We study some Diophantine properties of automatic real numbers and we present a method to derive irrationality measures for such numbers. As a consequence, we prove that the $b$-adic expansion of a Liouville number cannot be generated by a finite automaton, a conjecture due to Shallit.

We consider twisted multivariable zeta series associated to polynomials of several variables. We introduce a new class of polynomials, namely HDF, that contains strictly non-degenerate and hypoelliptic polynomials. For polynomials belonging to the HDF class, we show that we can extend holomorphically our series to $\mathbb{C}^T$. Then, thanks to a new principle called ‘the Exchange Lemma’, we give very simple formulae for the values of our series at $T$-tuples of negative integers. Finally, we make the $p$-adic interpolation of those values. Thus, we have generalized the results of Cassou-Noguès (that she used to construct the $p$-adic $L$-functions of totally real fields) in two ways: we consider multivariable series and our series are associated to more general polynomials. In addition, our proof is completely different.

We investigate the large weight ($k\to\infty$) limiting statistics for the low lying zeros of a GL(4) and a GL(6) family of $L$-functions, $\{L(s,\phi \times f): f \in H_k\}$ and $\{L(s,\phi \times {\rm sym}^2 f): f \in H_k\}$; here $\phi$ is a fixed even Hecke–Maass cusp form and $H_k$ is a Hecke eigenbasis for the space $H_k$ of holomorphic cusp forms of weight $k$ for the full modular group. Katz and Sarnak conjecture that the behavior of zeros near the central point should be well modeled by the behavior of eigenvalues near 1 of a classical compact group. By studying the 1- and 2-level densities, we find evidence of underlying symplectic and SO(even) symmetry, respectively. This should be contrasted with previous results of Iwaniec–Luo–Sarnak for the families $\{L(s,f): f\in H_k\}$ and $\{L(s,{\rm sym}^2f): f\in H_k\}$, where they find evidence of orthogonal and symplectic symmetry, respectively. The present examples suggest a relation between the symmetry type of a family and that of its twistings, which will be further studied in a subsequent paper. Both the GL(4) and the GL(6) families above have all even functional equations, and neither is naturally split from an orthogonal family. A folklore conjecture states that such families must be symplectic, which is true for the first family but false for the second. Thus, the theory of low lying zeros is more than just a theory of signs of functional equations. An analysis of these families suggest that it is the second moment of the Satake parameters that determines the symmetry group.

We give a purity theorem in the manner of Deligne's ‘Weil II’ theorem for rigid cohomology with coefficients in an overconvergent $F$-isocrystal; the proof mostly follows Laumon's Fourier-theoretic approach, transposed into the setting of arithmetic $\mathcal{D}$-modules. This yields in particular a complete, purely $p$-adic proof of the Weil conjectures when combined with recent results on $p$-adic differential equations by André, Christol, Crew, Kedlaya, Matsuda, Mebkhout and Tsuzuki.

In this paper we prove a formula relating the Faltings height of an abelian variety $A$ over $\bar{\mathbb{Q}}$ and the Néron–Tate height of a theta divisor on $A$.

We prove that an automorphism of the group of birational transformations of the complex projective plane is the composition of an interior automorphism and an automorphism of the field $\mathbb{C}$. The proof is based on a study of maximal abelian subgroups of the Cremona group.

Our main result is the description of generators of the total coordinate ring of the blow-up of $\mathbb{P}^n$ in any number of points that lie on a rational normal curve. As a corollary we show that the algebra of invariants of the action of a two-dimensional vector group introduced by Nagata is finitely generated by certain explicit determinants. We also prove the finite generation of the algebras of invariants of actions of vector groups related to T-shaped Dynkin diagrams introduced by Mukai.

Using multigraded Castelnuovo–Mumford regularity, we study the equations defining a projective embedding of a variety $X$. Given globally generated line bundles $B_{1}, \dotsc, B_{\ell}$ on $X$ and $m_{1}, \dotsc, m_{\ell} \in \mathbb{N}$, consider the line bundle $L := B_{1}^{m_{1}} \otimes \dotsb \otimes B_{\ell}^{m_{\ell}}$. We give conditions on the $m_{i}$ which guarantee that the ideal of $X$ in $\mathbb{P}(H^{0}(X,L)^{*})$ is generated by quadrics and that the first $p$ syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.

King's conjecture states that on every smooth complete toric variety $X$ there exists a strongly exceptional collection which generates the bounded derived category of $X$ and which consists of line bundles. We give a counterexample to this conjecture. This example is just the Hirzebruch surface $\mathbb{F}_2$ iteratively blown up three times, and we show by explicit computation of cohomology vanishing that there exist no strongly exceptional sequences of length 7 which consist of line bundles.

We study the Chow group of zero-dimensional cycles for projective homogeneous varieties of semisimple algebraic groups. We show that in many cases this group has no torsion.

Our main result says that the generic rank of the Baum–Bott map for foliations of degree $d,\ d\ge 2$, of the projective plane is $d^2+d$. This answers a question of Gómez-Mont and Luengo and shows that are no other universal relations between the Baum–Bott indexes of a foliation of $\mathbb P^2$ besides the Baum–Bott formula. We also define the Camacho–Sad field for foliations on surfaces and prove its invariance under the pull-back by meromorphic maps. As an application we prove that a generic foliation of degree $d\ge 2$ is not the pull-back of a foliation of smaller degree. In Appendix A we show that the monodromy of the singular set of the universal foliation with very ample cotangent bundle is the full symmetric group.

We show that holomorphic singular codimension one foliations of the complex projective space with a Kupka singular set of radial type and verifying some global hypotheses have rational first integral. The generic elements of such pencils are Calabi–Yau.

We construct bases for the stable branching algebras for the symmetric pairs $(\mathrm{GL}_{2n},\mathrm{Sp}_{2n}),\ (\mathrm{Sp}_{2(n+m)}, \mathrm{Sp}_{2n}\times\mathrm{Sp}_{2m})$ and $(\mathrm{O}_{2n},\mathrm{GL}_{n})$. Each basis element is expressed as a sum of products of pfaffians.