Die Theorie der Zahlstrahlen.
Differential and functional identities for the elliptic trilogarithm.
Differential overconvergence
We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation". This property can be viewed as a special kind of overconvergence property. One can also go in the opposite direction by using differential functions that arise in a ramified situation to construct "new" (unramified) differential functions.
Dimensions of some affine Deligne–Lusztig varieties
Diophantine approximation and deformation
Diophantine approximation on algebraic varieties
We present an overview of recent advances in diophantine approximation. Beginning with Roth's theorem, we discuss the Mordell conjecture and then pass on to recent higher dimensional results due to Faltings-Wustholz and to Faltings respectively.
Diophantine equations of matching games II
Diophantine -tuples and elliptic curves
A Diophantine -tuple is a set of positive integers such that the product of any two of them is one less than a perfect square. In this paper we study some properties of elliptic curves of the form , where , is a Diophantine triple. In particular, we consider the elliptic curve defined by the equation where and , denotes the -th Fibonacci number. We prove that if the rank of is equal to one, or , then all integer points on are given by
Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
Let be a one-variable function field over a field of constants of characteristic 0. Let be a holomorphy subring of , not equal to . We prove the following undecidability results for : if is recursive, then Hilbert’s Tenth Problem is undecidable in . In general, there exist such that there is no algorithm to tell whether a polynomial equation with coefficients in has solutions in .
Diophantische Analysis und Modulfunktionen.
Dirichlet motives via modular curves
Discrete universality of the -functions of elliptic curves.
Discrete Value -- Distribution of L-functions of Elliptic Curves
Distribution des points de petite hauteur dans les groupes multiplicatifs
We prove a new lower bound for the height of points on a subvariety of a multiplicative torus, which lie outside the union of torsion subvarieties of . Although lower bounds for the heights of these points where already known (decreasing multi-exponential function of the degree for Scmhidt and Bombieri–Zannier, [Sch], [Bo-Za], and inverse monomial in the degree by the second author of this note and P. Philippon, [Da-Phi]), our method provesup to an the sharpest conjectures that can be formulated....
Distribution of Mordell-Weil ranks of families of elliptic curves
We discuss the distribution of Mordell-Weil ranks of the family of elliptic curves y² = (x + αf²)(x + βbg²)(x + γh²) where f,g,h are coprime polynomials that parametrize the projective smooth conic a² + b² = c² and α,β,γ are elements from ℚ̅. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.
Distribution of the traces of Frobenius on elliptic curves over function fields
Doubles mélanges des polylogarithmes multiples aux racines de l’unité
Drinfeld modules of rank 1 and algebraic curves with many rational points. II
Drinfeld shtukas and applications. (Chtoucas de Drinfeld et applications.)