Some Artin-Schreier type function fields over finite fields with prescribed genus and number of rational places

We give existence and characterization results for some Artin-Schreier type function fields over finite fields with prescribed genus and number of rational places simultaneously.     

On Jacquet-Langlands isogeny over function fields

We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of D-elliptic sheaves. The kernel of the isogeny is a subgroup of the cuspidal divisor group constructed by examining the canonical maps from the cuspidal divisor group into the component groups.     

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.     

The Brauer-Manin obstruction for sections of the fundamental group

We introduce the notion of a Brauer-Manin obstruction for sections of the fundamental group extension and establish Grothendieck’s section conjecture for an open subset of the Reichardt-Lind curve.     

On the independence of Heegner points associated to distinct quadratic imaginary fields

Let E/Q be an elliptic curve with no CM and a fixed modular parametrization and let be Heegner points attached to the rings of integers of distinct quadratic imaginary fields k₁,…,kr. We prove that if the odd parts of the class numbers of k₁,…,kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in .     

Rational places in extensions and sequences of function fields of Kummer type

In this paper we prove results on the number of rational places in extensions of Kummer type over finite fields and give sufficient conditions for non-trivial lower bounds on the number of rational places at each step of sequences of function fields over a finite field, that we call (a, b)-sequences. In the case of a prime field, we apply these results to the study of rational places in certain sequences of function fields of Kummer type.     

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.     

On primitive points of elliptic curves with complex multiplication

Let E be an elliptic curve defined over Q and P∈E(Q) a rational point of infinite order. Suppose that E has complex multiplication by an order in the imaginary quadratic field k. Denote by M_E,P the set of rational primes ? such that ? splits in k, E has good reduction at ?, and P is a primitive point modulo ?. Under the generalized Riemann hypothesis, we can determine the positivity of the density of the set M_E,P explicitly.     

On Certain Subcovers of the Hermitian Curve

We present a simple construction that gives explicit equations for certain subcovers of the Hermitian curve. We show that maximal curves with a certain type of defining equations are covered by the Hermitian curve.     

On the number of stable quiver representations over finite fields

We prove a new formula for the generating function of polynomials counting absolutely stable representations of quivers over finite fields. The case of irreducible representations is studied in more detail.     

On the number of rational points of a plane algebraic curve

The number of F_q -rational points of a plane non-singular algebraic curve defined over a finite field F_q is computed, provided that the generic point of is not an inflexion and that is Frobenius non-classical with respect to conics. Received: 18 March 2003     

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.     

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.     

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     

A quotient curve of the Fermat curve of degree twenty-three attaining the Serre bound

The curve in the title is a non-maximal curve of genus 11, which has a defining equation analogous to the Klein quartic's. We study its properties and show how to apply it to coding theory.     

Constructing one-parameter families of elliptic curves with moderate rank

We give several new constructions for moderate rank elliptic curves over Q(T). In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over Q using polynomials of degree two in T. While our method generates linearly independent points, we are able to show the rank is exactly 6 without having to verify the points are independent. The method generalizes; however, the higher rank surfaces are not rational, and we need to check that the constructed points are linearly independent.