Related papers: Geometric primality tests using curves of genus 1 …
We derive a local formula for the parity of the $2^{\infty}$-Selmer rank of Jacobians of curves of genus $2$ or $3$ with a $K$-rational $2$-torsion point. We give an explicit example to show how this local formula gives rank parity…
We show how the output of the algorithm to compute modular Galois representations described in our previous article can be certified. We have used this process to compute certified tables of such Galois representations obtained thanks to an…
Let X be an Abelian surface and C a holomorphic curve in X representing a primitive homology class. The space of genus g curves in the class of C is g dimensional. We count the number of such curves that pass through g generic points and we…
We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the…
We show that finite fields over which there is a curve of a given genus g with its Jacobian having a small exponent, are very rare. This extends a recent result of W. Duke in the case g=1. We also show when g=1 or g=2 that our bounds are…
We consider expansion and property testing in the language of incidence geometry, covering both simplicial and cubical complexes in any dimension. We develop a general method for passing from an explicit description of the cohomology group,…
We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.
It is well known that the Gauss map for a complex plane curve is birational, whereas the Gauss map in positive characteristic is not always birational. Let $q$ be a power of a prime integer. We study a certain plane curve of degree…
We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with "probability-one", whether or not two…
We give a complete answer to the question of which polynomials occur as the characteristic polynomials of Frobenius for genus-2 curves over finite fields.
In this paper, we will study the arithmetic of the Eisenstein part of the modular Jacobians. In the first section, we introduce some general preliminaries of the arithmetic theory of modular curves that we will need later. In the second…
We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of…
We study hyperbolic curves and their Jacobians over finite fields in the context of anabelian geometry.
We survey the theory of the compactified Jacobian associated to a singular curve. We focus on describing low genus examples using the Abel map.
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…
In this paper, we investigate primeness of groupoid graded rings. We provide a set of necessary and sufficient conditions for primeness of a nearly-epsilon strongly groupoid graded ring. Furthermore, we apply our main result to get a…
This note gives the complete projective classification of rational, cuspidal plane curves of degree at least 6, and having only weighted homogeneous singularities. It also sheds new light on some previous characterizations of free and…
We consider cyclic unramified coverings of degree d of irreducible complex smooth genus 2 curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order d. The rich…
Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of…
The class-invariant homomorphism allows one to measure the Galois module structure of extensions obtained by dividing points on abelian varieties. In this paper, we consider the case when the abelian variety is the Jacobian of a Fermat…