Real analytic elliptic surfaces

Here, we give an upper bound for the number of connected components of the real locus of several smooth complex compact elliptic surfaces defined over R in terms of the type of the singular fibers of their elliptic fibration.     

Existence of solutions to quasi-linear elliptic problems with generalized Robin boundary conditions

We characterize the L^q-solvability of a class of quasi-linear elliptic equations involving the p-Laplace operator with generalized nonlinear Robin type boundary conditions on bad domains. Some uniqueness results are also given.     

On the moduli of surfaces admitting genus two fibrations over elliptic curves

In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that a surface admitting a smooth fibration as above is elliptic, and we employ results on the moduli of polarized elliptic surfaces to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes of morphisms of degree n from elliptic curves to the modular curve X(d), d ≥ 3. Ultimately, we show that the moduli spaces in the nonsmooth case are fiber spaces over the affine line with fibers determined by the components of . Received: 30 August 2006     

Compact Kähler manifolds with elliptic homotopy type

Simply connected compact Kähler manifolds of dimension up to three with elliptic homotopy type are characterized in terms of their Hodge diamonds. For surfaces there are only two possibilities, namely h1,1?2 with hp,q=0 for p≠q. For threefolds, there are three possibilities, namely h1,1?3 with hp,q=0 for p≠q. This characterization in terms of the Hodge diamonds is applied to explicitly classify the simply connected Kähler surfaces and Fano threefolds with elliptic homotopy type.     

On the classification of elliptic surfaces withq=1

We classify, up to isomorphism, elliptic surfaces with irregularity one having exactly one singular fiber (necessarily of typeI ₆ ^* ). All of them turn out to be elliptic modular surfaces (Shioda [11]), so that the problem is indirectly equivalent to classifying certain subgroups ofSL ₂(Z). These surfaces are then used to produce examples of (elliptic) surfaces withq=1, anyp _g 1, which have maximal Picard number (see Persson [7] for the caseq=0). Finally, the classification yields some interesting relationships between hypergeometric functions, theta functions, and certain automorphic forms.Supported in part by NSF DMS-8501724     

Parity of ranks for elliptic curves with a cyclic isogeny

Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p?3 and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for such curves. We prove the analogous results for p=2 under the additional assumption that E is not supersingular at primes above 2.     

Explicit identities for invariants of elliptic curves

New explicit formulas are given for the supersingular polynomial ssp(t) and the Hasse invariant of an elliptic curve E in characteristic p. These formulas are used to derive identities for the Hasse invariants of elliptic curves En in Tate normal form with distinguished points of order n. This yields a proof that and are projective invariants (mod p) for the octahedral group and the icosahedral group, respectively; and that the set of fourth roots λ1/4 of supersingular parameters of the Legendre normal form Y²=X(X−1)(X−λ) in characteristic p has octahedral symmetry. For general n?4, the field of definition of a supersingular En is determined, along with the field of definition of the points of order n on En.     

Fully maximal and fully minimal abelian varieties

We introduce and study a new way to categorize supersingular abelian varieties defined over a finite field by classifying them as fully maximal, mixed or fully minimal. The type of A depends on the normalized Weil numbers of A and its twists. We analyze these types for supersingular abelian varieties and curves under conditions on the automorphism group. In particular, we present a complete analysis of these properties for supersingular elliptic curves and supersingular abelian surfaces in arbitrary characteristic, and for a one-dimensional family of supersingular curves of genus 3 in characteristic 2.     

On the parity of supersingular Weil polynomials

Let q be an odd power of a prime p and let \({A/\mathbb{F}_q}\) be a supersingular abelian variety of dimension g. We show that if \({p > 2g+1}\), then the characteristic polynomial of the q-Frobenius is an even polynomial. This generalizes the well-known result on the vanishing of the trace of the p-Frobenius when \({p > 3}\) for supersingular elliptic curves over \({\mathbb{F}_p}\).     

Non-classical Godeaux surfaces

A non-classical Godeaux surface is a minimal surface of general type with χ = K ² = 1 but with h ⁰¹ ≠ 0. We prove that such surfaces fulfill h ⁰¹ = 1 and they can exist only over fields of positive characteristic at most 5. Like non-classical Enriques surfaces they fall into two classes: the singular and the supersingular ones. We give a complete classification in characteristic 5 and compute their Hodge-, Hodge–Witt- and crystalline cohomology (including torsion). Finally, we give an example of a supersingular Godeaux surface in characteristic 5.     

Selmer groups of elliptic curves that can be arbitrarily large

In this article, it is shown that certain kinds of Selmer groups of elliptic curves can be arbitrarily large. The main result is that if p is a prime at least 5, then p-Selmer groups of elliptic curves can be arbitrarily large if one ranges over number fields of degree at most g+1 over the rationals, where g is the genus of X₀(p). As a corollary, one sees that p-Selmer groups of elliptic curves over the rationals can be arbitrarily large for p=5,7 and 13 (the cases p?7 were already known). It is also shown that the number of elements of order N in the N-Selmer group of an elliptic curve over the rationals can be arbitrarily large for N=9,10,12,16 and 25.     

New generalized Jacobi elliptic function rational expansion method

In this work, a new generalized Jacobi elliptic function rational expansion method is based upon twenty-four Jacobi elliptic functions and eight double periodic Weierstrass elliptic functions, which solve the elliptic equation ?^′2=r+p?²+q?⁴, is described. As a consequence abundant new Jacobi-Weierstrass double periodic elliptic functions solutions for (3+1)-dimensional Kadmtsev-Petviashvili (KP) equation are obtained by using this method. We show that the new method can be also used to solve other nonlinear partial differential equations (NPDEs) in mathematical physics.     

Isogenies of elliptic curves and the Morava stabilizer group

Let S₂ be the p-primary second Morava stabilizer group, C a supersingular elliptic curve over , O the ring of endomorphisms of C, and ? a topological generator of (or if p=2). We show that for p>2 the group Γ⊆O[1/?]^× of quasi-endomorphisms of degree a power of ? is dense in S₂. For p=2, we show that Γ is dense in an index 2 subgroup of S₂.     

Irregularity of certain algebraic fiber spaces

LetX, Y be smooth complex projective varieties, andf: X →Y be a fiber space whose general fiber is a curve of genusg. Denote byq _f the relative irregularity off. It is proved thatq _f ≤5g+1 / 6, iff is not generically trivial; moreover, if either a)f is non-constant and the general fiber is either hyperelliptic or bielliptic or b)q(Y)=0, thenq _f ≤g+1 / 2, and the bound is best possible. A classification of fiber surfaces of genus 3 withq _f =2 is also given in this note. Project supported by China Postdoctoral Science Foundation     

Nonlinear gradient estimates for elliptic equations of general type

We study a very general model of nonvariational elliptic equations of p-Laplacian type. We prove the W ^1,q -estimates for each q > p regarding such a nonlinear elliptic equation in divergence form under the assumption that for each point and for each sufficiently small scale the nonlinearity is suitably close to a vector of the gradient in BMO space. In the same spirit our results can be easily extended to Orlicz spaces as well.     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.     

The p-torsion subgroup scheme of an elliptic curve

Let k be a field of positive characteristic p.

Question. Does every twisted form of μp over k occur as subgroup scheme of an elliptic curve over k?     

False Hyperelliptic Surfaces with Section

Let X be a nonsingular relatively minimal projective surface over an algebraically closed field of characteristic p > 0. We call X a false hyperelliptic surface if X satisfies the following conditions: (1) c₂(X) = 0, c₁(X)² = 0, dim Alb (X) = 1, and (2) All fibres of the Albanese mapping of X are rational curves with only one cusp of type xp^v + yⁿ = 0. In this article, we consider a false hyperelliptic surface whose Albanese mapping has a cross-section. We prove that every false hyperellyptic surface with section arises from an elliptic ruled surface and that every false hyperelliptic surface has an elliptic fibration with multiple fibre. Moreover, we construct an example of false hyperelliptic surface with section, whose elliptic fibration has a multiple fibre of supersingular elliptic curve of multiplicity p^v (v > 1).     

Renormalized solutions of degenerate elliptic problems

We consider a general class of degenerate elliptic problems of the form Au+g(x,u,Du)=f, where A is a Leray-Lions operator from a weighted Sobolev space into its dual. We assume that g(x,s,ξ) is a Caratheodory function verifying a sign condition and a growth condition on ξ. Existence of renormalized solutions is established in the L¹-setting.     

Stable bundles with torsion Chern classes on non-Kählerian elliptic surfaces

If π:X→B is a non-Kählerian elliptic surface with generic fibreF, the moduli space of stable holomorphic vector bundles with torsion Chern classes onX has an induced fibred structure with base Pic^o(F) and the moduli space of stable parabolic bundles onB ^orb as fibre. This is specific to the non-Kähler case.