Rational Points on Genus 3 Real Hyperelliptic Curves

We compute rational points on real hyperelliptic curves of genus 3 defined on whose Jacobian have Mordell-Weil rank r=0. We present an implementation in sagemath of an algorithm which describes the birational transformation of real hyperelliptic curves into imaginary hyperelliptic curves and the Chabauty-Coleman method to find C (). We run the algorithms in Sage on 47 real hyperelliptic curves of genus 3.

Remarks on the Number of Rational Points on a Class of Hypersurfaces over Finite Fields

Let Fq be the finite field of q elements and f be a nonzero polynomial over Fq. For each b ∈ Fq, let Nq（f = b） denote the number of Fq-rational points on the affine hypersurface f = b. We obtain the formula of Nq（f= b） for a class of hypersurfaces over Fq by using the greatest invariant factors of degree matrices under certain cases, which generalizes the previously known results. We also give another simple direct proof to the known results.

Explicit Classification for Torsion Subgroups of Rational Points of Elliptic Curves

We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups E_(tors)(Q). More precisely, we classify those elliptic curves with E_(tors)(Q) being cyclic with even orders. We also give explicit formulas for generators of E_(tors)(Q). These results, together with the recent results of K. Ono for the non-cyclic E_(tors)(Q), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2.

Rational Points of Elliptic Curve y2 = x3＋ k3

Let E be an elliptic curve defined over the field of rational numbers ~. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely many square-free integers d such that the rank of Ed（Q） is zero. Let E（k） denote the elliptic curve y2 = x3 ＋ k. Then the quadratic twist E（1）d of E（1） by d is the elliptic curve E（d3）： y2 = x3＋ k3. Let r = 1, 2, 5, 10, 13, 14, 17, 22. Ono proved that there are infinitely many square-free integers d = r （rood 24） such that rankE（-d3）（Q） = 0, using the theory of modular forms. In this paper, we use the class number of quadratic field and Pell equation to describe these square-free integers k such that E（k3）（Q） has rank zero.

Rational double points on Enriques surfaces

We classify,up to some lattice-theoretic equivalence,all possible configurations of rational double points that can appear on a surface whose minimal resolution is a complex Enriques surface.

Rational maps as Schwarzian primitives

We examine when a meromorphic quadratic differential φ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of φ around each pole c, the most singular term should take the form(1- d2)/(2(z- c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles(i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko(2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by φ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative.Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.

Note on the Number of Solutions of Cubic Diagonal Equations over Finite Fields

Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of Hasse-Davenport theorem,an explicit formula for the number of solutions of cubic diagonal equation x_(1)^(3)+x_(2)^(3)+…+x_(n)^(3)=c over Fqis given,where c∈F_(q)^(*)and p≡1(mod 3).This extends earlier results.

On a certain non-split cubic surface

This paper establishes an asymptotic formula with a power-saving error term for the number of rational points of bounded height on the singular cubic surface of P3 Qgiven by the following equation x0（x12+ x22）-x33= 0 in agreement with the Manin-Peyre conjectures.

Gorenstein Q-homology projective planes

We present the complete list of all singularity types on Gorenstein Q-homology projective planes,i.e.,normal projective surfaces of second Betti number one with at worst rational double points.The list consists of 58 possible singularity types,each except two types supported by an example.

The Number of Solutions of Certain Equations over Finite Fields

Let s be a positive integer,p be an odd prime,q=p^(s),and let F_(q)be a finite field of q elements.Let N_(q)be the number of solutions of the following equations:(x_(1)^(m_(1))+x_(2)^(m_(2))+…+x_(n)^(m_(n)))^(k)=x_(1)x_(2)…x_(n)x^(k_(n+1))_(n+1)…x^(k_(t))_(t)over the finite field F_(q),with n≥2,t>n,k,and k_(j)(n+1≤j≤t),m_(i)(1≤i≤n)are positive integers.In this paper,we find formulas for N_(q)when there is a positive integer l such that dD|(p^(l)+1),where D=1 cm[d_(1),…,d_(n)],d=gcd(n∑i=1M/m_(i)-kM,(q-1)/D),M=1 cm[m_(1),…,m_(n)],d_(j)=gcd(m_(j),q-1),1≤j≤n.And we determine N_(q)explicitly under certain cases.This extends Markoff-Hurwitz-type equations over finite field.

Counting Multiplicities in a Hypersurface over Number Fields

We fix a counting function of multiplicities of algebraic points in a projective hypersurface over a number field, and take the sum over all algebraic points of bounded height and fixed degree. An upper bound for the sum with respect to this counting function will be given in terms of the degree of the hypersurface, the dimension of the singular locus, the upper bounds of height, and the degree of the field of definition.

The value of polynomials represented by norm forms

Let P(t) be a product of(possibly repeated) linear factors over Q and K/Q an abelian extension. Under a strict condition, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one for any smooth proper model of the variety over Q defined by P(t) = NK/Q(x).

Research on a Class of Equations overFinite Fields

Let Fq stand for the finite field of odd characteristic p with q elements（q=pn,n∈N）and Fq＊ denote the set of all the nonzero elements of Fq.In this paper,by using the augmented degree matrix and the result given by Cao,we obtain a formula for the number of rational points of the following equation over Fq：f（x 1,x 2,...,x n）=（a1 x1 x2d＋a2 x2 x3d...＋a（n-1）x（n-1）xnd＋an xn x1d）λ-bx1（d1）x2d2...xn（dn）,with ai,b∈Fq＊,n≥2,λ〉0 being positive integers,and d,di being nonnegative integers for 1≤i n.This technique can be applied to the polynomials of the form h1λ=h2 with λ being positive integer and h1,h2∈Fq[x 1,x 2,...,x n].It extends the results of the Markoff-Hurwitz-type equations.