The Gross conjecture over rational function fields

We study the Gross conjecture for the cyclotomic function field extension k(∧f)/k where k = Fq(t) is the rational function field and f is a monic polynomial in Fq[t]. We prove the conjecture in the Fermat curve case(i.e., when f = t(t - 1)) by a direct calculation. We also prove the case when f is irreducible, which is analogous to the Weil reciprocity law. In the general case, we manage to show the weak version of the Gross conjecture here.     

Stark's conjecture over complex cubic number fields

Systematic computation of Stark units over nontotally real base fields is carried out for the first time. Since the information provided by Stark's conjecture is significantly less in this situation than the information provided over totally real base fields, new techniques are required. Precomputing Stark units in relative quadratic extensions (where the conjecture is already known to hold) and coupling this information with the Fincke-Pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding Stark units in larger extensions (where the conjecture is still unproven). Stark's conjecture is verified in each case for these Stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields, including Hilbert class fields, are obtained from the minimal polynomials of these new Stark units.

     

Bogomolov conjecture over function fields for stable curves with only irreducible fibers

Let K be a function field and C a non-isotrivial curve of genus g2 overK. In this paper, we will show that if C has a global stable modelwith only geometrically irreducible fibers, then Bogomolov conjecture over function fields holds.     

CM points on products of Drinfeld modular curves

Let be a product of Drinfeld modular curves over a general base ring of odd characteristic. We classify those subvarieties of which contain a Zariski-dense subset of CM points. This is an analogue of the André-Oort conjecture. As an application, we construct non-trivial families of higher Heegner points on modular elliptic curves over global function fields.

     

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.     

Artin's primitive root conjecture for function fields revisited

Artin's primitive root conjecture for function fields was proved by Bilharz in his thesis in 1937, conditionally on the proof of the Riemann hypothesis for function fields over finite fields, which was proved later by Weil in 1948. In this paper, we provide a simple proof of Artin's primitive root conjecture for function fields which does not use the Riemann hypothesis for function fields but rather modifies the classical argument of Hadamard and de la Vallée Poussin in their 1896 proof of the prime number theorem.     

Boundedness of Mordell-Weil ranks of certain elliptic curves and Lang's conjecture

In this paper, we show that a special case of Lang's conjecture on rational points on surfaces of general type implies that there exist only finitely many elliptic curves, when the x-coordinates of n rational points are specified with n?8.     

A note on a question of J. Nekovár and the Birch and Swinnerton-Dyer conjecture

If is a square-free integer, then let denote the elliptic curve over given by the equation

Let denote the Hasse-Weil -function of , and let denote the `algebraic part' of the central critical value . Using a theorem of Sturm, we verify a congruence conjectured by J. Neková\v{r}. By his work, if denotes the 3-Selmer group of and is a square-free integer with , then we find that

     

The Brumer-Stark conjecture in some families of extensions of specified degree

As a starting point, an important link is established between Brumer's conjecture and the Brumer-Stark conjecture which allows one to translate recent progress on the former into new results on the latter. For example, if is an abelian extension of relative degree , an odd prime, we prove the -part of the Brumer-Stark conjecture for all odd primes with belonging to a wide class of base fields. In the same setting, we study the -part and -part of Brumer-Stark with no special restriction on and are left with only two well-defined specific classes of extensions that elude proof. Extensive computations were carried out within these two classes and a complete numerical proof of the Brumer-Stark conjecture was obtained in all cases.

     

Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields

With the goal of producing elliptic curves and higher-dimensional abelian varieties of large rank over function fields, we provide a geometric construction of towers of surfaces dominated by products of curves; in the case where the surface is defined over a finite field our construction yields families of smooth, projective curves whose Jacobians satisfy the conjecture of Birch and Swinnerton-Dyer. As an immediate application of our work we employ known results on analytic ranks of abelian varieties defined in towers of function field extensions, producing a one-parameter family of elliptic curves over Fq(t1/d) whose members obtain arbitrarily large rank as d→∞.     

Class group frequencies of real quadratic function fields: The degree 4 case

The distribution of ideal class groups of is examined for degree-four monic polynomials when is a finite field of characteristic greater than 3 with or and is irreducible or has an irreducible cubic factor. Particular attention is paid to the distribution of the -Sylow part of the class group, and these results agree with those predicted using the Cohen-Lenstra heuristics to within about 1 part in 10000. An alternative set of conjectures specific to the cases under investigation is in even sharper agreement.

     

An algebro-geometric proof of Witten's conjecture

We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the -sphere.

     

The Galois closure of Drinfeld modular towers

In this article we study Drinfeld modular curves X₀(pn) associated to congruence subgroups Γ₀(pn) of GL(2,Fq[T]) where p is a prime of Fq[T]. For n>r>0 we compute the extension degrees and investigate the structure of the Galois closures of the covers X₀(pn)→X₀(pr) and some of their variations. The results have some immediate implications for the Galois closures of two well-known optimal wild towers of function fields over finite fields introduced by Garcia and Stichtenoth, for which the modular interpretation was given by Elkies.