We prove the unbounded denominators conjecture in the theory of noncongruence modular forms for finite index subgroups of SL 2 (Z). Our result includes also Mason's generalization of the original conjecture to the setting of vector-valued modular forms, thereby supplying a new path to the congruence property in rational conformal field theory. The proof involves a new arithmetic holonomicity bound of a potential-theoretic flavor, together with Nevanlinna's second main theorem, the congruence subgroup property of SL 2 (Z[1/p]), and a close description of the Fuchsian uniformization D(0, 1)/Γ N of the Riemann surface C µ N .forms for SL 2 (Z), in particular resolving -in a sharper form, in fact -Mason's unbounded denominators conjecture [Mas12, KM08] on generalized modular forms.1.1. A sketch of the main ideas. Our proof of Theorem 1.0.1 follows a broad Diophantine analysis path known in the literature (see [Bos04,Bos13] or [Bos20, Chapter 10]) as the arithmetic algebraization method.1.1.1. The Diophantine principle. The most basic antecedent of these ideas is the following easy lemma:x which defines a holomorphic function on D(0, R) for some R > 1 is a polynomial.Lemma 1.1.1 follows upon combining the following two obsevations, fixing some 1 > η > R −1 :(1) The coefficients a n are either 0 or else ≥ 1 in magnitude.(2) The Cauchy integral formula gives a uniform upper bound |a n | = o(η n ). We shall refer to the first inequality as a Liouville lower bound, following its use by Liouville in his proof of the lower bound |α − p/q| ≫ 1/q n for algebraic numbers α = p/q of degree n ≥ 1. We shall refer to the second inequality as a Cauchy upper bound, following the example above where it comes from an application of the Cauchy integral formula. The first non-trivial generalization of Lemma 1.1.1 was Émile Borel's theorem [Bor94]. Dwork famously used a p-adic generalization of Borel's theorem in his p-adic analytic proof of the rationality of the zeta function of an algebraic variety over a finite field (see Dwork's account in the book [DGS94, Chapter 2]). The simplest non-trivial statement of Borel's theorem is that an integral formal power series f (x) ∈ Z x must already be a rational function as soon as it has a meromorphic representation as a quotient of two convergent complex-coefficients power series on some disc D(0, R) of a radius R > 1. The subject of arithmetic algebraization blossomed at the hands of many authors, including most prominently Carlson,

Consider a smooth, geometrically irreducible, projective curve of genus g ≥ 2 defined over a number field of degree d ≥ 1. It has at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of g, d, and the Mordell-Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounds, in g and d, for the number of geometric torsion points of the Jacobian which lie in the image of an Abel-Jacobi map. Both estimates generalize our previous work for 1-parameter families. Our proof uses Vojta's approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second-and third-named authors. Contents 1. Introduction 1 2. Betti map and Betti form 7 3. Setup and notation for the height inequality 13 4. Intersection theory and height inequality on the total space 16 5. Proof of the height inequality Theorem 1.6 22 6. Preparation for counting points 23 7. Néron-Tate distance between points on curves 29 8. Proof of Theorems 1.1, 1.2, and 1.4 31 Appendix A. The Silverman-Tate Theorem revisited 36 Appendix B. Full version of Theorem 1.6 40 References 47

Consider a one-parameter family of smooth, irreducible, projective curves of genus $g\ge 2$ defined over a number field. Each fiber contains at most finitely many rational points by the Mordell conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell–Weil rank of the fiber’s Jacobian. Our proof uses Vojta’s approach to the Mordell Conjecture furnished with a height inequality due to the 2nd- and 3rd-named authors. In addition we obtain uniform bounds for the number of torsion points in the Jacobian that lie in each fiber of the family.

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