Transcendental points on group varieties

Rational points on varieties

Nearest Points on Toric Varieties

We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the A-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point.

Abelian Points on Algebraic Varieties

We attempt to determine which classes of algebraic varieties over Q must have points in some abelian extension of Q. We give: (i) for every odd d > 1, an explicit family of degree d, dimension d − 2 diagonal hypersurfaces without Qab-points, (ii) for every number field K, a genus one curve C/Q with no K ab-points, and (iii) for every g ≥ 4 an algebraic curve C/Q of genus g with no Qab-points. I...

Points of Bounded Height on Algebraic Varieties

Introduction 1 1. Heights on the projective space 3 1.1. Basic height function 3 1.2. Height function on the projective space 5 1.3. Behavior under maps 7 2. Heights on varieties 9 2.1. Divisors 9 2.2. Heights 13 3. Conjectures 19 3.1. Zeta functions and counting 19 3.2. Height zeta function 20 3.3. Results and methods 22 3.4. Examples 24 4. Compactifications of Semi-Simple Groups 26 4.1. A Con...

Counting Rational Points on Algebraic Varieties

In these lectures we will be interested in solutions to Diophantine equations F (x1, . . . , xn) = 0, where F is an absolutely irreducible polynomial with integer coefficients, and the solutions are to satisfy (x1, . . . , xn) ∈ Z. Such an equation represents a hypersurface in A, and we may prefer to talk of integer points on this hypersurface, rather than solutions to the corresponding Diophan...