On isogeny graphs of supersingular elliptic curves over finite fields

We study the isogeny graphs of supersingular elliptic curves over finite fields, with an emphasis on the vertices corresponding to elliptic curves of j-invariant 0 and 1728.     

Lattices from elliptic curves over finite fields

In their well known book [6] Tsfasman and Vladut introduced a construction of a family of function field lattices from algebraic curves over finite fields, which have asymptotically good packing density in high dimensions. In this paper we study geometric properties of lattices from this construction applied to elliptic curves. In particular, we determine the generating sets, conditions for well-roundedness and a formula for the number of minimal vectors. We also prove a bound on the covering radii of these lattices, which improves on the standard inequalities.     

On generalizations of Fermat curves over finite fields and their automorphisms

Let 𝒳 be an irreducible algebraic curve defined over a finite field 𝔽_q of characteristic p>2. Assume that the 𝔽_q-automorphism group of 𝒳 admits a subgroup isomorphic to the direct product of two cyclic groups C_m and C_n of orders m and n prime to p, such that both quotient curves 𝒳∕C_n and 𝒳∕C_m are rational. In this paper, we provide a complete classification of such curves as well as a characterization of their full automorphism groups.     

Orchards in elliptic curves over finite fields

Consider a set of n points on a plane. A line containing exactly 3 out of the n points is called a 3-rich line. The classical orchard problem asks for a configuration of the n points on the plane that maximizes the number of 3-rich lines. In this note, using the group law in elliptic curves over finite fields, we exhibit several (infinitely many) group models for orchards wherein the number of 3-rich lines agrees with the expected number given by Green-Tao (or, Burr, Grünbaum and Sloane) formula for the maximum number of lines. We also show, using elliptic curves over finite fields, that there exist infinitely many point-line configurations with the number of 3-rich lines exceeding the expected number given by Green-Tao formula by two, and this is the only other optimal possibility besides the case when the number of 3-rich lines agrees with the Green-Tao formula.     

Zeta-functions of curves of genus 3 over finite fields

In this paper, we determine zeta-functions of some curves of genus 3 over finite fields by gluing three elliptic curves based on Xing＇s research, and the examples show that there exists a maximal curve of genus 3 over F49.     

Explicit maximal and minimal curves over finite fields of odd characteristics

In this work we present explicit classes of maximal and minimal Artin–Schreier type curves over finite fields having odd characteristics. Our results include the proof of Conjecture 5.9 given in [1] as a very special subcase. We use some techniques developed in [2], which were not used in [1].     

Galois cohomology in degree 3 of function fields of curves over number fields

Let k be a field of characteristic not equal to 2. For n≥1, let denote the nth Galois Cohomology group. The classical Tate's lemma asserts that if k is a number field then given finitely many elements , there exist such that α_i=(a)∪(b_i), where for any λ∈k∗, (λ) denotes the image of k∗ in . In this paper we prove a higher dimensional analogue of the Tate's lemma.     

Some remarks on the Picard curves over a finite field

In this paper we study the Newton polygon of the L ‐polynomial L (t) associate to the Picard curves y³ = x⁴ – 1, y³ = x⁴ – x defined over a finite field ??_p . In the former case we get a complete classification. In the latter case we obtained a partial result. As a consequence of our result we obtain a criterion to find a supersingular Picard curves for the above two cases. Our main results are stated in Theorems 3.1 and 4.1. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the number of curves of genus 2 over a finite field

In this paper we compute the number of curves of genus 2 defined over a finite field k of odd characteristic up to isomorphisms defined over k; the even characteristic case is treated in an ongoing work (G. Cardona, E. Nart, J. Pujolàs, Curves of genus 2 over field of even characteristic, 2003, submitted for publication). To this end, we first give a parametrization of all points in , the moduli variety that classifies genus 2 curves up to isomorphism, defined over an arbitrary perfect field (of zero or odd characteristic) and corresponding to curves with non-trivial reduced group of automorphisms; we also give an explicit representative defined over that field for each of these points. Then, we use cohomological methods to compute the number of k-isomorphism classes for each point in .     

Special values of hypergeometric functions over finite fields

For an odd prime p, define , where z is an integer (mod p) and the summands are Legendre symbols. The function H _p (z) was explicitly evaluated for z=1 by Evans (1981) and for z=−1 by Greene and Stanton (1986). Koike (1992) determined H _p (1/4) (mod p), and Ono (1998) evaluated H _p (z) for z=1/4,−1/8, and 1/64. This paper evaluates H _p (z) for infinitely many new classes of arguments z.      

Low-lying zeros in families of elliptic curve L-functions over function fields

We investigate the low-lying zeros in families of L-functions attached to quadratic and cubic twists of elliptic curves defined over ${\mathbb{F}}_q(T)$. In particular, we present precise expressions for the expected values of traces of high powers of the Frobenius class in these families with a focus on the lower order behavior. As an application we obtain results on one-level densities and we verify that these elliptic curve families have orthogonal symmetry type. In the quadratic twist families our results refine previous work of Comeau-Lapointe. Moreover, in this case we find a lower order term in the one-level density reminiscent of the deviation term found by Rudnick in the hyperelliptic ensemble. On the other hand, our investigation is the first to treat these questions in families of cubic twists of elliptic curves and in this case it turns out to be more complicated to isolate lower order terms due to a larger degree of cancellation among lower order contributions.     

On ring class eigenspaces of Mordell-Weil groups of elliptic curves over global function fields

If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional χ-eigenspace (with χ a complex ring class character) provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is non-zero. This represents the analogue in the function field setting of a theorem for elliptic curves over Q due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof employs Kolyvagin-type arguments, and the cohomological machinery is started up by the control on the Galois structure of the torsion of E provided by classical results of Igusa in positive characteristic.