Soit $A$ une variété abélienne définie sur un corps de nombres $K$. Le nombre de points de torsion définis sur une extension finie $L$ est borné polynomialement en terme du degré $\left[L:K\right]$ de $L$ sur $K$. Sous les trois hypothèses suivantes nous calculons l’exposant optimal dans cette borne, en terme de la dimension des sous-variétés abéliennes de $A$ et de leurs anneaux d’endomorphismes. Les trois hypothèses faites sur $A$ sont les suivantes : (1) $A$ est géométriquement isogène à un produit de variétés abéliennes simples de type I ou II dans la classification d’Albert ; (2) $A$ est de “type Lefschetz” c’est-à-dire que le groupe de Mumford–Tate est le groupe des similitudes symplectiques commutant aux endomorphismes ; (3) $A$ vérifie la conjecture de Mumford–Tate. Le résultat est notamment inconditionnel (i.e., ces trois hypothèses sont vérifiées) pour un produit de variétés abéliennes simples de type I ou II et de dimension relative impaire. Par ailleurs nous prouvons, en étendant des résultats de Serre, Pink et Hall, la conjecture de Mumford–Tate pour quelques nouveaux cas de variétés abéliennes de type Lefschetz.

Let $A$ be an abelian variety defined over a number field $K$. The number of torsion points that are rational over a finite extension $L$ is bounded polynomially in terms of the degree $\left[L:K\right]$ of $L$ over $K$. Under the following three conditions we compute the optimal exponent for this bound, in terms of the dimension of abelian subvarieties and their endomorphism rings. The three hypotheses are the following: (1) $A$ is geometrically isogenous to a product of simple abelian varieties of type I or II, according to the Albert classification; (2) $A$ is of “Lefschetz type,” that is, the Mumford–Tate group is the group of symplectic similitudes which commute with the endomorphism ring. (3) $A$ satisfies the Mumford–Tate conjecture. This result is unconditional (i.e., the three hypotheses are satisfied) for a product of simple abelian varieties of type I or II with odd relative dimension. Further, building on work of Serre, Pink and Hall, we also prove the Mumford–Tate conjecture for a few new cases of abelian varieties of Lefschetz type.

We discuss existence and abundance of Galois-generic points for adelic representations attached to Shimura varieties. First, we show that, for Shimura varieties of abelian type, $\ell$-Galois-generic points are Galois-generic; in particular, adelic representations attached to such Shimura varieties admit (“lots of”) closed Galois-generic points. Next, we investigate further the distribution of Galois-generic points and show the André–Pink conjecture for them: if $S$ is a connected Shimura variety associated to a $\mathbb{Q}$-simple reductive group, then every infinite subset of the generalized Hecke orbit of a Galois-generic point is Zariski-dense in $S$. Our proof follows the approach of Pink for Siegel Shimura varieties. Our main contribution consists in showing that there are only finitely many Hecke operators of bounded degree on (adelic and connected) Shimura varieties. Compared with other approaches of this result, our proof, which relies on Bruhat–Tits theory, is effective and works for arbitrary Shimura varieties.

The locus of genus-two curves with $n$ marked Weierstrass points has codimension $n$ inside the moduli space of genus-two curves with $n$ marked points, for $n\le 6$. It is well known that the class of the closure of the divisor obtained for $n=1$ spans an extremal ray of the cone of effective divisor classes. We generalize this result for all $n$: we show that the class of the closure of the locus of genus-two curves with $n$ marked Weierstrass points spans an extremal ray of the cone of effective classes of codimension $n$, for $n\le 6$. A related construction produces extremal nef curve classes in moduli spaces of pointed elliptic curves.

We prove that the moduli space ${\overline{\mathcal{ℛ}}}_{15}$ of Prym curves of genus $15$ is of general type. To this end we exhibit a virtual divisor ${\overline{\mathcal{D}}}_{15}$ on ${\overline{\mathcal{ℛ}}}_{15}$ as the degeneracy locus of a globalized multiplication map of sections of line bundles. We then proceed to show that this locus is indeed of codimension one and calculate its class. Using this class, we can conclude that $K_{{\overline{\mathcal{ℛ}}}_{15}}$ is big. This complements a 2010 result of Farkas and Ludwig: now the spaces ${\overline{\mathcal{ℛ}}}_g$ are known to be of general type for $g\ge 14$.

We give a criterion for the vanishing of the weight-one syzygies associated to a line bundle $B$ in a sufficiently positive embedding of a smooth complex projective variety of arbitrary dimension.

In this paper we study the cones of higher codimension (pseudo)effective cycles on point blowups of projective space. We determine bounds on the number of points for which these cones are generated by the classes of linear cycles and for which these cones are finitely generated. Surprisingly, we discover that for (very) general points the higher codimension cones behave better than the cones of divisors. For example, for the blowup $X_r^n$ of ${\mathbb{P}}^n$, $n>4$ at $r$ very general points, the cone of divisors is not finitely generated as soon as $r>n+3$, whereas the cone of curves is generated by the classes of lines if $r\le 2^n$. In fact, if $X_r^n$ is a Mori dream space then all the effective cones of cycles on $X_r^n$ are finitely generated.

Let $\mathcal{O}$ be the category of representations of the Borel subalgebra of a quantum affine algebra introduced by Jimbo and the first author. We show that the Grothendieck ring of a certain monoidal subcategory of $\mathcal{O}$ has the structure of a cluster algebra of infinite rank, with an initial seed consisting of prefundamental representations. In particular, the celebrated Baxter relations for the 6-vertex model get interpreted as Fomin–Zelevinsky mutation relations.