Let $A$ be an abelian variety over a global field $K$ of characteristic $p\geqslant 0$. If $A$ has nontrivial (respectively full) $K$-rational $l$-torsion for a prime $l\neq p$, we exploit the fppf cohomological interpretation of the $l$-Selmer group $\text{Sel}_{l}\,A$ to bound $\#\text{Sel}_{l}\,A$ from below (respectively above) in terms of the cardinality of the $l$-torsion subgroup of the ideal class group of $K$. Applied over families of finite extensions of $K$, the bounds relate the growth of Selmer groups and class groups. For function fields, this technique proves the unboundedness of $l$-ranks of class groups of quadratic extensions of every $K$ containing a fixed finite field $\mathbb{F}_{p^{n}}$ (depending on $l$). For number fields, it suggests a new approach to the Iwasawa ${\it\mu}=0$ conjecture through inequalities, valid when $A(K)[l]\neq 0$, between Iwasawa invariants governing the growth of Selmer groups and class groups in a $\mathbb{Z}_{l}$-extension.

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 the complete $L$-function associated to any cuspidal automorphic representation of $\operatorname{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$ has infinitely many simple zeros.

In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G defined over p-adic fields in order to understand their admissible representations. Each Hecke algebra is associated to a pair (JΣ, ρΣ) of an open compact subgroup JΣ and its irreducible representation ρΣ which is constructed from given data Σ = (Γ, P′0, ϱ). Here, Γ is a semisimple element in the Lie algebra of G, P′0 is a parahoric subgroup in the centralizer of Γ in G, and ϱ is a cuspidal representation on the finite reductive quotient of P′0. In this paper, we explicitly describe those Hecke algebras when P′0 is a minimal parahoric subgroup, ϱ is trivial and ρΣ is a character.

For an optimal modular parametrization $J_{0}(n){\twoheadrightarrow}E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^{0}(E,\unicode[STIX]{x1D6FA}^{1})$. Multiple authors generalized his conjecture to higher-dimensional newform quotients. We prove the Manin conjecture for semistable $E$, give counterexamples to all the proposed generalizations, and prove several semistable special cases of these generalizations. The proofs establish general relations between the integral $p$-adic étale and de Rham cohomologies of abelian varieties over $p$-adic fields and exhibit a new exactness result for Néron models.

Let X be the Fermat curve of degree q+1 over the field k of q$^2$ elements, where q is some prime power. Considering the Jacobian J of X as a constant abelian variety over the function field k(X), we calculate the multiplicities, in subfactors of the Shafarevich–Tate group, of representations associated with the action on X of a finite unitary group. J is isogenous to a power of a supersingular elliptic curve E, the structure of whose Shafarevich–Tate group is also described.

We study the variation of $\unicode[STIX]{x1D707}$-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the $p$-adic zeta function. This lower bound forces these $\unicode[STIX]{x1D707}$-invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When $U_{p}-1$ generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the $p$-adic $L$-function is simply a power of $p$ up to a unit (i.e. $\unicode[STIX]{x1D706}=0$). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

Let $X\subset \mathbb{P}^{4}$ be a terminal factorial quartic $3$-fold. If $X$ is non-singular, $X$ is birationally rigid, i.e. the classical minimal model program on any terminal $\mathbb{Q}$-factorial projective variety $Z$ birational to $X$ always terminates with $X$. This no longer holds when $X$ is singular, but very few examples of non-rigid factorial quartics are known. In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface $X\subset \mathbb{P}^{4}$. A singular point on such a hypersurface is of type $cA_{n}$ ($n\geqslant 1$), or of type $cD_{m}$ ($m\geqslant 4$) or of type $cE_{6},cE_{7}$ or $cE_{8}$. We first show that if $(P\in X)$ is of type $cA_{n}$, $n$ is at most $7$ and, if $(P\in X)$ is of type $cD_{m}$, $m$ is at most $8$. We then construct examples of non-rigid factorial quartic hypersurfaces whose singular loci consist (a) of a single point of type $cA_{n}$ for $2\leqslant n\leqslant 7$, (b) of a single point of type $cD_{m}$ for $m=4$ or $5$ and (c) of a single point of type $cE_{k}$ for $k=6,7$ or $8$.

Each finite dimensional irreducible rational representation V of the symplectic group Sp$_2g$(Q) determines a generically defined local system V over the moduli space M$_g$ of genus g smooth projective curves. We study H$^2$ (M$_g$; V) and the mixed Hodge structure on it. Specifically, we prove that if g [ges ] 6, then the natural map IH$^2$(M˜$_g$; V) → H$^2$(M$_g$; V) is an isomorphism where M˜$_g$ is the Satake compactification of M$_g$. Using the work of Saito we conclude that the mixed Hodge structure on H$^2$(M$_g$; V) is pure of weight 2+r if V underlies a variation of Hodge structure of weight r. We also obtain estimates on the weight of the mixed Hodge structure on H$^2$(M$_g$; V) for 3 [les ] g < 6. Results of this article can be applied in the study of relations in the Torelli group T$_g$.

Consider a vector bundle with connection on a $p$-adic analytic curve in the sense of Berkovich. We collect some improvements and refinements of recent results on the structure of such connections, and on the convergence of local horizontal sections. This builds on work from the author’s 2010 book and on subsequent improvements by Baldassarri and by Poineau and Pulita. One key result exclusive to this paper is that the convergence polygon of a connection is locally constant around every type 4 point.

Let $G$ be a complex reductive linear algebraic group and $G_0 \subseteq G$ a real form. Suppose $P$ is a parabolic subgroup of $G$ and assume that $P$ has a Levi factor $L$ such that $G_0 \cap L = L_0$ is a real form of $L$. Using the minimal globalization $V_{\min}$ of a finite length admissible representation for $L_0$, one can define a homogeneous analytic vector bundle on the $G_0$ orbit $S$ of $P$ in the generalized flag manifold $Y = G/P$. Let $A(P, V_{\min})$ denote the corresponding sheaf of polarized sections. In this article we analyze the $G_0$ representations obtained on the compactly supported sheaf cohomology groups $H^p_c(S,A(P, V_{\min}))$.

In this article, we study the homomorphisms between scalar generalized Verma modules. We conjecture that any homomorphism between scalar generalized Verma modules is a composition of elementary homomorphisms. The purpose of this article is to confirm the conjecture for some parabolic subalgebras under the assumption that the infinitesimal characters are regular.

Let ${E}$ be a nonconstant elliptic curve, over a global field ${K}$ of positive, odd characterisitc. Assuming the finiteness of the Shafarevich-Tate group of ${E}$, we show that the order of the Shafarevich-Tate group of ${E}$, is given by${O}({N}^{1/2+6\,\log(2)/\log({q})})$, where ${N}$ is the conductor of ${E}, {q}$ is the cardinality of the finite field of constants of ${K}$, and where the constant in the bound depends only on ${K}$. The method of proof is to work with the geometric analog of the Birch-Swinnerton Dyer conjecture for the corresponding elliptic surface over the finite field, as formulated by Artin-Tate, and to examine the geometry of this elliptic surface.

A theorem due to Ohkawa states that the collection of Bousfield equivalence classes of spectra is a set. We extend this result to arbitrary combinatorial model categories.

We study the étale cohomology of Hilbert modular varieties, building on the methods introduced by Caraiani and Scholze for unitary Shimura varieties. We obtain the analogous vanishing theorem: in the ‘generic’ case, the cohomology with torsion coefficients is concentrated in the middle degree. We also probe the structure of the cohomology beyond the generic case, obtaining bounds on the range of degrees where cohomology with torsion coefficients can be non-zero. The proof is based on the geometric Jacquet–Langlands functoriality established by Tian and Xiao and avoids trace formula computations for the cohomology of Igusa varieties. As an application, we show that, when $p$ splits completely in the totally real field and under certain technical assumptions, the $p$-adic local Langlands correspondence for $\mathrm {GL}_2(\mathbb {Q}_p)$ occurs in the completed homology of Hilbert modular varieties.

We show that, under conditions about the microcharacteristic variety of a coherent $\cal D$-module, the Cauchy problem is well-posed in the spaces of formal power series with Gevrey growth. We deduce that the filtration of the Irregularity Sheaf of a holonomic $\cal D$-module, which we defined in a previous work, is preserved under inverse image if some rather general geometric conditions are fullfilled.

Upper and lower bounds for the Seshadri constants of canonical bundles of compact hyperbolic spaces are given in terms of metric invariants. The lower bound is obtained by carrying out the symplectic blow-up construction for the Poincaré metric, and the upper bound is obtained by a convexity-type argument.

Let $F$ be a non-discrete non-Archimedean locally compact field and ${\mathcal{O}}_{F}$ the ring of integers in $F$ . The main results of this paper are the classification of ergodic probability measures on the space $\text{Mat}(\mathbb{N},F)$ of infinite matrices with entries in $F$ with respect to the natural action of the group $\text{GL}(\infty ,{\mathcal{O}}_{F})\times \text{GL}(\infty ,{\mathcal{O}}_{F})$ and the classification, for non-dyadic $F$ , of ergodic probability measures on the space $\text{Sym}(\mathbb{N},F)$ of infinite symmetric matrices with respect to the natural action of the group $\text{GL}(\infty ,{\mathcal{O}}_{F})$ .

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.

On établit une décomposition de l’homologie stable des groupes d’automorphismes des groupes libres à coefficients polynomiaux contravariants en termes d’homologie des foncteurs. Elle permet plusieurs calculs explicites, qui recoupent des résultats établis de manière indépendante par O. Randal-Williams et généralisent certains d’entre eux. Nos méthodes reposent sur l’examen d’extensions de Kan dérivées associées à plusieurs catégories de groupes libres, la généralisation d’un critère d’annulation homologique à coefficients polynomiaux dû à Scorichenko, le théorème de Galatius identifiant l’homologie stable des groupes d’automorphismes des groupes libres à celle des groupes symétriques, la machinerie des $\unicode[STIX]{x1D6E4}$-espaces et le scindement de Snaith.