over a finite field in order to shorten the loop of the Weil and Tate pairings algorithms. This extends preceding results about ate and twisted ate pairings to all abelian varieties. Then building upon the two preceding ingredients, we obtain a variant of optimal pairings on abelian varieties. Finally, by introducing new addition formulas, we explain how to compute optimal pairings on Kummer varieties. We compare in term of performance the resulting algorithms to the algorithms already known in the genus one and two case.

The complete principal divisor lattices

In this paper, we introduce a concept of principal divisor lattice and describe the structure of its elements. We first give a necessary and sufficient condition for the existence of irredundant join irreducible decompositions in complete principal divisor distributive lattices, and prove that the complete lower continuous, principal divisor lattices have irredundant join irreducible decompositions. In the end, we show the descriptions of lattices that have unique (resp. replaceable) irredundant join irreducible decompositions in complete lower continuous principal divisor lattices.

Consecutive Integers with Equally Many Principal Divisors

Classifying the positive integers as primes, composites, and the unit, is so familiar that it seems inevitable. However, other classifications can bring interesting relationships to our attention. In that spirit, let us classify positive integers by the number o? principal divisors they possess, where we define a principal divisor of a positive integer n to be any prime-power divisor pa n which is maximal (so p is prime, a is a positive integer, and pa+l is not a divisor of n). The standard notation pa \\n can be read as /?fl is a principal divisor of n. The Fundamental Theorem of Arithmetic is usually stated in a form emphasizing how primes enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite multiset of primes. (Recall that a multiset is a collection of elements in which multiple occurrences are permitted.) Alternatively, the Fundamental Theorem of Arithmetic can be stated in a form that focuses on how maximal prime powers enter the structure of the positive integers, such as: Every positive integer is the product of a unique finite set of powers of distinct primes. Consequently every positive integer is the product of its principal divisors, and every finite set of powers of distinct primes is the set of principal divisors of a unique positive integer. Of course, the number of principal divisors of n is equal to the number of distinct prime factors of n, but here the principal divisors are the simple structural components of n, whereas the distinct prime factors are but a shadow of that structure. Readers who find the present paper of interest might find similar interest in [6], where upper bounds on the sum of principal divisors of n are established by elementary means. For each integer n > 0, let Pn be the set of all positive integers with exactly n principal divisors, so Pq = {1}, and

A sequence of one-point codes from a tower of function fields

We construct a sequence of one-point codes from a tower of function fields whose relative minimum distances have a positive limit. Our tower is characterized by principal divisors. We determine completely the minimum distance of the codes from the first field of our tower. These results extend those of Stichtenoth [IEEE Trans Inform Theory (1988), 34(15):1345---1348], Yang and Kumar [Lecture Notes in Mathematics, 1518, (1991), Springer-Verlag, Berlin Hidelberg New York, pp. 99---107], and Garcia [Comm. Algebra, 20(12): 3683---3689]. As an application, we show that the minimum distance corresponds to the Feng---Rao bound.

Upper Bounds on the Sum of Principal Divisors of an Integer

A prime-power is any integer of the form p, where p is a prime and a is a positive integer. Two prime-powers are independent if they are powers of different primes. The Fundamental Theorem of Arithmetic amounts to the assertion that every positive integer N is the product of a unique set of independent prime-powers, which we call the principal divisors of N. For example, 3 and 4 are the principal divisors of 12, while 2, 5 and 9 are the principal divisors of 90. The case N = 1 fits this description, using the convention that an empty set has product equal to 1 (and sum equal to 0). In a recent invited lecture, Brian Alspach noted [1]: Any odd integer N > 15 that is not a prime-power is greater than twice the sum of its principal divisors. For instance, 21 is more than twice 3 plus 7, and 35 is almost three times 5 plus 7, but 15 falls just short of twice 3 plus 5. Alspach asked for a nice (elegant and satisfying) proof of this observation, which he used in his lecture to prove a result about cyclic decomposition of graphs. Responding to the challenge, we prove Alspach's observation by a very elementary argument. You, the reader, must be the judge of whether our proof qualifies as nice. We also show how the same line of reasoning leads to several stronger yet equally elegant upper bounds on sums of principal divisors. Perhaps surprisingly, our methods will not focus on properties of integers. Rather, we consider properties of finite sequences of positive real numbers, and use a classical elementary inequality between the product and sum of any such sequence. But first, let us put Alspach's observation in its number theoretic context.

Upper Bounds on the Sum of Principal Divisors of an Integer

Click to increase image sizeClick to decrease image size This article is part of the following collections: Carl B. Allendoerfer Prize: 2000s

Coverings over $d$-gonal curves

Let M be a compact Riemann surface and / be a meromorphic function on M. Let (/) be the principal divisor associated to / and (/)≪,be the polar divisor of /. We call/ a meromorphic function ofdegree d if rf=degree (/)≪. If d is the minimal integer in which a meromorphic function of degree d existson M, then we callM a d-gonal curve. Now we assume that M is d-gonal,and considera covering map it':M'^M that M' stillremains rf-gonal. The purpose of thispaper is to show how such %' can be characterized. The case that it'is a normal covering and d=2 (i.e.,M is hyperelliptic) has been already studied([2], [3], [4] and [7]). In this case the existence of the hyperellipticinvolution v' on M' plays an important role. More precisely, as v' commutes with each element of the Galois group G―Gal{M''/M), v'induces the hyperellipticinvolution v on M and we can reduce rc'to a normal covering x: P'^Pi with Galois group G, where P[ and Px are Riemann spheres isomorphic to quotient Riemann surfaces M'/ and M/ respectively. On the other hand itisknown thatfinitesubgroups of thelinear transformation group are cyclic, dihedral, tetrahedral,octahedral and icosahedral. Horiuchi [3] decided all the different normal coverings it':M'-*M over a hyperellipticcurve M that M' stillremains a hyperellipticcurve by investigating each of above fivetypes. Let M be a rf-gonal curve. In this paper we willshow at firstthat a covering map %': M'―>M (not necessarily normal) with rf-gona!M' canonically induces some covering map n: P[-^PX (Theorem 2.1§2). Moreover if both M and M' have unique linearsystem gldand it'is normal, then we can see that itis also normal (Cor. 2.3). In §3,§4 and §5 we assume that M is a cyclic/>-gonalcurve fora prime number p. We will determine allramification types of normal coverings it':M'^M with 6-gonal M' by thesame way as Horiuchi did in case ￡=2(§4),

Algebraic cycles relative to a smooth principal divisor

Algebraic cycles relative to a smooth principal divisor

Rings which resemble rings of entire functions

Since Helmer's 1940 paper [9] laid the foundations for the study of the ideal theory of the ring A(ℂ) of entire functions, many interesting results have been obtained for the rings A(X) of analytic functions on non-compact connected Riemann surfaces. For example, the partially ordered set Spec (A(ℂ) of prime ideals of A(ℂ) has been described by Henrikson and others [2], [10], [11]. Also, it has been shown by Ailing [4] that Spec(A(ℂ))sSpec(A(X)) as topological spaces for any non-compact connected Riemann surface X. Many results on the valuation theory of A(X) have also been obtained [1], [2]. In this note we show that a large portion of the results on the rings A(X) extend to the W-rings with complete principal divisor space which were defined by J. Klingen in [15], [16]. Therefore, many properties of A(ℂ) are shared by its non-archimedian counterparts studied by M. Lazard, M. Krasner, and others [8], [17], [18].

On Homeomorphisms Preserving Principal Divisors

On Homeomorphisms Preserving Principal Divisors