K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.     

Pesenti-Szpiro inequality for optimal elliptic curves

We study Pesenti-Szpiro inequality in the case of elliptic curves over F_q(t) which occur as subvarieties of Jacobian varieties of Drinfeld modular curves. In general, we obtain an upper-bound on the degrees of minimal discriminants of such elliptic curves in terms of the degrees of their conductors and q. In the special case when the level is prime, we bound the degrees of discriminants only in terms of the degrees of conductors. As a preliminary step in the proof of this latter result we generalize a construction (due to Gekeler and Reversat) of 1-dimensional optimal quotients of Drinfeld Jacobians.     

Defining equations of modular curves

We obtain defining equations of modular curves X₀(N), X₁(N), and X(N) by explicitly constructing modular functions using generalized Dedekind eta functions. As applications, we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup. We also use our model of X₀(37) to find explicit modular parameterization of rational elliptic curves of conductor 37.     

On the torsion of Jacobians of principal modular curves of level 3<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We demonstrate that the 3-power torsion points of the Jacobians of the principal modular curves X(3ⁿ) are fixed by the kernel of the canonical outer Galois representation of the pro-3 fundamental group of the projective line minus three points. The proof proceeds by demonstrating the curves in question satisfy a two-part criterion given by Anderson and Ihara. Two proofs of the second part of the criterion are provided; the first relies on a theorem of Shimura, while the second uses the moduli interpretation. Received: 30 September 2005     

Torsion points on elliptic curves over function fields and a theorem of Igusa

If F   is a global function field of characteristic p>3$p>3$, we employ Tate's theory of analytic uniformization to give an alternative proof of a theorem of Igusa describing the image of the natural Galois representation on torsion points of non-isotrivial elliptic curves defined over F. Along the way, using basic properties of Faltings heights of elliptic curves, we offer a detailed proof of the function field analogue of a classical theorem of Shafarevich according to which there are only finitely many F-isomorphism classes of admissible elliptic curves defined over F with good reduction outside a fixed finite set of places of F. We end the paper with an application to torsion points rational over abelian extensions of F.     

On additive polynomials and certain maximal curves

We show that a maximal curve over F_q² given by an equation A(X)=F(Y), where A(X)∈F_q²[X] is additive and separable and where F(Y)∈F_q²[Y] has degree m prime to the characteristic p, is such that all roots of A(X) belong to F_q². In the particular case where F(Y)=Y^m, we show that the degree m is a divisor of q+1.     

Automorphic forms and cohomology theories on Shimura curves of small discriminant

We apply Lurie's theorem to produce spectra associated to 1-dimensional formal group laws on the Shimura curves of discriminants 6, 10, and 14. We compute rings of automorphic forms on these curves and the homotopy of the associated spectra. At p=3, we find that the curve of discriminant 10 recovers much the same as the topological modular forms spectrum, and the curve of discriminant 14 gives rise to a model of a truncated Brown-Peterson spectrum as an E_∞ ring spectrum.     

On plane models for Drinfeld modular curves

In this work, we find plane models for certain Drinfeld modular curves X₀(n) which have better properties than the plane models derived from the usual Drinfeld modular equations. As an application, we construct ring class fields over imaginary quadratic fields by using singular values of generators of the function field of X₀(n).     

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form Y ²=f ₃ X ⁶+f ₂ X ⁴+f ₁ X ²+f ₀, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field ℚα. If each of these elliptic curves has rank less than the degree of ℚα : ℚ, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over ℚα) on the elliptic curves, for which x∈ℚ. This in turn allows us to find all ℚ-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over ℚ), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over ℚ. Received: 27 November 1998 / Revised version: 4 June 1999     

Visualizing elements of order two in the Weil-Châtelet group

Let E be an elliptic curve over an infinite field K with characteristic ≠2, and σ∈H¹(G_K,E)[2] a two-torsion element of its Weil-Châtelet group. We prove that σ is always visible in infinitely many abelian surfaces up to isomorphism, in the sense put forward by Cremona and Mazur in their article (J. Exp. Math. 9(1) (2000) 13). Our argument is a variant of Mazur's proof, given in (Asian J. Math. 3(1) (1999) 221), for the analogous statement about three-torsion elements of the Shafarevich-Tate group in the setting where K is a number field. In particular, instead of the universal elliptic curve with full level-three-structure, our proof makes use of the universal elliptic curve with full level-two-structure and an invariant differential.     

Bielliptic modular curves X 1(M,N)

In this work, we determine all modular curves X₁(M,N) which are bielliptic, and then we also discuss the problem of finding the curves X₁(M,N) which admit infinitely many quartic points over .     

Rational torsion of J0(N) for hyperelliptic modular curves and families of Jacobians of genus 2 and genus 3 curves with a rational point of order 5,7 or 10

We describe a way of constructing Jacobians of hyperelliptic curves of genus g ≥ 2, defined over a number field, whose Jacobians have a rational point of order of some (well chosen) integer l ≥ g + 1; the method is based on a polynomial identity. Using this approach we construct new families of genus 2 curves defined over — which contain the modular curves X₀(31) (and X₀(22) as a by-product) and X₀(29), the Jacobians of which have a rational point of order 5 and 7 respectively. We also construct a new family of hyperelliptic genus 3 curves defined over —, which contains the modular curve X₀(41), the Jacobians of which have a rational point of order 10. Finally we show that all hyperelliptic modular curves X₀(N) with N a prime number fit into the described strategy, except for N = 37 in which case we give another explanation. The authors thank the FNR (project FNR/04/MA6/11) for their support.     

Galois closure of essentially finite morphisms

Let X be a reduced connected k-scheme pointed at a rational point x∈X(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:Y→X satisfying the condition H⁰(Y,O_Y)=k; this Galois closure is a torsor dominating f by an X-morphism and universal for this property. Moreover, we show that is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f:Y→X under a finite group scheme satisfying the condition H⁰(Y,O_Y)=k, Y has a fundamental group scheme π₁(Y,y) fitting in a short exact sequence with π₁(X,x).     

On Chow and Brauer groups of a product of Mumford curves

Let C₁,···,C_d be Mumford curves defined over a finite extension of and let X=C₁×···×C_d. We shall show the following: (1) The cycle map CH₀(X)/n → H^2d(X, μ_n^⊗d) is injective for any non-zero integer n. (2) The kernel of the canonical map CH₀(X)→Hom(Br(X),) (defined by the Brauer-Manin pairing) coincides with the maximal divisible subgroup in CH₀(X).     

Serre-Taubes duality for pseudoholomorphic curves

According to Taubes, the Gromov invariants of a symplectic four-manifold X with b₊>1 satisfy the duality Gr(α)=±Gr(κ−α), where κ is Poincaré dual to the canonical class. Extending joint work with Simon Donaldson, we interpret this result in terms of Serre duality on the fibres of a Lefschetz pencil on X, by proving an analogous symmetry for invariants counting sections of associated bundles of symmetric products. Using similar methods, we give a new proof of an existence theorem for symplectic surfaces in four-manifolds with b₊=1 and b₁=0. This reproves another theorem due to Taubes: two symplectic homology projective planes with negative canonical class and equal volume are symplectomorphic.     

Distribution of Selmer groups of quadratic twists of a family of elliptic curves

We study the distribution of the size of the Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with full 2-torsion points in Q. We show that one of these Selmer groups is almost always bounded, while the 2-rank of the other follows a Gaussian distribution. This provides us with a small Tate-Shafarevich group and a large Tate-Shafarevich group. When combined with a result obtained by Yu [G. Yu, On the quadratic twists of a family of elliptic curves, Mathematika 52 (1-2) (2005) 139-154 (2006)], this shows that the mean value of the 2-rank of the large Tate-Shafarevich group for square-free positive integers n less than X is , as X→∞.     

On the tensor rank of the multiplication in the finite fields

First, we prove the existence of certain types of non-special divisors of degree g−1 in the algebraic function fields of genus g defined over Fq. Then, it enables us to obtain upper bounds of the tensor rank of the multiplication in any extension of quadratic finite fields Fq by using Shimura and modular curves defined over Fq. From the preceding results, we obtain upper bounds of the tensor rank of the multiplication in any extension of certain non-quadratic finite fields Fq, notably in the case of F₂. These upper bounds attain the best asymptotic upper bounds of Shparlinski-Tsfasman-Vladut [I.E. Shparlinski, M.A. Tsfasman, S.G. Vladut, Curves with many points and multiplication in finite fields, in: Lecture Notes in Math., vol. 1518, Springer-Verlag, Berlin, 1992, pp. 145-169].     

Wildly ramified covers with large genus

We study wildly ramified G-Galois covers branched at B (defined over an algebraically closed field of characteristic p). We show that curves Y of arbitrarily high genus occur for such covers even when G, X, B and the inertia groups are fixed. The proof relies on a Galois action on covers of germs of curves and formal patching. As a corollary, we prove that for any nontrivial quasi-p group G and for any sufficiently large integer σ with p?σ, there exists a G-Galois étale cover of the affine line with conductor σ above the point ∞.