ON THE COHOMOLOGY OF A SPACE CURVE CONTAINING A PLANE CURVE

In this paper we study the Hartshorne–Rao module of curves in P ³ of degree d and genus g, containing plane curves of degree d ? p, p ≥ 1. We prove an optimal upper bound for the Rao function of these curves and we show that the curves attaining the bound are obtained from an extremal curve by an elementary biliaison of height min(p, d ? p) ? 1 on a quadric surface.     

On Moduli Spaces,Equidistribution, Estimates,and Rational Points of Algebraic Curves

We consider the moduli spaces of hyperelliptic curves, Artin–Schreier coverings, and some other families of curves of this type over fields of characteristic p. By using the Postnikov method, we obtain expressions for the Kloosterman sums. The distribution of angles of the Kloosterman sums was investigated on a computer. For small prime p, we study rational points on curves y ² = f(x). We consider the problem of the accuracy of estimates of the number of rational points of hyperelliptic curves and the existence of rational points of curves of the indicated type on the moduli spaces of these curves over a prime finite field.     

Hyperelliptic curves with prescribed p-torsion

In this paper, we show that there exist families of curves (defined over an algebraically closed field k of characteristic p>2) whose Jacobians have interesting p-torsion. For example, for every 0≤f≤g, we find the dimension of the locus of hyperelliptic curves of genus g with p-rank at most f. We also produce families of curves so that the p-torsion of the Jacobian of each fibre contains multiple copies of the group scheme α_p. The method is to study curves which admit an action by (ℤ/2)ⁿ so that the quotient is a projective line. As a result, some of these families intersect the hyperelliptic locus .     

Milnor K-Groups and Zero-Cycles on Products of Curves over p-Adic Fields

We investigate the Chow groups of zero cycles of products of curves over a p-adic field by means of the Milnor K-groups of their Jacobians as introduced by Somekawa. We prove some finiteness results for CH ₀(X)/m for X a product of curves over a p-adic field.     

Rigid Analytic Uniformization of Curves and the Study of Isogenies

Complex uniformization of curves is a popular tool in Number Theory. There are, however, some arithmetic and computational advantages in the use of p-adic uniformization. This paper compares the two theories and discusses how they can be used to study isogenies, with explicit examples of p-adic uniformization of hyperelliptic curves.      

Class field theory for a product of curves over a local field

We prove that the kernel of the reciprocity map for a product of curves over a p-adic field with split semi-stable reduction is divisible. We also consider the K ₁ of a product of curves over a number field.      

The p-rank strata of the moduli space of hyperelliptic curves

We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p?3. This yields a strong technique that allows us to analyze the stratum of hyperelliptic curves of genus g and p-rank f. Using this, we prove that the endomorphism ring of the Jacobian of a generic hyperelliptic curve of genus g and p-rank f is isomorphic to Z if g?4. Furthermore, we prove that the Z/?-monodromy of every irreducible component of is the symplectic group Sp2g(Z/?) if g?3, and ?≠p is an odd prime (with mild hypotheses on ? when f=0). These results yield numerous applications about the generic behavior of hyperelliptic curves of given genus and p-rank over finite fields, including applications about Newton polygons, absolutely simple Jacobians, class groups and zeta functions.     

Construction techniques for Galois coverings of the affine line

For constructing un ramified coverings of the affine line in characteristicp, a general theorem about good reductions modulop of coverings of characteristic zero curves is proved. This is applied to modular curves to realize SL(2, ℤ/nℤ)/±1, with GCD(n, 6) = 1, as Galois groups of unramified coverings of the affine line in characteristicp, for p = 2 or 3. It is applied to the Klein curve to realize PSL(2, 7) for p = 2 or 3, and to the Macbeath curve to realize PSL(2, 8) for p = 3. By looking at curves with big automorphism groups, the projective special unitary groups PSU(3, p^v) and the projective special linear groups PSL(2, p^v) are realized for allp, and the Suzuki groups Sz(2^2v+1) are realized for p = 2. Jacobian varieties are used to realize certain extensions of realizable groups with abelian kernels.     

A Practical Criterion For Some Submanifolds To Be Totally Geodesic

Our main theorem is a characterization of a totally geodesic K?hler immersion of a complex n-dimensional K?hler manifold M _n into an arbitrary complex (n + p)-dimensional K?hler manifold [(M)\tilde]_n+p\tilde{M}_{n+p} by observing the extrinsic shape of K?hler Frenet curves on the submanifold M _n . Those curves are closely related to the complex structure of M _n .     

Implementing the 4-dimensional GLV method on GLS elliptic curves with <Emphasis Type="Italic">j</Emphasis>-invariant 0

The Gallant–Lambert–Vanstone (GLV) method is a very efficient technique for accelerating point multiplication on elliptic curves with efficiently computable endomorphisms. Galbraith et al. (J Cryptol 24(3):446–469, 2011) showed that point multiplication exploiting the 2-dimensional GLV method on a large class of curves over \mathbbF_p²{\mathbb{F}_{p^2}} was faster than the standard method on general elliptic curves over \mathbbF_p{\mathbb{F}_{p}} , and left as an open problem to study the case of 4-dimensional GLV on special curves (e.g., j (E) = 0) over \mathbbF_p²{\mathbb{F}_{p^2}} . We study the above problem in this paper. We show how to get the 4-dimensional GLV decomposition with proper decomposed coefficients, and thus reduce the number of doublings for point multiplication on these curves to only a quarter. The resulting implementation shows that the 4-dimensional GLV method on a GLS curve runs in about 0.78 the time of the 2-dimensional GLV method on the same curve and in between 0.78 − 0.87 the time of the 2-dimensional GLV method using the standard method over \mathbbF_p{\mathbb{F}_{p}} . In particular, our implementation reduces by up to 27% the time of the previously fastest implementation of point multiplication on x86-64 processors due to Longa and Gebotys (CHES2010).     

Convex hulls of generalized moment curves

We consider the family of curves inR ⁴: {fx115-1}, wherep andq are positive integers, and determine the facial structure of the convex hull of these curves.     

Normal Rational Curves Over Prime Fields

It is shown that in the projective spaces PG(n,p),p prime, 2 n p-2, the normal rational curves are the only (p+1)-arcs fixed by a projective group G isomorphic to PSL(2,p).     

Torsors under finite and flat group schemes of rank <Emphasis Type="Italic">p</Emphasis> with Galois action

In this note we study the geometry of torsors under flat and finite commutative group schemes of rank p above curves in characteristic p, and above relative curves over a complete discrete valuation ring of inequal characteristic. In both cases we study the Galois action of the Galois group of the base field on these torsors. We also study the degeneration of _p -torsors, from characteritic 0 to characteristic p, and show that this degeneration is compatible with the Galois action. We then discuss the lifting of torsors under flat and commutative group schemes of rank p from positive to zero characteristics. Finally, for a proper and smooth curve X over a complete discrete valuation field, of inequal characteristic, which has good reduction, we show the existence of a canonical Galois equivariant filtration, on the first étale cohomology group of the geometric fibre of X, with values in _p .     

Nodal Curves with General Moduli on K3 Surfaces

We investigate the modular properties of nodal curves on a low genus K3 surface. We prove that a general genus g curve C is the normalization of a δ-nodal curve X sitting on a primitively polarized K3 surface S of degree 2p ? 2, for 2 ≤ g = p ? δ < p ≤ 11. The proof is based on a local deformation-theoretic analysis of the map from the stack of pairs (S, X) to the moduli stack of curves ? _g that associates to X the isomorphism class [C] of its normalization.     

Morse theory for analytic functions on surfaces

In this paper we deal with analytic functions defined on a compact two dimensional Riemannian surface S whose critical points are semi degenerated (critical points having a non identically vanishing Hessian). To any element p of the set of semi degenerated critical points Q we assign an unique index which can take the values −1, 0 or 1, and prove that Q is made up of finitely many (critical) points with non zero index and embedded circles. Further, we generalize the famous Morse result by showing that the sum of the indexes of the critical points of f equals χ (S), the Euler characteristic of S. As an intermediate result we locally describe the level set of f near a point p ∈Q. We show that the level set f ⁻¹(f (p)) is either a) the set {p}, or b) the graph of a smooth curve passing through p, or c) the graphs of two smooth curves tangent at p or d) the graphs of two smooth curves building at p a cusp shape.     

Construction of elliptic curves with large Iwasawa $${\lambda}$$ -invariants and large Tate-Shafarevich groups

In this paper we construct elliptic curves defined over the rationals with arbitrarily large Iwasawa λ-invariants for primes p satisfying or p = 13. We use this to obtain that the p-rank of the Tate-Shafarevich group can be arbitrarily large for such primes p.     

Konjugierte Netze mit p-ebenen u-Kurven,welche von den v-Kurven projektiv aufeinander bezogen werden

In this paper we consider conjugate nets in projective space P ⁿwith the following property: The u-curves of the net belong to p-dimensional subspaces of P ⁿand are projectively related to each other by the v-curves. We show that these nets form two classes. The first class consists of conjugate nets for which the u-curves from conic shadow boundaries. The u-curves of the nets of the second class, which we call C _p-nets, are rational normal curves of order p. Each u-curve possesses the characteristic (p–1)-space of its ambient p-space as a (p–1)-dimensional osculating space. This generalizes a result found by Degen [3] in the case p=2, n=3. By means of the Laplace transformation we get a construction of C _p-nets without integration.     

Uniform boundedness of <Emphasis Type="Italic">p</Emphasis>-primary torsion of abelian schemes

Let k be a field finitely generated over ℚ and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d=1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d=1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E→S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E _s , s∈S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves.     

Explicit Rational Functions on Fermat Curves and a Theorem of Greenberg

This paper is concerned with the arithmetic of curves of the form v^p=u^s(1-u), where p is a prime with p 5 and s is an integer such that 1 s p-2. The Jacobians of these curves admit complex multiplication by a primitive p-th root of unity . We find explicit rational functions on these curves whose divisors are p-multiples of divisors representing (1-)² - and (1-)³-division points on the corresponding Jacobians. This also gives an effective version of a theorem of Greenberg.     

Completeness of Normal Rational Curves

The completeness of normal rational curves, considered as (q + 1)-arcs in PG(n, q), is investigated. Previous results of Storme and Thas are improved by using a result by Kovács. This solves the problem completely for large prime numbers q and odd nonsquare prime powers q = p ^2h+1 with p prime, , where p ₀(h) is an odd prime number which depends on h.