If $E$ is an elliptic curve over ${\bb Q}$, then let $E(D)$ denote the $D$-quadratic twist of $E$. It is conjectured that there are infinitely many primes $p$ for which $E(p)$ has rank 0, and that there are infinitely many primes $l$ for which $E(l)$ has positive rank. For some special curves $E$ we show that there is a set $S$ of primes $p$ with density $\frac{1}{3}$; for which if $D = \Pi_{pj}$ is a squarefree integer where $pj \in S$, then $E(D)$ has rank 0. In particular $E(p)$ has rank 0 for every $p \in S$. As an example let $E_1$ denote the curve \[ E_1: y^2 = x^3 + 44x^2 -19360x + 1682384. \]Then its associated set of primes $S_1$ consists of the prime 11 and the primes $p$ for which the order of the reduction of $X0(11)$ modulo $p$ is odd. To obtain the general result we show for primes $p \in S$ that the rational factor of $L(E(p),1)$ is nonzero which implies that $E(p)$ has rank 0. These special values are related to surjective ${\bb Z}/2{\bb Z}$ Galois representations that are attached to modular forms. Another example of this result is given, and we conclude with some remarks regarding the existence of positive rank prime twists via polynomial identities.

In this paper we take into consideration a conjecture of Bloch equating in a suitable range and under some standard conjectures, the rank of the motivic cohomology of the special fiber of a semistable degeneration over the ring of integers of a number field with the order of zero of the local Euler factor of the L-function over the semistable prime corresponding to the special fiber. We then develop, following the philosophy that the fiber ‘at infinity’ of an arithmetic variety should be considered as ‘maximally degenerate’, a construction that goes in parallel to the one we use for the non-archimedean fiber. Namely, a definition of a double complex and a weighted operator N on it that plays the role of the logarithm of the (local) monodromy map at infinity. We like to see this archimedean theory as the analogue of the limiting mixed Hodge structure theory of a degeneration of projective varieties over a disc. In particular, this yields to a description of the (motivic) Deligne cohomology as homology of the mapping cone of N. The main result arising from this construction is the proof of a conjecture of Deninger. Namely we show that the archimedean Γ-factor of the Zeta-function of the archimedean fiber can be seen as the characteristic polynomial of an archimedean Frobenius acting on the subgroup of the invariants of N into the hypercohomology of our double complex.

In 1987, Sabbah proved the existence of Bernstein–Sato polynomials associated with several analytic functions. The purpose of this paper is to give a more elementary and constructive proof of Sabbah's result based on the notion of the analytic Gröbner fan of a $\mathcal D$-module.

We prove that the Newton polygons of Frobenius on the crystalline cohomology of proper smooth varieties satisfy a symmetry that results, in the case of projective smooth varieties, from Poincaré duality and the hard Lefschetz theorem. As a corollary, we deduce that the Betti numbers in odd degrees of any proper smooth variety over a field are even (a consequence of Hodge symmetry in characteristic zero), answering an old question of Serre. Then we give a generalization and a refinement for arbitrary varieties over finite fields, in response to later questions of Serre and of Katz.

We show that a linear subspace of a reductive Lie algebra $\operatorname{\mathfrak g}$ that consists of nilpotent elements has dimension at most $\frac{1}{2}(\dim\operatorname{\mathfrak g}-\operatorname{rk}\operatorname{\mathfrak g})$, and that any nilpotent subspace attaining this upper bound is equal to the nilradical of a Borel subalgebra of $\operatorname{\mathfrak g}$. This generalizes a classical theorem of Gerstenhaber, which states this fact for the algebra of $(n\times n)$-matrices.

Let $X$ be a curve over a number field $K$ with genus $g\geq 2$, $\mathfrak{p}$ a prime of ${ \mathcal{O} }_{K} $ over an unramified rational prime $p\gt 2r$, $J$ the Jacobian of $X$, $r= \mathrm{rank} \hspace{0.167em} J(K)$, and $\mathscr{X}$ a regular proper model of $X$ at $\mathfrak{p}$. Suppose $r\lt g$. We prove that $\# X(K)\leq \# \mathscr{X}({ \mathbb{F} }_{\mathfrak{p}} )+ 2r$, extending the refined version of the Chabauty–Coleman bound to the case of bad reduction. The new technical insight is to isolate variants of the classical rank of a divisor on a curve which are better suited for singular curves and which satisfy Clifford’s theorem.

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$.

In this paper, we present a conjecture on a relation between the Zelevinski involutions for reductive groups over a p-adic field and the conjectural A-packets. As evidence for this conjecture, we prove that the Zelevinski involutions, which are regarded as operators on the space of virtual characters, commute with the endoscopic transfers under the assumption of the fundamental lemma for groups and for Lie algebras.

In this paper, we study representations of a Galois group of a local field absolutely unramified whose residue field is perfect of characteristic $p$ and give two ways of recognize crystalline representations with weights between $r$ and $r+p-1$. The first one is the description of terms of weakly admissible filtered modules proved by J.-M. Fontaine and G. Laffaille (Ann Sc ENS 1982): here a new proof of this result is proposed (second chapter). The second criterion (third chapter) is the equivalence between crystalline representations with weights between $r$ and $r+p-1$ and representations of finite ‘cr-height’ ${\leqslant} p-1$, result announced by J.-M. Fontaine in a paper edited a few years ago where he introduced the category of $(\varphi, \Gamma)$-modules.

We prove a sharp upper bound on the L2 norm of a GL3 Maass form restricted to GL2×ℝ+.

We prove the existence of weak solutions of complex $m$ -Hessian equations on compact Hermitian manifolds for the non-negative right-hand side belonging to $L^{p}$ , $p>n/m$ ( $n$  is the dimension of the manifold). For smooth, positive data the equation has recently been solved by Székelyhidi and Zhang. We also give a stability result for such solutions.

We introduce some blow-analytic invariants of real analytic function-germs and discuss their properties. As a consequence, we obtain, for instance, the multiplicity of function-germs is a blow-analytic invariant.

In this paper we prove that surfaces of general type with irregularity $q\geq 3$ are rationally cohomologically rigidified, and so are minimal surfaces $S$ with $q(S)= 2$ unless ${ K}_{S}^{2} = 8\chi ({ \mathcal{O} }_{S} )$. Here a surface $S$ is said to be rationally cohomologically rigidified if its automorphism group $\mathrm{Aut} (S)$ acts faithfully on the cohomology ring ${H}^{\ast } (S, \mathbb{Q} )$. As examples we give a complete classification of surfaces isogenous to a product with $q(S)= 2$ that are not rationally cohomologically rigidified.

We study the $p$-adic variation of triangulations over $p$-adic families of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules. In particular, we study certain canonical sub-filtrations of the pointwise triangulations and show that they extend to affinoid neighborhoods of crystalline points. This generalizes results of Kedlaya, Pottharst and Xiao and (independently) Liu in the case where one expects the entire triangulation to extend. We also study the ramification of weight parameters over natural $p$-adic families.

Consider a smooth quasi-projective variety $X$ equipped with a $\mathbb{C}^{\ast }$-action, and a regular function $f:X\rightarrow \mathbb{C}$ which is $\mathbb{C}^{\ast }$-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of $f$ on proper components of the critical locus of $f$, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work by Kontsevich and Soibelman, Nagao, and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary.

For an arbitrary connected reductive group $G$, we consider the motivic integral over the arc space of an arbitrary $ \mathbb{Q} $-Gorenstein horospherical $G$-variety ${X}_{\Sigma } $ associated with a colored fan $\Sigma $ and prove a formula for the stringy $E$-function of ${X}_{\Sigma } $ which generalizes the one for toric varieties. We remark that, in contrast to toric varieties, the stringy $E$-function of a Gorenstein horospherical variety ${X}_{\Sigma } $ may be not a polynomial if some cones in $\Sigma $ have nonempty sets of colors. Using the stringy $E$-function, we can formulate and prove a new smoothness criterion for locally factorial horospherical varieties. We expect that this smoothness criterion holds for arbitrary spherical varieties.

Let ${\rm\Gamma}(n,p)$ denote the binomial model of a random triangular group. We show that there exist constants $c,C>0$ such that if $p\leqslant c/n^{2}$, then asymptotically almost surely (a.a.s.) ${\rm\Gamma}(n,p)$ is free, and if $p\geqslant C\log n/n^{2}$, then a.a.s. ${\rm\Gamma}(n,p)$ has Kazhdan’s property (T). Furthermore, we show that there exist constants $C^{\prime },c^{\prime }>0$ such that if $C^{\prime }/n^{2}\leqslant p\leqslant c^{\prime }\log n/n^{2}$, then a.a.s. ${\rm\Gamma}(n,p)$ is neither free nor has Kazhdan’s property (T).

Let $k=\mathbb{C}$ be the field of complex numbers. In this article we construct Hodge realization functors defined on the triangulated categories of étale motives with rational coefficients. Our construction extends to every smooth quasi-projective $k$-scheme, the construction done by Nori over a field, and relies on the original version of the basic lemma proved by Beĭlinson. As in the case considered by Nori, the realization functor factors through the bounded derived category of a perverse version of the Abelian category of Nori motives.

We analyzes a notion of strong semistability of principal G-bundles by including reduction to nonreduced parabolic subgroup schemes. It turns out that strong semistability is equivalent to the Frobenius semistability of Ramanan and Rananathan. We also give a bound for nonstrongly semsitability of a semistable GL(n)-bundle improving a previous result of Shepherd-Barron.

We consider suspension hypersurface singularities of type g = f(x, y) + zn, where f is an irreducible plane curve singularity. For such germs, we prove that the link of g determines completely the Newton pairs of f and the integer n except for two pathological cases, which can be completely described. Even in the pathological cases, the link and the Milnor number of g determine uniquely the Newton pairs of f and n. In particular, for such g, we verify Zariski's conjecture about the multiplicity. The result also supports the following conjecture formulated in this paper: if the link of an isolated hypersurface singularity is a rational homology 3-sphere, then it determines the equisingularity type, the embedded topological type, the equivariant Hodge numbers and the multiplicity of the singularity. The conjecture is verified for weighted homogeneous singularities too.