The concept of hyperellipticity on surfaces with nodes

A surface with nodes X is hyperelliptic if there exists an involution such that the genus of X/〈h〉 is 0. We prove that this definition is equivalent, as in the category of surfaces without nodes, to the existence of a degree 2 morphism satisfying an additional condition where the genus of Y is 0. Other question is if the hyperelliptic involution is unique or not. We shall prove that the hyperelliptic involution is unique in the case of stable Riemann surfaces but is not unique in the case of Klein surfaces with nodes. Finally, we shall prove that a complex double of a hyperelliptic Klein surface with nodes could not be hyperelliptic.     

The fluctuations in the number of points on a hyperelliptic curve over a finite field

The number of points on a hyperelliptic curve over a field of q elements may be expressed as q+1+S where S is a certain character sum. We study fluctuations of S as the curve varies over a large family of hyperelliptic curves of genus g. For fixed genus and growing q, Katz and Sarnak showed that is distributed as the trace of a random 2g×2g unitary symplectic matrix. When the finite field is fixed and the genus grows, we find that the limiting distribution of S is that of a sum of q independent trinomial random variables taking the values ±1 with probabilities 1/2(1+q⁻¹) and the value 0 with probability 1/(q+1). When both the genus and the finite field grow, we find that has a standard Gaussian distribution.     

On unramified normal coverings of hyperelliptic curves

It is well known that the number of unramified normal coverings of an irreducible complex algebraic curve C with a group of covering transformations isomorphic to Z₂⊕Z₂ is (2^4g−3⋅2^2g+2)/6. Assume that C is hyperelliptic, say . Horiouchi has given the explicit algebraic equations of the subset of those covers which turn out to be hyperelliptic themselves. There are of this particular type. In this article, we provide algebraic equations for the remaining ones.     

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.     

Non-hyperelliptic curves of genus three over finite fields of characteristic two

Let k be a finite field of even characteristic. We obtain in this paper a complete classification, up to k-isomorphism, of non-singular quartic plane curves defined over k. We find explicit rational models and closed formulas for the total number of k-isomorphism classes. We deduce from these computations the number of k-rational points of the different strata by the Newton polygon of the non-hyperelliptic locus of the moduli space M₃ of curves of genus 3. By adding to these computations the results of Oort [Moduli of abelian varieties and Newton polygons, C.R. Acad. Sci. Paris 312 (1991) 385-389] and Nart and Sadornil [Hyperelliptic curves of genus three over finite fields of characteristic two, Finite Fields Appl. 10 (2004) 198-200] on the hyperelliptic locus we obtain a complete picture of these strata for M₃.     

Double coverings for quadratic extensions and function fields

A double covering of a Galois extension K/k in the sense of P. Das (2000) [4] is an extension of degree ?2 such that is Galois. In this paper we determine explicitly all double coverings of any quadratic extensions, the Carlitz cyclotomic extensions of the rational function field over a finite field, and their maximal real subfields. In the case of quadratic extensions, we get the result by Hilbert Theorem 90 and in the function field cases, we get the results by using the method in G.W. Anderson (2002) [1] with a modification and using the results in S. Bae et al. (2003) [2] and S. Bae and L. Yin (2004) [3]. We also construct explicitly a large kind of (q−1)-th coverings of cyclotomic function fields.     

A characterization of double covers of curves in terms of the ample cone of second symmetric product

We investigate the nef cone spanned by the diagonal and the fibre classes of second symmetric product of a curve of genus g. This 2-dimensional nef cone gives a characterization of double covers of curves of genus . This is a generalization of a result by Debarre [Olivier Debarre, Seshadri constants of abelian varieties, in: The Fano Conference, Univ. Torino, Turin, 2004, pp. 379-394, Proposition 8].     

Chow groups of the moduli spaces of weighted pointed stable curves of genus zero

The moduli space of weighted pointed stable curves of genus zero is stratified according to the degeneration types of such curves. We show that the homology groups of are generated by the strata of and give all additive relations between them. We also observe that the Chow groups and the homology groups are isomorphic. This generalizes Kontsevich-Manin's and Losev-Manin's theorems to arbitrary weight data A.     

Quasi-periodic solutions of mixed AKNS equations

A hierarchy of new nonlinear evolution equations, which are composed of the positive and negative AKNS flows, is proposed. On the basis of the theory of algebraic curves, the corresponding flows are straightened using the Abel-Jacobi coordinates. The meromorphic function ?, the Baker-Akhiezer vector , and the hyperelliptic curve K_n are introduced and, by using these, quasi-periodic solutions of the first three nonlinear evolution equations in the hierarchy are constructed according to the asymptotic properties and the algebro-geometric characters of ?, and K_n.     

Complete classification of torsion of elliptic curves over quadratic cyclotomic fields

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In a previous paper Najman (in press) [9], the author examined the possible torsions of an elliptic curve over the quadratic fields Q(i) and . Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over Q(i) and , we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves over Q(i) and .

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=VPhCkJTGB_o.     

Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups II

Let R be a compact, connected, orientable surface of genus g with p boundary components. Let C(R) be the complex of curves on R and be the extended mapping class group of R. Suppose that either g=2 and p?2 or g?3 and p?0. We prove that a simplicial map is superinjective if and only if it is induced by a homeomorphism of R. As a corollary, we prove that if K is a finite index subgroup of and is an injective homomorphism, then f is induced by a homeomorphism of R and f has a unique extension to an automorphism of . This extends the author's previous results about closed connected orientable surfaces of genus at least 3, to the surface R.     

The orientable genus of some joins of complete graphs with large edgeless graphs

In an earlier paper the authors showed that with one exception the nonorientable genus of the graph with m≥n−1, the join of a complete graph with a large edgeless graph, is the same as the nonorientable genus of the spanning subgraph . The orientable genus problem for with m≥n−1 seems to be more difficult, but in this paper we find the orientable genus of some of these graphs. In particular, we determine the genus of when n is even and m≥n, the genus of when n=2^p+2 for p≥3 and m≥n−1, and the genus of when n=2^p+1 for p≥3 and m≥n+1. In all of these cases the genus is the same as the genus of K_m,n, namely ⌈(m−2)(n−2)/4⌉.     

Hecke curves and Hitchin discriminant

Let C be a smooth projective curve of genus g?4 over the complex numbers and be the moduli space of stable vector bundles of rank r with a fixed determinant of degree d. In the projectivized cotangent space at a general point E of , there exists a distinguished hypersurface SE consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at E, there exists a distinguished subvariety CE consisting of vectors tangent to Hecke curves in through E. Our main result establishes that the hypersurface SE and the variety CE are dual to each other. As an application of this duality relation, we prove that any surjective morphism , where C^′ is another curve of genus g, is biregular. This confirms, for , the general expectation that a Fano variety of Picard number 1, excepting the projective space, has no non-trivial self-morphism and that morphisms between Fano varieties of Picard number 1 are rare. The duality relation also gives simple proofs of the non-abelian Torelli theorem and the result of Kouvidakis-Pantev on the automorphisms of .     

Moduli spaces of metric graphs of genus 1 with marks on vertices

In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces , which parametrize the isometry classes of metric graphs of genus 1 with n marks on vertices are homotopy equivalent to the spaces TM1,n, which are the moduli spaces of tropical curves of genus 1 with n marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.     

The hyperelliptic limit cycles of the Liénard systems

For Liénard systems , with fm and gn real polynomials of degree m and n respectively, in [H. Zoladek, Algebraic invariant curves for the Liénard equation, Trans. Amer. Math. Soc. 350 (1998) 1681-1701] the author showed that if m?3 and m+1<n<2m there always exist Liénard systems which have a hyperelliptic limit cycle. Llibre and Zhang [J. Llibre, Xiang Zhang, On the algebraic limit cycles of Liénard systems, Nonlinearity 21 (2008) 2011-2022] proved that the Liénard systems with m=3 and n=5 have no hyperelliptic limit cycles and that there exist Liénard systems with m=4 and 5<n<8 which do have hyperelliptic limit cycles. So, it is still an open problem to characterize the Liénard systems which have an algebraic limit cycle in cases m>4 and m+1<n<2m. In this paper we will prove that there exist Liénard systems with m=5 and m+1<n<2m which have hyperelliptic limit cycles.     

Normalizers of non-split Cartan subgroups, modular curves, and the class number one problem

Let be the open non-cuspidal locus of the modular curve associated to the normalizer of a non-split Cartan subgroup of level n. As Serre pointed out, an imaginary quadratic field of class number one gives rise to an integral point on for suitably chosen n. In this note, we give a genus formula for the modular curves and we give three new solutions to the class number one problem using the modular curves for n=16,20,21. These are the only such modular curves of genus ?2 that had not yet been exploited.     

Superinjective simplicial maps of complexes of curves and injective homomorphisms of subgroups of mapping class groups

Let S be a closed, connected, orientable surface of genus at least 3, be the complex of curves on S and be the extended mapping class group of S. We prove that a simplicial map, , preserves nondisjointness (i.e. if α and β are two vertices in and i(α,β)≠0, then i(λ(α),λ(β))≠0) iff it is induced by a homeomorphism of S. As a corollary, we prove that if K is a finite index subgroup of and is an injective homomorphism, then f is induced by a homeomorphism of S and f has a unique extension to an automorphism of .