We propose applying transcendence techniques to curves and abelian varieties in a family and to classical period maps to deduce transcendence properties for functions which are solutions of a Picard-Fuchs equation. This leads naturally to considering the distribution of points with special properties on subvarieties of Shimura varieties. These special properties can be interpreted as meaning that the points lie on the intersection of the subvariety with certain Shimura subvarieties. This leads to studying open conjectures in the theory of Shimura varieties. We also propose an approach independent of these conjectures which would involve studying the locus of ``irreducible'' points on subvarieties of Shimura varieties. We propose developing a transcendence theory for more general varieties in a family, and for more general period maps as well as for periods of higher order forms.
A fundamental problem in transcendence theory is the determination of the exceptional set of a function, that is the set of algebraic numbers at which the function assumes algebraic values. A classical result is that the exceptional set of the exponential function consists only of the origin. This implies that e and pi are transcendental. In previous work, we determine the exceptional set of the classical hypergemetric function and its generalization to several complex variables by relating these functions to families of algebraic curves. We intend to extend this work to a wider class of functions related to more general families of algebraic curves and varieties.
The research in this NSF project is in the field of transcendental number theory. An algebraic number is a number which is the root of a non-zero polynomial with coefficients in the integers. The integers are the positive and negative counting numbers 0, 1, -1, 2, -2,.... For example, the algebraic number "the square root of 2" is a solution of the quadratic x2-2=0. Algebraic numbers may also be imaginary (complex) numbers. For example, the square root of (-1), or i, is a solution of x2+1=0. The operations of addition, subtraction, multiplication and division between algebraic numbers yield other algebraic numbers. These are the so-called ``field'' properties of algebraic numbers, and we talk of the ``field of algebraic numbers''. Numbers which are not algebraic numbers are called transcendental numbers. They do not form a field, which is one of the reasons that they are difficult to study. The search for a rigorous proof that a given number is transcendental goes back to the ancient Greeks. They asked whether it is possible to construct with straight edge and compass a square with area equal the area of the unit circle. This is equivalent to asking whether pi is transcendental. It was not until 1844 that Liouville constructed the first demonstrably transcendental number, but his construction was very artifical. The transcendence of pi was finally proved by Lindemann in 1882, using a method based on ideas of Hermite, who proved the transcendence of e in 1873. The ideas of Hermite pervade the study of transcendental numbers to this day. The numbers e and pi are related to the exponential function exp(z). The number e is the value of exp(z) at z=1, a so-called ``special value'' since 1 is clearly an algebraic number, and 2(pi)i is a ``period'' of exp(z) since exp(z+2(pi)i)=exp(z). This period is pi times the algebraic number 2i, so is transcendental if and only if pi is transcendental. Most of the results of this NSF project concern the transcendence of special values (values at algebraic numbers) of classical functions and the transcendence of numbers related to periods of functions generalizing the exponential function. Iterating the relation exp(z+2(pi)i)=exp(z), we see that exp(z+2n(pi)i)=exp(z) for all integers n, so that the entire set {2n(pi)i: n is an integer} consists of periods of exp(z) and is called a ``period lattice'' in complex dimension 1, roughly speaking because it consists of isolated points which are all integer multiples of one complex number. Certain period lattices in higher dimensional spaces are related to ``varieties over the algebraic numbers'', namely the loci of solutions of polynomial equations in several variables with coefficients in the field of algebraic numbers. The existence of non-trivial mappings of the higher dimensional spaces that preserve the period lattices translates into the existence of self-mappings of the associated varieties. These mappings are expressed by linear relations between the periods involving algebraic numbers. Lattices which are preserved under many non-trivial mappings are extremely important in number theory. Although not obviously related to transcendental number theory, a basic question, that is answered in many cases using techniques from that theory, is to ask whether all linear relations between periods involving algebraic numbers are those coming from such mappings. In all known cases, the answer is ``yes''. The PI has contributed to this particular question and the current project, in particular, examines this question in a much more general context.
