When the branch character has root number $- 1$ , the corresponding anticyclotomic Katz $p$ -adic $L$ -function vanishes identically. For this case, we determine the $\mu $ -invariant of the cyclotomic derivative of the Katz $p$ -adic $L$ -function. The result proves, as an application, the non-vanishing of the anticyclotomic regulator of a self-dual CM modular form with root number $- 1$ . The result also plays a crucial role in the recent work of Hsieh on the Eisenstein ideal approach to a one-sided divisibility of the CM main conjecture.

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor–Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.

Given a prime $p\gt 2$ , an integer $h\geq 0$ , and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $ , we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$ , to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $ -coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$ -adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope  $\leq h$ .

We study fiber functors on Tannakian categories which are equipped with a grading or a filtration. Our goal is to give a comprehensive set of foundational results about such functors. A main result is that each filtration on a fiber functor can be split by a grading fpqc-locally on the base scheme.

We prove that groups definable in o-minimal structures have Cartan subgroups, and only finitely many conjugacy classes of such subgroups. We also delineate with precision how these subgroups cover the ambient group.

Let $\mathfrak{F}$ be a locally compact nonarchimedean field with residue characteristic $p$, and let $\mathrm{G} $ be the group of $\mathfrak{F}$-rational points of a connected split reductive group over $\mathfrak{F}$. For $k$ an arbitrary field of any characteristic, we study the homological properties of the Iwahori–Hecke $k$-algebra ${\mathrm{H} }^{\prime } $ and of the pro-$p$ Iwahori–Hecke $k$-algebra $\mathrm{H} $ of $\mathrm{G} $. We prove that both of these algebras are Gorenstein rings with self-injective dimension bounded above by the rank of $\mathrm{G} $. If $\mathrm{G} $ is semisimple, we also show that this upper bound is sharp, that both $\mathrm{H} $ and ${\mathrm{H} }^{\prime } $ are Auslander–Gorenstein, and that there is a duality functor on the finite length modules of $\mathrm{H} $ (respectively ${\mathrm{H} }^{\prime } $). We obtain the analogous Gorenstein and Auslander–Gorenstein properties for the graded rings associated to $\mathrm{H} $ and ${\mathrm{H} }^{\prime } $.

When $k$ has characteristic $p$, we prove that in ‘most’ cases $\mathrm{H} $ and ${\mathrm{H} }^{\prime } $ have infinite global dimension. In particular, we deduce that the category of smooth $k$-representations of $\mathrm{G} = {\mathrm{PGL} }_{2} ({ \mathbb{Q} }_{p} )$ generated by their invariant vectors under the pro-$p$ Iwahori subgroup has infinite global dimension (at least if $k$ is algebraically closed).

We study the consequences of stationary and semi-stationary set reflection. We show that the semi-stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of the weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss, and prove that they follow from stationary and semi-stationary set reflection augmented with a weak form of Martin’s Axiom. We also show that there are some differences between the two reflection principles, which suggests that stationary set reflection is analogous to supercompactness, whereas semi-stationary set reflection is analogous to strong compactness.

In this work, we describe a method to construct the generic braid monodromy of the preimage of a curve by a Kummer cover. This method is interesting since it combines two techniques, namely, the construction of a highly non-generic braid monodromy and a systematic method to go from a non-generic to a generic braid monodromy. The latter process, called generification, is independent from Kummer covers, and it can be applied in more general circumstances since non-generic braid monodromies appear more naturally and are oftentimes much easier to compute. Explicit examples are computed using these techniques.

We deal with the incompressible Navier–Stokes equations with vortex patches as initial data. Such data describe an initial configuration for which the vorticity is discontinuous across a hypersurface. We give an asymptotic expansion of the solutions in the vanishing viscosity limit which exhibits an internal layer where the fluid vorticity has a sharp variation. This layer moves with the flow of the Euler equations.

We prove uniform Sobolev estimates $\Vert u\Vert _{{L}^{p\prime } } \leq C\Vert (\Delta - \alpha )u\Vert _{{L}^{p} } $ for $\alpha \in \mathbb{C} $ and $p= 2n/ (n+ 2),{p}^{\prime } = 2n/ (n- 2)$ on non-trapping asymptotically conic manifolds of dimension $n\geq 3$ , generalizing to non-constant coefficient Laplacians a result of Kenig, Ruiz and Sogge [13].

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.

In this paper, we introduce variants of formal nearby cycles for a locally noetherian formal scheme over a complete discrete valuation ring. If the formal scheme is locally algebraizable, then our nearby cycle gives a generalization of Berkovich’s formal nearby cycle. Our construction is entirely scheme theoretic, and does not require rigid geometry. Our theory is intended for applications to the local study of the cohomology of Rapoport–Zink spaces.

Consider a pseudogroup on $( \mathbb{C} , 0)$ generated by two local diffeomorphisms having analytic conjugacy classes a priori fixed in $\mathrm{Diff} \hspace{0.167em} ( \mathbb{C} , 0)$ . We show that a generic pseudogroup as above is such that every point has a (possibly trivial) cyclic stabilizer. It also follows that these generic groups possess infinitely many hyperbolic orbits. This result possesses several applications to the topology of leaves of foliations, and we shall explicitly describe the case of nilpotent foliations associated to Arnold’s singularities of type ${A}^{2n+ 1} $ .

This paper studies ‘pro-excision’ for the $K$ -theory of one-dimensional, usually semi-local, rings and its various applications. In particular, we prove Geller’s conjecture for equal characteristic rings over a perfect field of finite characteristic, give results towards Geller’s conjecture in mixed characteristic, and we establish various finiteness results for the $K$ -groups of singularities, covering both orders in number fields and singular curves over finite fields.

We study the global well-posedness of 3D Navier–Stokes equations for a class of large initial data. This type of data slowly varies in the vertical direction (expressed as a function of $\varepsilon {x}_{3} $ ), and it is ill-prepared in the sense that its norm in ${C}^{- 1} $ will blow up at the rate ${\varepsilon }^{- \alpha } $ for $\frac{1}{2} \lt \alpha \lt 1$ as $\varepsilon $ tends to zero.