On the variation of Tate-Shafarevich groups of elliptic curves over hyperelliptic curves

Let E be an elliptic curve over F=F_q(t) having conductor (p)·∞, where (p) is a prime ideal in F_q[t]. Let d∈F_q[t] be an irreducible polynomial of odd degree, and let . Assume (p) remains prime in K. We prove the analogue of the formula of Gross for the special value L(E⊗_FK,1). As a consequence, we obtain a formula for the order of the Tate-Shafarevich group Ш(E/K) when L(E⊗_FK,1)≠0.     

Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k=Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q−1 and q+1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is defined to be self-dual if it is k-isomorphic to its own hyperelliptic twist.     

Indices of hyperelliptic curves over p-adic fields

Let k be a p-adic field of odd residue characteristic and let C be a hyperelliptic (or elliptic) curve defined by the affine equation Y ²=h(X). We discuss the index of C if h(X)=ɛf(X), where ɛ is either a non-square unit or a uniformising element in O _k and f(X) a monic, irreducible polynomial with integral coefficients. If a root θ of f generates an extension k(θ) with ramification index a power of 2, we completely determine the index of C in terms of data associated to θ. Theorem 3.11 summarizes our results and provides an algorithm to calculate the index for such curves C. Received: 14 July 1997 / Revised version: 16 February 1998     

Explicit Mumford isomorphism for hyperelliptic curves

Using an explicit version of the Mumford isomorphism on the moduli space of hyperelliptic curves we derive a closed formula for the Arakelov-Green function of a hyperelliptic Riemann surface evaluated at its Weierstrass points.     

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.     

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.     

Higher syzygies of hyperelliptic curves

Let X be a hyperelliptic curve of arithmetic genus g and let f:X→P¹ be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d≤2g. Note that the minimal free resolution of X of degree ≥2g+1 is already completely known. Let A=f^∗O_P¹(1), and let L be a very ample line bundle on X of degree d≤2g. For , we call the pair (m,d−2m)the factorization type ofL. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by |L| are precisely determined by the factorization type of L.     

Elliptic subcovers of hyperelliptic curves

The main result of this paper states that if C is a hyperelliptic curve of even genus over an arbitrary field K , then there is a natural bijection between the set of equivalence classes of elliptic subcovers of and the set of elliptic subgroups of its Jacobian .     

Dyson's theorem for curves

Let K be a number field and X₁ and X₂ two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson theorem for the product X₁×X₂. If Xi=P₁ we find the classical Dyson theorem. In general, it will imply a self contained and easy proof of Siegel theorem on integral points on hyperbolic curves and it will give some insight on effectiveness. This proof is new and avoids the use of Roth and Mordell-Weil theorems, the theory of Linear Forms in Logarithms and the Schmidt subspace theorem.     

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→∞.     

Exhibiting SHA[2] on hyperelliptic Jacobians

We discuss approaches to computing in the Shafarevich-Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2-Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of genus 2, we also demonstrate a connection with degree 4 del Pezzo surfaces, and show how the Brauer-Manin obstruction on these surfaces can be used to compute members of the Shafarevich-Tate group of Jacobians. We derive an explicit parametrised infinite family of genus 2 curves whose Jacobians have nontrivial members of the Shafarevich-Tate group. Finally, we prove that under certain conditions, the visualisation dimension for order 2 cocycles of Jacobians of certain genus 2 curves is 4 rather than the general bound of 32.     

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.     

Divisibility of L-polynomials for a family of Artin–Schreier curves

In this paper we consider the curves $C_k^{(p,a)}:y^p?y=x^{p^k+1}+ax$ defined over ${\mathbb{F}}_p$ and give a positive answer to a conjecture about a divisibility condition on L-polynomials of the curves $C_k^{(p,a)}$. Our proof involves finding an exact formula for the number of ${\mathbb{F}}_{p^n}$-rational points on $C_k^{(p,a)}$ for all n, and uses a result we proved elsewhere about the number of rational points on supersingular curves.     

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.     

Application of covering techniques to families of curves

Much success in finding rational points on curves has been obtained by using Chabauty's Theorem, which applies when the genus of a curve is greater than the rank of the Mordell-Weil group of the Jacobian. When Chabauty's Theorem does not directly apply to a curve , a recent modification has been to cover the rational points on by those on a covering collection of curves , obtained by pullbacks along an isogeny to the Jacobian; one then hopes that Chabauty's Theorem applies to each . So far, this latter technique has been applied to isolated examples. We apply, for the first time, certain covering techniques to infinite families of curves. We find an infinite family of curves to which Chabauty's Theorem is not applicable, but which can be solved using bielliptic covers, and other infinite families of curves which even resist solution by bielliptic covers. A fringe benefit is an infinite family of Abelian surfaces with non-trivial elements of the Tate-Shafarevich group killed by a bielliptic isogeny.     

Computing complex and real tropical curves using monodromy

Tropical varieties capture combinatorial information about how coordinates of points in a classical variety approach zero or infinity. We present algorithms for computing the rays of a complex and real tropical curve defined by polynomials with constant coefficients. These algorithms rely on homotopy continuation, monodromy loops, and Cauchy integrals. Several examples are presented which are computed using an implementation that builds on the numerical algebraic geometry software Bertini.     

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].     

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” we introduce are extensions by groups of analytic types of the usual extension groups attached to OX-modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X - and we especially consider the case when X is an “arithmetic curve”, namely the spectrum SpecOK of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers.Namely, for any two hermitian vector bundles and over X:=SpecOK, we attach a logarithmic size to any element α of , and we give an upper bound on in terms of slope invariants of and . We further illustrate this notion by relating the sizes of restrictions to points in P¹(Z) of the universal extension over to the geometry of PSL₂(Z) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K^′): when the base field K is Q, we establish that the size, which cannot increase under base change, is actually invariant when the field K^′ is an abelian extension of K, or when is a direct sum of root lattices and of lattices of Voronoi's first kind.The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.