Automorphisms of curves fixing the order two points of the Jacobian

Let X be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic different from two. If X admits a nontrivial automorphism σ that fixes pointwise all the order two points of Pic⁰(X), then we prove that X is hyperelliptic with σ being the unique hyperelliptic involution. As a corollary, if a nontrivial automorphisms of X fixes pointwise all the theta characteristics on X, then X is hyperelliptic with being its hyperelliptic involution.      

Lie Algebras of Heat Operators in a Nonholonomic Frame

Mathematical Notes - We construct the Lie algebras of systems of $$2g$$ graded heat operators $$Q_0,Q_2,\dots,Q_{4g-2}$$ that determine the sigma functions $$\sigma(z,\lambda)$$ of hyperelliptic...     

Improved Complexity Bounds for Counting Points on Hyperelliptic Curves

Foundations of Computational Mathematics - We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus g defined over $${\mathbb {F}}_q$$...     

Periodicity Criterion for Continued Fractions of Key Elements in Hyperelliptic Fields

Doklady Mathematics - The periodicity and quasi-periodicity of functional continued fractions in the hyperelliptic field $$L = \mathbb{Q}(x)(\sqrt f )$$ has a more complex nature than the...     

Z_2-商的秩为r的超椭圆纤维化曲面的斜率

设Ｓ是一般型的相对极小曲面，ｆ：Ｓ→Ｃ是亏格ｇ的超椭圆纤维化．本文中我们证明了如果 Ｓ的代数基本群的垂直部分的极大挠 ２商为，那么其斜率且等号成立仅当 Ｓ上的超椭圆对合所诱导的二次复盖的分歧除子 Ｒ仅有（ｒ＋１→，＋１）（当ｒ为偶数）型奇点，或（ｒ＋２→ｒ＋２）（当ｒ为奇数）型奇点．     

Extremal Problems on Components and Loops in Graphs

The authors recently defined a new graph invariant denoted by Ω(G) only in terms of a given degree sequence which is also related to the Euler characteristic. It has many important combinatorial applications in graph theory and gives direct information compared to the better known Euler characteristic on the realizability, connectedness, cyclicness, components, chords, loops etc. Many similar classification problems can be solved by means of Ω. All graphs G so that Ω(G) ≤-4 are shown to be disconnected, and if Ω(G) ≥-2, then the graph is potentially connected. It is also shown that if the realization is a connected graph and Ω(G) =-2, then certainly the graph should be a tree.Similarly, it is shown that if the realization is a connected graph G and Ω(G) ≥ 0, then certainly the graph should be cyclic. Also, when Ω(G) ≤-4, the components of the disconnected graph could not all be cyclic and if all the components of G are cyclic, then Ω(G) ≥ 0. In this paper, we study an extremal problem regarding graphs. We find the maximum number of loops for three possible classes of graphs.We also state a result giving the maximum number of components amongst all possible realizations of a given degree sequence.     

On the Rogers--Ramanujan Periodic Continued Fraction

In the paper, the convergence properties of the Rogers--Ramanujan continued fraction $$1 + \frac{qz}{1 + \tfrac{q^2 z}{1 + \cdots}}$$ are studied for q = exp (2 π i τ), where τ is a rational number. It is shown that the function H _q to which the fraction converges is a counterexample to the Stahl conjecture (the hyperelliptic version of the well-known Baker--Gammel--Wills conjecture). It is also shown that, for any rational τ, the number of spurious poles of the diagonal Padé approximants of the hyperelliptic function H _q does not exceed one half of its genus.     

Tropical hyperelliptic curves

We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. We define a harmonic morphism of metric graphs and prove that a metric graph is hyperelliptic if and only if it admits a harmonic morphism of degree 2 to a metric tree. This generalizes the work of Baker and Norine on combinatorial graphs to the metric case. We then prove that the locus of 2-edge-connected genus g tropical hyperelliptic curves is a (2g?1)-dimensional stacky polyhedral fan whose maximal cells are in bijection with trees on g?1 vertices with maximum valence 3. Finally, we show that the Berkovich skeleton of a classical hyperelliptic plane curve satisfying a certain tropical smoothness condition is a standard ladder of genus g.     

Twofold unbranched coverings of genus two 3-manifolds are hyperelliptic

A closed 3-dimensional manifold is hyperelliptic if it admits an involution such that the quotient space of the manifold by the action of the involution is homeomorphic to the 3-sphere. We prove that a twofold unbranched covering of a genus two 3-manifold is hyperelliptic. This result is reminiscent of a theorem, which seems to have first appeared in a paper by Enriques and which has been reproved more recently by Farkas and Accola, which states that a twofold unbranched covering of a Riemann surface of genus two is hyperelliptic.     

On hyperelliptic modular curves over function fields

We prove that there are only finitely many modular curves of -elliptic sheaves over which are hyperelliptic. In odd characteristic we give a complete classification of such curves. The author was supported in part by NSF grant DMS-0801208 and Humboldt Research Fellowship.     

The integral monodromy of hyperelliptic and trielliptic curves

We compute the and monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group . We prove that the monodromy of the moduli space of trielliptic curves with signature (r,s) is the special unitary group . Rachel Pries was partially supported by NSF grant DMS-04-00461.     

A conjecture on the lengths of filling pairs

Geometriae Dedicata - A pair $$(\alpha , \beta )$$ of simple closed geodesics on a closed and oriented hyperbolic surface $$M_g$$ of genus g is called a filling pair if the complementary components...     

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.     

Fields of moduli and definition of hyperelliptic covers

In this paper, for all genera g>1, g ≡ 1 mod 4, we construct an explicit hyperelliptic curve whose field of moduli is and such that the minimum subfield of over which it can be hyperelliptically defined is a degree three extension of . These examples are related to previous work by Earle, Shimura, and Mestre and to a recent conjecture by Shaska. Received: 19 January 2005     

Isomorphism classes of hyperelliptic curves of genus 2 over finite fields with characteristic 2

In this paper we study the computation of the number of isomorphism classes of hyperelliptic curves of genus 2 over finite fields Fq with q even. We show the formula of the number of isomorphism classes, that is, for q = 2m, if 4 m, then the formula is 2q3 q2 - q; if 4 | m, then the formula is 2q3 q2 - q 8. These results can be used in the classification problems and the hyperelliptic curve cryptosystems.     

Seshadri constants on Jacobian of curves

We compute the Seshadri constants on the Jacobian of hyperelliptic curves, as well as of curves with genus three and four. For higher genus curves we conclude that if the Seshadri constants of their Jacobian are less than 2, then the curves must be hyperelliptic.

     

On football manifolds of E. Molnár

A closed 3-manifold M is said to be hyperelliptic if it has an involution τ such that the quotient space of M by the action of τ is homeomorphic to the standard 3-sphere. We show that the hyperbolic football manifolds of Emil Molnár [12] are hyperelliptic. Then we determine the isometry groups of such manifolds. Another consequence is that the unique hyperbolic dodecahedral and icosahedral 3-space forms with first homology group ℤ₃₅ (constructed by I. Prok in [16], on the basis of a principal algorithm due to Emil Molnár [13], and by Richardson and Rubinstein in [18]) are also hyperelliptic.     

Complete nonorientable minimal surfaces in R3

We consider complete nonorientable minimal immersionsx(M)⊇R ³. Assuming the double coverN ofM has finite total curvature, we generalize an argument of Lopez/Ros to give a sufficient condition for the instability ofx(M) in terms of the total curvature ofM and the genus γ of . We apply this condition to prove that if the immersion is regular thenx(M) is unstable. We also consider the case where the immersion is finitely branched, and we classify the possibilities under the assumption the is hyperelliptic.