It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ . Such a bound is obtained by comparing the walk on $G_n$ to the walk on d-regular tree $\mathcal{T}_d$ . If one can map another transitive graph $\mathcal{G} $ onto $G_n$ , then we can improve the strategy by using a comparison with the random walk on $\mathcal{G} $ (instead of that of $\mathcal{T} _d$ ), and we obtain a lower bound of the form $\frac {1}{\mathfrak{h} }\log n$ , where $\mathfrak{h} $ is the entropy rate associated with $\mathcal{G} $ . We call this the entropic lower bound.

It was recently proved that in the case $\mathcal{G} =\mathcal{T} _d$ , this entropic lower bound (in that case $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$ ) is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibits cutoff at the entropic time. In this article, we provide a generalisation of the result by providing a sufficient condition on the spectra of the random walks on $G_n$ under which the random walk exhibits cutoff at the entropic time. It applies notably to anisotropic random walks on random d-regular graphs and to random walks on random n-lifts of a base graph (including nonreversible walks).

Using the $\ell $-invariant constructed in our previous paper we prove a Mazur–Tate–Teitelbaum-style formula for derivatives of $p$-adic $L$-functions of modular forms at trivial zeros. The novelty of this result is to cover the near-central point case. In the central point case our formula coincides with the Mazur–Tate–Teitelbaum conjecture proved by Greenberg and Stevens and by Kato, Kurihara and Tsuji at the end of the 1990s.

Following Losik’s approach to Gelfand’s formal geometry, certain characteristic classes for codimension-one foliations coming from the Gelfand-Fuchs cohomology are considered. Sufficient conditions for nontriviality in terms of dynamical properties of generators of the holonomy groups are found. The nontriviality for the Reeb foliations is shown; this is in contrast with some classical theorems on the Godbillon-Vey class; for example, the Mizutani-Morita-Tsuboi theorem about triviality of the Godbillon-Vey class of foliations almost without holonomy is not true for the classes under consideration. It is shown that the considered classes are trivial for a large class of foliations without holonomy. The question of triviality is related to ergodic theory of dynamical systems on the circle and to the problem of smooth conjugacy of local diffeomorphisms. Certain classes are obstructions for the existence of transverse affine and projective connections.

We prove upper bounds for Hecke–Laplace eigenfunctions on certain Riemannian manifolds $X$ of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are $d$-fold products of $2$-spheres or $3$-spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a (‘Weyl-type’) saving of $\mathrm{vol} \hspace{0.167em} (X)^{- 1/ 6+ \varepsilon } $.

In this paper we demonstrate that non-commutative localizations of arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category $\text{}\underline{C}$ by a set $S$ of morphisms in the heart $\text{}\underline{Hw}$ of a weight structure $w$ on it one obtains a triangulated category endowed with a weight structure $w^{\prime }$. The heart of $w^{\prime }$ is a certain version of the Karoubi envelope of the non-commutative localization $\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ (of $\text{}\underline{Hw}$ by $S$). The functor $\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$ is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of $S$ invertible. For any additive category $\text{}\underline{A}$, taking $\text{}\underline{C}=K^{b}(\text{}\underline{A})$ we obtain a very efficient tool for computing $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that $\text{}\underline{A}[S^{-1}]_{\mathit{add}}$ coincides with the ‘abstract’ localization $\text{}\underline{A}[S^{-1}]$ (as constructed by Gabriel and Zisman) if $S$ contains all identity morphisms of $\text{}\underline{A}$ and is closed with respect to direct sums. We apply our results to certain categories of birational motives $DM_{gm}^{o}(U)$ (generalizing those defined by Kahn and Sujatha). We define $DM_{gm}^{o}(U)$ for an arbitrary $U$ as a certain localization of $K^{b}(Cor(U))$ and obtain a weight structure for it. When $U$ is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general $U$ the result is completely new. The existence of the corresponding adjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over $U$.

Since Faltings proved Mordell’s conjecture in [16] in 1983, we have known that the sets of rational points on curves of genus at least $2$ are finite. Determining these sets in individual cases is still an unsolved problem. Chabauty’s method (1941) [10] is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the Jacobian, the closure of the Mordell–Weil group with the p-adic points of the curve. Under the condition that the Mordell–Weil rank is less than the genus, Chabauty’s method, in combination with other methods such as the Mordell–Weil sieve, has been applied successfully to determine all rational points in many cases.

Minhyong Kim’s nonabelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Besser, Dogra, Müller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both $3$ ).

This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only ‘simple algebraic geometry’ (line bundles over the Jacobian and models over the integers).

In our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces $X$ attached to real orthogonal groups of type $(p, q)$. This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in $X$ with coefficients.

In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification $ \overline{X} $ of $X$. However, for the $ \mathbb{Q} $-split case for signature $(p, p)$, we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of $ \overline{X} $ is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.

As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of $X$. In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for ${L}^{2} $-cohomology.

(Torsion in the cohomology of Kottwitz–Harris–Taylor Shimura varieties) When the level at $l$ of a Shimura variety of Kottwitz–Harris–Taylor is not maximal, its cohomology with coefficients in a $\overline{\mathbb{Z}}_{l}$-local system isn’t in general torsion free. In order to prove torsion freeness results of the cohomology, we localize at a maximal ideal $\mathfrak{m}$ of the Hecke algebra. We then prove a result of torsion freeness resting either on $\mathfrak{m}$ itself or on the Galois representation $\overline{\unicode[STIX]{x1D70C}}_{\mathfrak{m}}$ associated to it. Concerning the torsion, in a rather restricted case than Caraiani and Scholze (« On the generic part of the cohomology of compact unitary Shimura varieties », Preprint, 2015), we prove that the torsion doesn’t give new Satake parameters systems by showing that each torsion cohomology class can be raised in the free part of the cohomology of a Igusa variety.

This article deals with kinetic Fokker–Planck equations with essentially bounded coefficients. A weak Harnack inequality for nonnegative super-solutions is derived by considering their log-transform and adapting an argument due to S. N. Kružkov (1963). Such a result rests on a new weak Poincaré inequality sharing similarities with the one introduced by W. Wang and L. Zhang in a series of works about ultraparabolic equations (2009, 2011, 2017). This functional inequality is combined with a classical covering argument recently adapted by L. Silvestre and the second author (2020) to kinetic equations.

The Beauville–Voisin conjecture for a hyperkähler manifold $X$ states that the subring of the Chow ring $A^{\ast }(X)$ generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of $X$. We prove a weak version of this conjecture when $X$ is the Hilbert scheme of points on a K3 surface for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn’s formula and the Li–Qin–Wang $W_{1+\infty }$ algebra action from cohomology to Chow groups for the Hilbert scheme of an arbitrary smooth projective surface $S$.

Let $X$ be a smooth proper curve over a finite field of characteristic $p$. We prove a product formula for $p$-adic epsilon factors of arithmetic $\mathscr{D}$-modules on $X$. In particular we deduce the analogous formula for overconvergent $F$-isocrystals, which was conjectured previously. The $p$-adic product formula is a counterpart in rigid cohomology of the Deligne–Laumon formula for epsilon factors in $\ell$-adic étale cohomology (for $\ell \neq p$). One of the main tools in the proof of this $p$-adic formula is a theorem of regular stationary phase for arithmetic $\mathscr{D}$-modules that we prove by microlocal techniques.

In this paper we define a p-adic analogue of the Borel regulator for the K-theory of p-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this p-adic regulator to the Bloch–Kato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups. We also show that the Soulé regulator is induced by continuous and even analytic classes.

Given two arbitrary sequences $({\lambda }_{j} )_{j\geq 1} $ and $({\mu }_{j} )_{j\geq 1} $ of real numbers satisfying

$$\begin{eqnarray*}\displaystyle \vert {\lambda }_{1} \vert \gt \vert {\mu }_{1} \vert \gt \vert {\lambda }_{2} \vert \gt \vert {\mu }_{2} \vert \gt \cdots \gt \vert {\lambda }_{j} \vert \gt \vert {\mu }_{j} \vert \rightarrow 0, &&\displaystyle\end{eqnarray*}$$

$c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $

${\Gamma }_{c} $

${\Gamma }_{\tilde {c} } $

$c= ({c}_{n} )_{n\geq 0} $

$\tilde {c} = ({c}_{n+ 1} )_{n\geq 0} $

${\ell }^{2} ({ \mathbb{Z} }_{+ } )$

$({\lambda }_{j} )_{j\geq 1} $

$({\mu }_{j} )_{j\geq 1} $

$c$

${\Gamma }_{c} $

${\Gamma }_{\tilde {c} } $

$({\lambda }_{j} )_{j\geq 1} $

$({\mu }_{j} )_{j\geq 1} $

$({\rho }_{j} )_{j\geq 1} $

$({\sigma }_{j} )_{j\geq 1} $

$$\begin{eqnarray*}\displaystyle {\rho }_{1} \gt {\sigma }_{1} \gt {\rho }_{2} \gt {\sigma }_{2} \gt \cdots \gt {\rho }_{j} \gt {\sigma }_{j} \rightarrow 0, &&\displaystyle\end{eqnarray*}$$

$c= ({c}_{n} )_{n\in { \mathbb{Z} }_{+ } } $

${\Gamma }_{c} $

${\Gamma }_{\tilde {c} } $

${\ell }^{2} ({ \mathbb{Z} }_{+ } )$

$({\rho }_{j} )_{j\geq 1} $

$({\sigma }_{j} )_{j\geq 1} $

We prove that the class of reflexive asymptotic-$c_{0}$ Banach spaces is coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a reflexive asymptotic-$c_{0}$ space $Y$, then $X$ is also reflexive and asymptotic-$c_{0}$. In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-$c_{0}$ space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.

We study the arithmetic of abelian varieties over where is an arbitrary field. The main result relates Mordell–Weil groups of certain Jacobians over to homomorphisms of other Jacobians over . Our methods also yield completely explicit points on elliptic curves with unbounded rank over and a new construction of elliptic curves with moderately high rank over .

Following a recent paper by Faltings, we study the integral points on $\bm{P}_2\setminus\mathcal{D}$, where $\mathcal{D}$ is the branch locus of a projection from a surface $\mathcal{X}$; a crucial point in the analysis is that the pull-back of $\mathcal{D}$ in the Galois closure of the projection often splits into several components. As in the paper by Faltings, under certain assumptions we obtain finiteness of the integral points (Theorem 3.1); for instance, we shall find that it suffices if the projection is sufficiently general and if $\mathcal{X}$ has Kodaira number $\ge0$ (Corollary 4.1). We have borrowed freely from Faltings’s paper, for the whole geometrical setting. As to the arithmetic, our method is in part different, relying on the recent paper by Corvaja and Zannier and leading to apparently new conditions. We shall also use a more elementary approach to study a similar situation in arbitrary dimension, where the projection is taken from a hypersurface (Theorem 2.1).

In concrete terms, these results deal with certain diophantine equations $F(x_0,\dots,x_n)=c$.

AMS 2000 Mathematics subject classification: Primary 11D72; 11G35; 11G99

Suppose that $G$ is a connected reductive algebraic group defined over $\mathbf{R}$, $G(\mathbf{R})$ is its group of real points, ${\it\theta}$ is an automorphism of $G$, and ${\it\omega}$ is a quasicharacter of $G(\mathbf{R})$. Kottwitz and Shelstad defined endoscopic data associated to $(G,{\it\theta},{\it\omega})$, and conjectured a matching of orbital integrals between functions on $G(\mathbf{R})$ and its endoscopic groups. This matching has been proved by Shelstad, and it yields a dual map on stable distributions. We express the values of this dual map on stable tempered characters as a linear combination of twisted characters, under some additional hypotheses on $G$ and ${\it\theta}$.

It is shown that the Grayson tower for $K$-theory of smooth algebraic varieties is isomorphic to the slice tower of $S^{1}$-spectra. We also extend the Grayson tower to bispectra, and show that the Grayson motivic spectral sequence is isomorphic to the motivic spectral sequence produced by the Voevodsky slice tower for the motivic $K$-theory spectrum $\mathit{KGL}$. This solves Suslin’s problem about these two spectral sequences in the affirmative.

We prove a compactness theorem for metrics with bounded integral curvature on a fixed closed surface $\unicode[STIX]{x1D6F4}$. As a corollary we obtain a new convergence result for sequences of metrics with conical singularities, where an accumulation of singularities is allowed.

We develop a strategy making extensive use of tent spaces to study parabolic equations with quadratic nonlinearities as for the Navier–Stokes system. We begin with a new proof of the well-known result of Koch and Tataru on the well-posedness of Navier–Stokes equations in $\mathbb{R}^{n}$ with small initial data in $\mathit{BMO}^{-1}(\mathbb{R}^{n})$. We then study another model where neither pointwise kernel bounds nor self-adjointness are available.