Mathematics

Watkins conjectured that for an elliptic curve E over Q of Mordell-Weil rank r , the modular degree of E is divisible by 2 r . If E has non-trivial rational 2 -torsion, we prove the conjecture for all the quadratic twists of E by squarefree integers with sufficiently many prime factors.

We propose a formulation of the relative Bogomolov conjecture and show that it gives an affirmative answer to a question of Mazur's concerning the uniformity of the Mordell-Lang conjecture for curves. In particular we show that the relative Bogomolov conjecture implies the uniform Manin-Mumford conjecture for curves. The proof is built up on our previous work "Uniformity in Mordell-Lang for curves".

This note is concerned with the set of integral solutions of the equation x 2 +3 y 2 =12n+4 , where n is a positive integer. We will describe a parametrization of this set using the 3-core partitions of n. In particular we construct a crank using the action of a suitable subgroup of the isometric group of the plane that we connect with the unit group of the ring of Eisenstein integers. We also show that the process goes in the reverse direction: from the solutions of the equation and the crank, we can describe the 3-core partitions of n. As a consequence we describe an explicit bijection between 3 -core partitions and ideals of the ring of Eisenstein integers, explaining a result of G. Han and K. Ono obtained using modular forms.

In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric structures. In particular, we give a proof of the full Ax-Schanuel Theorem with derivatives for uniformizers of any Fuchsian group of the first kind and any genus. Our techniques combine tools from differential geometry, differential algebra and the model theory of differentially closed fields. The proof is very similar in spirit to Ax's proof of the theorem in the case of the exponential function.

The three distance theorem (also known as the three gap theorem or Steinhaus problem) states that, for any given real number α and integer N , there are at most three values for the distances between consecutive elements of the Kronecker sequence α,2α,…,Nα mod 1. In this paper we consider a natural generalisation of the three distance theorem to the higher dimensional Kronecker sequence α → ,2 α → ,…,N α → modulo an integer lattice. We prove that in two dimensions there are at most five values that can arise as a distance between nearest neighbors, for all choices of α → and N . Furthermore, for almost every α → , five distinct distances indeed appear for infinitely many N and hence five is the best possible general upper bound. In higher dimensions we have similar explicit, but less precise, upper bounds. For instance in three dimensions our bound is 13, though we conjecture the truth to be seven. We furthermore study the number of possible distances from a point to its nearest neighbor in a restricted cone of directions. This may be viewed as a generalisation of the gap length in one dimension. For large cone angles we use geometric arguments to produce explicit bounds directly analogous to the three distance theorem. For small cone angles we use ergodic theory of homogeneous flows in the space of unimodular lattices to show that the number of distinct lengths is (a) unbounded for almost all α → and (b) bounded for α → that satisfy certain Diophantine conditions.

The well-known fundamental identity in number theory expresses the degree of an extension of global fields in terms of local information. In this article we modify the fundamental identity so that it holds for arbitrary Dedekind domains. We also give necessary and sufficient conditions for the ground Dedekind domain such that the fundamental identity holds true. In particular, a Dedekind domain is excellent if and only if it is Japanese.

In this paper we study the derivatives of Frobenius and the derivatives of Hodge-Tate weights for families of Galois representations with triangulations. We give a generalization of the Fontaine-Mazur L-invariant and use it to build a formula which is a generalization of the Colmez-Greenberg-Stevens formula.

This is a sequel of the paper "A generalization of the Ross symbols in higher K-groups and hypergeometric functions I" where we introduced higher Ross symbols in higher K -groups of the hypergeometric schemes, and discussed the Beilinson regulators. In this paper we give its p-adic counterpart and an application to the p -adic Beilinson conjecture for K3 surfaces of Picard number 20.

The Chabauty-Coleman method is a p -adic method for finding all rational points on curves of genus g whose Jacobians have Mordell-Weil rank r<g . Recently, Edixhoven and Lido developed a geometric quadratic Chabauty method that was adapted by Spelier to cover the case of geometric linear Chabauty. We compare the geometric linear Chabauty method and the Chabauty-Coleman method and show that geometric linear Chabauty can outperform Chabauty-Coleman in certain cases. However, as Chabauty-Coleman remains more practical for general computations, we discuss how to strengthen Chabauty-Coleman to make it theoretically equivalent to geometric linear Chabauty.

Assuming the weak Bombieri-Lang conjecture, we prove that a generalization of Hilbert's irreducibility theorem holds for families of geometrically mordellic varieties (for instance, families of hyperbolic curves). As an application we prove that, assuming Bombieri-Lang, there are no polynomial bijections Q?Q?�Q .