We develop a theory to connect the following three areas: (a) the mean field equations on flat tori, (b) the classical Lame equations and (c) modular forms. A major theme in part I is a classification of developing maps f attached to solutions of the mean field equation according to the types of transformation laws (or monodromy) with respect to the period lattice satisfied by f.

We count the number$S$($x$) of quadruples $ {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} $ for which $$ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $$ is a prime number and satisfying the determinant condition:$x$_{1}$x$_{4}−$x$_{2}$x$_{3}= 1. By means of the sieve, one shows easily the upper bound$S$($x$) ≪$x$/log$x$. Under a hypothesis about prime numbers, which is stronger than the Bombieri–Vinogradov theorem but is weaker than the Elliott–Halberstam conjecture, we prove that this order is correct, that is$S$($x$) ≫$x$/log$x$.

We show that Q-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable Q-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.

The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$. Results obtained so far are rather fragmentary and often treated case by case. In this work, we shall unify all the known results on the subject by showing that the irrationality exponents of large classes of automatic numbers and Mahler numbers (which are transcendental) are exactly equal to $2$. Our classes contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the Fermat numbers, the regular paperfolding numbers, which have been previously considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new classes such as the Stern numbers and so on. Among other ingredients, our proofs use results on Hankel determinants obtained recently by Han.

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