Analogue of the degree conjecture over function fields

Under a certain assumption, similar to Manin's conjecture, we prove an upper bound on the degree of modular parametrizations of elliptic curves by Drinfeld modular curves, which is the function field analogue of the conjectured bound over the rational numbers.

     

Abelian subvarieties of Drinfeld Jacobians and congruences modulo the characteristic

We relate the existence of Frobenius morphisms into the Jacobians of Drinfeld modular curves to the existence of congruences between cusp forms.     

On eigenvalues of <Emphasis Type="Italic">p</Emphasis>-adic curvature

We study the eigenvalues of the p-adic curvature transformationson buildings. In particular, we determine the maximal eigenvalues ofthese transformations.     

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.     

On hyperelliptic modular curves over function fields

We prove that there are only finitely many modular curves of -elliptic sheaves over which are hyperelliptic. In odd characteristic we give a complete classification of such curves. The author was supported in part by NSF grant DMS-0801208 and Humboldt Research Fellowship.     

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.     

Endomorphisms of exceptional {\mathcal{D}} -elliptic sheaves

We relate the endomorphism rings of certain D{\mathcal{D}} -elliptic sheaves of finite characteristic to hereditary orders in central division algebras over function fields.     

Genus formula for modular curves of {\mathcal{D}}-elliptic sheaves

We prove a genus formula for modular curves of -elliptic sheaves. We use this formula to show that the reductions of modular curves of -elliptic sheaves attain the Drinfeld-Vladut bound as the degree of the discriminant of tends to infinity. Received: 14 October 2008 The author was supported in part by NSF grant DMS-0801208 and Humboldt Research Fellowship.     

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.