We consider F-theory compactifications on a mirror pair of elliptic Calabi–Yau threefolds. This yields two different six-dimensional theories, each of them being nonperturbatively equivalent to some compactification of heterotic strings on a K3 surface S with certain bundle data E → S. We find evidence for a transformation of S together with the bundle that takes one heterotic model to the othe...

A new technique for proving uniqueness of martingale problems is introduced. The method is illustrated in the context of elliptic diffusions in Rd.

grows exponentially. Thus, monodromy groups of elliptic fibrations over P constitute a small, but still very significant fraction of all subgroups of finite index in SL(2,Z). Our goal is to introduce some structure on the set of monodromy groups of elliptic fibrations which would help to answer some natural questions. For example, we show how to describe the set of groups corresponding to ratio...

We explain how the Elliptic Curve Primality Proving algorithm can be implemented in a distributed way. Applications are given to the certiication of large primes (more than 500 digits). As a result, we describe the successful attempt at proving the primality of the 1065-digit (2 3539 +1)=3, the rst ordinary Titanic prime.

We prove that there are at most finitely many complex λ = 0, 1 such that two points on the Legendre elliptic curve Y2 = X(X − 1)(X − λ) with coordinates X = 2, 3 both have finite order. This is a very special case of some conjectures on unlikely intersections in semiabelian schemes.

Let E/Q be an elliptic curve. Silverman and Stange define primes p and q to be an elliptic amicable pair if #E(Fp) = q and #E(Fq) = p. More generally, they define the notion of aliquot cycles for elliptic curves. Here we study the same notion in the case that the elliptic curve is defined over a number field K. We focus on proving the existence of an elliptic curve E/K with aliquot cycle (p1, ....

We discuss microscopic origin of integrability in Seiberg-Witten theory, following the results of [1], as well as present their certain extension and consider several explicit examples. In particular, we discuss in more detail the theory with the only switched on higher perturbationin the ultraviolet, where extra explicit formulas are obtained using bosonization and elliptic uniformization of t...

A celebrated theorem of Bogomolov asserts that the l-adic Lie algebra attached to the Galois action on the Tate module of an abelian variety over a number field contains all homotheties. This is not the case in characteristic p: a “counterexample” is provided by an ordinary elliptic curve defined over a finite field. In this note we discuss (and explicitly construct) more interesting examples o...

In this paper we reformulate the question of whether the ranks of the quadratic twists of an elliptic curve over Q are bounded, into the question of the whether certain infinite series converge. Our results were inspired by ideas in a paper of Gouvêa and Mazur [2]. Fix a, b, c ∈ Z such that f(x) = x + ax + bx+ c has 3 distinct complex roots, and let E be the elliptic curve y = f(x). For D ∈ Z−{...

This paper proposes new explicit formulae for the point doubling, tripling and addition on ordinary Weierstrass elliptic curves with a point of order 3 over finite fields of characteristic three. The cost of basic point operations is lower than that of all previously proposed ones. The new doubling, mixed addition and tripling formulae in projective coordinates require 3M + 2C, 8M + 1C + 1D and...