We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and matrices of size $O(g^2)\times O(g)$. When the base field $k$ is finite, or if $C$ has a rational point over $k$, these give algorithms for working on the Jacobian of $C$ that require $O(g^4)$ field operations, arising from the Gaussian elimination. Our point of view is strongly geometric, and our representation of points on the Jacobian is fairly simple to work with; in particular, none of our algorithms involves arithmetic with polynomials. We note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in Hess' 1999 Ph.D. thesis, which works with function fields as extensions of $k[x]$. However, for special classes of curves, Hess' algorithms are asymptotically more efficient than ours, generalizing other known efficient algorithms for special classes of curves, such as hyperelliptic curves (Cantor), superelliptic curves (Galbraith, Paulus, and Smart), and $C_{ab}$ curves (Harasawa and Suzuki); in all those cases, one can attain a complexity of $O(g^2)$.
The volume of a Cartier divisor on a projective variety is a nonnegative real number that measures the asymptotic growth of sections of multiples of the divisor. It is known that the set of these numbers is countable and has the structure of a multiplicative semigroup. At the same time it still remains unknown which nonnegative real algebraic numbers arise as volumes of Cartier divisors on some variety. Here we extend a construction first used by Cutkosky, and use the theory of real multiplication on abelian varieties to obtain a large class of examples of algebraic volumes. We also show that …
Let F/F_q be an algebraic function field of genus g defined over a finite field F_q. We obtain new results on the existence, the number and the density of dimension zero divisors of degree g-k in F where k is a positive integer. In particular, for q=2,3 we prove that there always exists a dimension zero divisor of degree γ-1 where γis the q-rank of F. We also give a necessary and sufficient condition for the existence of a dimension zero divisor of degree g-k for a hyperelliptic field F in terms of its Zeta function.
We study the existence of non-special divisors of degree $g$ and $g-1$ for algebraic function fields of genus $g\geq 1$ defined over a finite field $\F_q$. In particular, we prove that there always exists an effective non-special divisor of degree $g\geq 2$ if $q\geq 3$ and that there always exists a non-special divisor of degree $g-1\geq 1$ if $q\geq 4$. We use our results to improve upper and upper asymptotic bounds on the bilinear complexity of the multiplication in any extension $\F_{q^n}$ of $\F_q$, when $q=2^r\geq 16$.
Let $Z \to X$ be a finite branched Galois cover of normal projective geometrically integral varieties of dimension $d \geq 2$ over a perfect field $k$. For such a cover, we prove a Chebotarev-type density result describing the decomposition behaviour of geometrically integral Cartier divisors. As an application, we classify Galois covers among all finite branched covers of a given normal geometrically integral variety $X$ over $k$ by the decomposition behaviour of points of a fixed codimension $r$ with $0 < r < \dim X$.
Let $k$ be a field with algebraic closure $\bar{k}$ and $D \subset \mathbb{P}^{1}_{\bar{k}}$ a reduced, effective divisor of degree $n \ge 3$, write $k(D)$ for the field of moduli of $D$. A. Marinatto proved that when $n$ is odd, or $n = 4$, $D$ descends to a divisor on $\mathbb{P}^{1}_{k(D)}$. We analyze completely the problem of when $D$ descends to a divisor on a smooth, projective curve of genus $0$ on $k(D)$, possibly with no rational points. In particular, we study the remaining cases $n \ge 6$ even, and we obtain conceptual proofs of Marinatto's results and of a theorem by B. Huggins ab…
We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations can be done efficiently in this setting: decomposing divisors into prime divisors; computing pull-backs and push-forwards of divisors under finite morphisms, and hence Picard and Albanese maps on Jacobians; generating uniformly random divisors and points on Jacobians; computing Frobenius maps and Kummer maps; and finding a basis for the $l$-torsion of the Pic…
We compute the arithmetic intersection numbers of certain Heegner divisors on integral models of Shimura curves over Q. Our formulas generalize the formulas of Gross-Kohnen-Zagier for intersection numbers of Heegner divisors on integral models of modular curves.
Let $(G, ω)$ be a hyperelliptic vertex-weighted graph of genus $g \geq 2$. We give a characterization of $(G, ω)$ for which there exists a smooth projective curve $X$ of genus $g$ over a complete discrete valuation field with reduction graph $(G, ω)$ such that the ranks of any divisors are preserved under specialization. We explain, for a given vertex-weighted graph $(G, ω)$ in general, how the existence of such $X$ relates the Riemann--Roch formulae for $X$ and $(G, ω)$, and also how the existence of such $X$ is related to a conjecture of Caporaso.
Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered as the simplest model. Explicit formulas defining reduced divisors for some particular cases are found. The reduced divisors are obtained in the form of solution of the Jacobi inversion problem which provides the way of computing Abelian functions on arbitrary non-special divisors. An effective reduction algorithm is proposed, which has the advantage that it involve…
We study the behavior of the greatest common divisor of a^k-1 and b^k-1, where a,b are fixed integers or polynomials, and k varies. In the integer case, we conjecture that when a and b are multiplicatively independent and in addition a-1 and b-1 are coprime, then a^k-1 and b^k-1 are coprime infinitely often. In the polynomial case, we prove a strong version of this conjecture. To do this we use a result of Lang's on the finiteness of torsion points on algebraic curves. We also give a matrix analogue of these results, where for a unimodular matrix A, we look at the greatest common divisor of th…
In a one-page fragment published with his lost notebook, Ramanujan stated two double series identities associated, respectively, with the famous Gauss Circle and Dirichlet Divisor problems. The identities contain an "extra" parameter, and it is possible that Ramanujan derived these identities with the intent of attacking these famous problems. Similar famous unsolved problems are connected with $f_K(n)$, the number of integral ideals of norm $n$ in an algebraic number field $K$. In this paper we establish Riesz sum identities containing an "extra" parameter and involving $f_K(n)$, or divisor f…
We study the existence of a Zsigmondy bound for a sequence of divisors associated to points on an elliptic curve over a function field. More precisely, let $k$ be an algebraically closed field, let $\mathcal{C}$ be a nonsingular projective curve over $k$, and let $K$ denote the function field of $\mathcal{C}$. Suppose $E$ is an ordinary elliptic curve over $K$ and suppose there does not exist an elliptic curve $E_0$ defined over $k$ that is isomorphic to $E$ over $K$. Suppose $P\in E(K)$ is a non-torsion point and $Q\in E(K)$ is a torsion point of order $r$. The sequence of points $\{nP+Q\}\su…
We show that the values of a certain family of weakly holomorphic modular functions at points in the divisors of any meromorphic modular form with algebraic Fourier coefficients are algebraic. We use this to extend the classical result of Schneider by proving that zeros or poles of any non-zero meromorphic modular form with algebraic Fourier coefficients are either transcendental or imaginary quadratic irrational.
Let $ \mathfrak{C}^+_{ns}(p) $ be the Cuspidal Divisor Class Group of the modular curves $X^+_{ns}(p) $ associated to the normalizer of a non-split Cartan subgroup of level $ p$. I study the $ p-$primary part of $ \mathfrak{C}^+_{ns}(p) $ and estimate the order of growth of $ |\mathfrak{C}^+_{ns}(p)| $.
As a consequence of our recently established generalized Schmidt's subspace theorem for closed subschemes in general position, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and mak…
In a previous joint article with F. Abu Salem, we gave efficient algorithms for Jacobian group arithmetic of "typical" divisor classes on C_{3,4} curves, improving on similar results by other authors. At that time, we could only state that a generic divisor was typical, and hence unlikely to be encountered if one implemented these algorithms over a very large finite field. This article pins down an explicit characterization of these typical divisors, for an arbitrary smooth projective curve of genus g >= 1 having at least one rational point. We give general algorithms for Jacobian group arithm…
We give a formula for divisors of modular units on $X_1(N)$ and use it to prove that the $\mathbb{Q}$-gonality of the modular curve $X_1(N)$ is bounded above by $\left[\frac{11N^2}{840}\right]$, where $[\bullet]$ denotes the nearest integer.
We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive prime divisor, one that has not appeared in any earlier term. Proofs of this are known in only a few cases. Weaker results in the general direction are given, using a strong form of Siegel's Theorem and some congruence arguments. Our main result is applied to the study of prime divisors of Somos sequences.
For a given curve X and divisor class C, we give lower bounds on the degree of a divisor A such that A and A-C belong to specified semigroups of divisors. For suitable choices of the semigroups we obtain (1) lower bounds for the size of a party A that can recover the secret in an algebraic geometric linear secret sharing scheme with adversary threshold C, and (2) lower bounds for the support A of a codeword in a geometric Goppa code with designed minimum support C. Our bounds include and improve both the order bound and the floor bound. The bounds are illustrated for two-point codes on general…
We deal with the distributions of holomorphic curves and integral points off divisors. We will simultaneouly prove an optimal dimension estimate from above of a subvariety W off a divisor D which contains a Zariski dense entire holomorphic curve, or a Zariski dense D-integral point set, provided that in the latter case everything is defined over a number field. Then, if the number of components of D is large, the estimate leads to the constancy of such a holomorphic curve or the finiteness of such an integral point set. At the begining, we extend logarithmic Bloch-Ochiai's Theorem to the Kaehl…
I find an explicit description of modular units in terms of Siegel functions for the modular curves $X^+_{ns}(p^k)$ associated to the normalizer of a non-split Cartan subgroup of level $p^k$ where $p\not=2,3$ is a prime. The Cuspidal Divisor Class Group $\mathfrak{C}^+_{ns}(p^k)$ on $X^+_{ns}(p^k)$ is explicitly described as a module over the group ring $R = \mathbb{Z}[(\mathbb{Z}/p^k\mathbb{Z})^*/\{\pm 1\}]$. In this paper I give a formula involving generalized Bernoulli numbers $B_{2,χ}$ for $|\mathfrak{C}^+_{ns}(p^k)|$.
Let $p$ be a prime number such that the modular curve $X_0(p)$ has genus at least two. We show that the only points of the reduction mod $p$ of $X_0(p)$ with image in the reduction mod $p$ of $J_0(p)$ in the cuspidal group are the two cusps. This answers a question of Robert Coleman. For the proof we give a description of the special fibre of the Néron model of the jacobian of a semi-stable curve in terms of divisors. We also study to what extent the morphism from a semistable curve with given base point to the Néron model of its jacobian is a closed immmersion. Implicitly, logarithmic structu…
In this paper we propose a direct and explicit realization of addition of divisors by means of an iterative reduction algorithm. Each iteration of the algorithm is the reduction of a degree $g+1$ divisor to a divisor of degree~$g$. Such an approach allows to carry out all computations explicitly in a symbolic form, which is done for curves $C_{3,4}$, $C_{3,5}$ in this paper, and also for curves of higher genera up to $C_{3,13}$, $C_{3,14}$.
We establish a quantitative adelic equidistribution theorem for a sequence of algebraic zeros divisors on the projective line over the separable closure of a product formula field having small diagonals and small $g$-heights with respect to an adelic normalized weight $g$ in arbitrary characteristic and in possibly non-separable setting, and obtain local proximity estimates between the iterations of a rational function $f\in k(z)$ of degree $>1$ and a rational function $a\in k(z)$ of degree $>0$ over a product formula field $k$ of characteristic $0$, applying this quantitative adelic equidistr…
We study the degeneration dimension of non-archimedean analytic maps into the complement of hypersurface divisors of smooth projective varieties. We also show that there exist no non-archimedean analytic maps into $P^n\setminus\cup_{i= 1}^n D_i$ where $D_i, 1\le i\le n$, are hypersurfaces of degree at least 2 in general position and intersecting transversally. Moreover, we prove that there exist no non-archimedean analytic maps into $P^2\setminus\cup_{i=1}^2 D_i$ when $D_1, D_2$ are generic plane curves with deg$D_1+$deg$D_2\ge 4$.
For a prime number $p$ we study the zeros modulo $p$ of divisor polynomials of rational elliptic curves $E$ of conductor $p$. Ono made the observation that these zeros of are often $j$-invariants of supersingular elliptic curves over $\overline{\mathbb{F}_p}$. We show that these supersingular zeros are in bijection with zeros modulo $p$ of an associated quaternionic modular form $v_E$. This allows us to prove that if the root number of $E$ is $-1$ then all supersingular $j$-invariants of elliptic curves defined over $\mathbb{F}_{p}$ are zeros of the corresponding divisor polynomial. If the roo…
Inspired by the analogy between the group of units $\mathbb{F}_p^{\times}$ of the finite field with $p$ elements and the group of points $E(\mathbb{F}_p)$ of an elliptic curve $E/\mathbb{F}_p$, E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum $\sum_{p \leq x} τ(p + a) \sim C x$. In this paper, we present a comprehensive study of the constants $C(E)$ emerging in the asymptotic study of these elliptic curve divisor sums. Specifically, by analyzing the division fields of an ellipt…
