Torsion bounds for elliptic curves and Drinfeld modules

We derive upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations, due to Serre, Pink and Rütsche. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.     

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).     

Some formulas for the coefficients of Drinfeld modular forms

We obtain some formulas for t-expansion coefficients of meromorphic Drinfeld modular forms for GL₂(F_q[T]). Let j(z) be the Drinfeld modular invariant. As an application we show that the values of j(z) at points in the divisor of Drinfeld modular forms for GL₂(F_q[T]) are algebraic over F_q(T).     

The distribution of class groups of function fields

For any sufficiently general family of curves over a finite field F_q and any elementary abelian ?-group H with ? relatively prime to q, we give an explicit formula for the proportion of curves C for which Jac(C)[?](F_q)≅H. In doing so, we prove a conjecture of Friedman and Washington.     

Pure Anderson motives over finite fields

In the arithmetic of function fields Drinfeld modules play the role that elliptic curves take on in the arithmetic of number fields. As higher dimensional generalizations of Drinfeld modules, and as the appropriate analogues of abelian varieties, G. Anderson introduced pure t-motives. In this article we study the arithmetic of the latter. We investigate which pure t-motives are semisimple, that is, isogenous to direct sums of simple ones. We give examples for pure t-motives which are not semisimple. Over finite fields the semisimplicity is equivalent to the semisimplicity of the endomorphism algebra, but also this fails over infinite fields. Still over finite fields we study the Zeta function and the endomorphism rings of pure t-motives and criteria for the existence of isogenies. We obtain answers which are similar to Tate's famous results for abelian varieties.     

Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold

We present simple trace formulas for Hecke operators Tk(p) for all p>3 on Sk(Γ₀(3)) and Sk(Γ₀(9)), the spaces of cusp forms of weight k and levels 3 and 9. These formulas can be expressed in terms of special values of Gaussian hypergeometric series and lend themselves to recursive expressions in terms of traces of Hecke operators on spaces of lower weight. Along the way, we show how to express the traces of Frobenius of a family of elliptic curves equipped with a 3-torsion point as special values of a Gaussian hypergeometric series over Fq, when . As an application, we use these formulas to provide a simple expression for the Fourier coefficients of η⁸(3z), the unique normalized cusp form of weight 4 and level 9, and then show that the number of points on a certain threefold is expressible in terms of these coefficients.     

Elliptic units in K2

We present certain norm-compatible systems in K₂ of function fields of some CM elliptic curves. We demonstrate that these systems have some properties similar to elliptic units.     

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].     

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.     

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.     

Root numbers and ranks in positive characteristic

For a global field K and an elliptic curve E_η over K(T), Silverman's specialization theorem implies rank(E_η(K(T)))?rank(E_t(K)) for all but finitely many t∈P¹(K). If this inequality is strict for all but finitely many t, the elliptic curve E_η is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial.Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K=κ(u) over any finite field κ with characteristic ≠2, we construct an explicit 2-parameter family E_c,d of non-isotrivial elliptic curves over K(T) (depending on arbitrary c,d∈κ^×) such that, under the parity conjecture, each E_c,d has elevated rank.     

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.     

Proof of an exceptional zero conjecture for elliptic curves over function fields

Based on the analogy between number fields and function fields of one variable over finite fields, we formulate and prove an analogue of the exceptional zero conjecture of Mazur, Tate and Teitelbaum for elliptic curves defined over function fields. The proof uses modular parametrization by Drinfeld modular curves and the theory of non-archimedean integration. As an application we prove a refinement of the Birch-Swinnerton-Dyer conjecture if the analytic rank of the elliptic curve is zero.     

A Weil-Barsotti formula for Drinfeld modules

We study the group of extensions in the category of Drinfeld modules and Anderson's t-modules, and we show in certain cases that this group can itself be given the structure of a t-module. Our main result is a Drinfeld module analogue of the Weil-Barsotti formula for abelian varieties. Extensions of general t-modules are also considered, in particular extensions of tensor powers of the Carlitz module. We motivate these results from various directions and compare to the situation of elliptic curves.     

On the low-lying zeros of Hasse-Weil L-functions for elliptic curves

In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. This upper bound for the average rank enables us to deduce that, under the same assumption, a positive proportion of elliptic curves have algebraic ranks equaling their analytic ranks and finite Tate-Shafarevich group. Statements of this flavor were known previously [M.P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (1) (2005) 205-250] under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper.     

Weil Pairing for Drinfeld Modules

As is well-known, there exists a Weil pairing for elliptic curves which is a perfect bilinear form from the m-torsion of the elliptic curve E to the m-th roots of unity. In this paper we will show how Andersons paper [1] gives rise to an analogue of this pairing for Drinfeld modules.The author was supported by NWO Grant 613.007.040. The author would like to thank G. Böckle and S. J. Edixhoven for their comments.     

Positivity of certain functions associated with analysis on elliptic surfaces

In this paper, we study functions of one variable that are called boundary terms of two-dimensional zeta integrals established in recent works of Ivan Fesenko?s two-dimensional adelic analysis attached to arithmetic elliptic surfaces. It is known that the positivity of the fourth log derivatives of boundary terms around the origin is a sufficient condition for the Riemann hypothesis of Hasse-Weil L-functions of elliptic curves. We show that such positivity is also a necessary condition under some reasonable technical assumptions.     

On the uniform boundedness conjecture for Drinfeld modules

 In analogy with the famous theorems of Mazur and Merel on the torsion subgroups of elliptic curves, one can formulate similar conjectures for the torsion points of Drinfeld modules. We prove some partial results for rank 2 Drinfeld 𝔽 _q [T]-modules, for example the uniform boundedness of the 𝔭-primary torsion. Received: 22 May 2001; in final form 4 September 2002 / Published online: 1 April 2003     

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.     

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.