Nachdiplomvorlesungen


Lecture notes


Lecture 1 (September 28)

Historical motivation: the Hermite-Lindemann-Weierstrass theorem and Bourget's hypothesis. Definition and examples of arithmetic Gevrey series. Summary of results and questions we will address: arithmetic nature of special values (proof of the Siegel-Shidlovsky-André-Beukers theorem), structure of the minimal differential operator annihilating an E-function and a G-function, integral representations coming from geometry.

Lecture 2 (October 5)

The modified Laplace transform of a formal power series of exponential type. Equivalence of the Hermite-Lindemann-Weierstrass theorem with a statement about exponential polynomials vanishing at 1. A proof of the former using the structure theorems for minimal differential equations of E-functions.

Lecture 3 (October 12)

Galochkin's condition and definition of a $G$-operator. Statement of Chudnovsky's theorem. Road map to the proof of the theorem that G-operator have regular singularities, rational exponents, and a basis of solutions given by G-functions with monodromy around every point.


References

[1] Y. André, Séries Gevrey de type arithmétique. I. Théorèmes de pureté et de dualité, Ann. of Math.151 (2000), no. 2, 705-740.

[2] F. Beukers, J.P. Bézivin, P. Robba, An alternative proof of the Lindemann-Weierstrass theorem, Amer. Math. Monthly 97 (1990), no. 3, 193-197.

[3] D. V. Chudnovsky, G. V. Chudnovsky, Applications of Padé approximations to Diophantine inequalities in values of G-functions, Number theory (New York, 1983–84), Lecture Notes in Math. 1135 (1985) 9-51.