We calculate l-adic nearby cycles in the étale cohomology for families with log smooth reduction using log étale cohomology. In particular, nearby cycles for log smooth families coincide with tame nearby cycles, as L. Illusie expected, and nearby cycles for semistable families depend only on the first infinitesimal neighborhood of the special fiber.

We prove the finiteness of class numbers and Tate–Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semi-simple groups via systematic use of the recently developed structure theory and classification of pseudo-reductive groups.

Let $X$ be a compact Kähler manifold and $\{\unicode[STIX]{x1D703}\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $\unicode[STIX]{x1D703}$-plurisubharmonic functions with full mass are the same as those of a current with minimal singularities. Second, given another big and nef class $\{\unicode[STIX]{x1D702}\}$, we show the inclusion ${\mathcal{E}}(X,\unicode[STIX]{x1D702})\cap \operatorname{PSH}(X,\unicode[STIX]{x1D703})\subset {\mathcal{E}}(X,\unicode[STIX]{x1D703})$. Third, we characterize big classes whose full mass currents are ‘additive’. Our techniques make use of a characterization of full mass currents in terms of the envelope of their singularity type. As an essential ingredient we also develop the theory of weak geodesics in big cohomology classes. Numerous applications of our results to complex geometry are also given.

We study the Rankin $L$-series of a cuspidal automorphic representation of GL(2) of even weight over the rational numbers, twisted by a character of a real quadratic field. When the sign of the functional equation is <formula form="inline" disc="math" id="ffm004"><formtex notation="AMSTeX">$+1$, we give an explicit formula for the central value of the $L$-series, analogous to the formulae obtained by Gross, Zhang, and Xue in the imaginary case. The proof uses a version of the Rankin–Selberg method in which the theta correspondence plays an important role. We give two applications, to computing the order of the Tate–Shafarevich group of the base change to a real quadratic field of an elliptic curve defined over the rationals, and to proving the equidistribution of individual closed geodesics on modular curves.

We introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne–Mumford stability. For every pair of integers 1≤m<n, we prove that the moduli problem of n-pointed m-stable curves of arithmetic genus one is representable by a proper irreducible Deligne–Mumford stack . We also consider weighted variants of these stability conditions, and construct the corresponding moduli stacks . In forthcoming work, we will prove that these stacks have projective coarse moduli and use the resulting spaces to give a complete description of the log minimal model program for .

The chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the ‘nondegenerate spectrality’ axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and apply to any Floer-type theory (including Novikov homology) satisfying certain standard formal properties. The key ingredient is a theorem about the existence of best approximations of arbitrary elements of finitely generated free modules over Novikov rings by elements of prescribed submodules with respect to a certain family of non-Archimedean metrics.

We study the back stable Schubert calculus of the infinite flag variety. Our main results are:

• – a formula for back stable (double) Schubert classes expressing them in terms of a symmetric function part and a finite part;

• – a novel definition of double and triple Stanley symmetric functions;

• – a proof of the positivity of double Edelman–Greene coefficients generalizing the results of Edelman–Greene and Lascoux–Schützenberger;

• – the definition of a new class of bumpless pipedreams, giving new formulae for double Schubert polynomials, back stable double Schubert polynomials, and a new form of the Edelman–Greene insertion algorithm;

• – the construction of the Peterson subalgebra of the infinite nilHecke algebra, extending work of Peterson in the affine case;

• – equivariant Pieri rules for the homology of the infinite Grassmannian;

• – homology divided difference operators that create the equivariant homology Schubert classes of the infinite Grassmannian.

In an earlier paper we developed an algorithm for computing all integral points on elliptic curves over the rationals Q. Here we illustrate our method by applying it to Mordell‘s Equation y$^2$=x$^3+$k for 0 ≠ k ∈ Z and draw some conclusions from our numerical findings. In fact we solve Mordell‘s Equation in Z for all integers k within the range 0 < | k | [les ] 10 000 and partially extend the computations to 0 < | k | [les ] 100 000. For these values of k, the constant in Hall‘s conjecture turns out to be C=5. Some other interesting observations are made concerning large integer points, large generators of the Mordell–Weil group and large Tate–Shafarevič groups. Three graphs illustrate the distribution of integer points in dependence on the parameter k. One interesting feature is the occurrence of lines in the graphs.

We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker–Fourier coefficients of automorphic representations. For $\text{GL}_{n}(\mathbb{F})$ this implies that a smooth admissible representation $\unicode[STIX]{x1D70B}$ has a generalized Whittaker model ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ corresponding to a nilpotent coadjoint orbit ${\mathcal{O}}$ if and only if ${\mathcal{O}}$ lies in the (closure of) the wave-front set $\operatorname{WF}(\unicode[STIX]{x1D70B})$. Previously this was only known to hold for $\mathbb{F}$ non-archimedean and ${\mathcal{O}}$ maximal in $\operatorname{WF}(\unicode[STIX]{x1D70B})$, see Moeglin and Waldspurger [Modeles de Whittaker degeneres pour des groupes p-adiques, Math. Z. 196 (1987), 427–452]. We also express ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ as an iteration of a version of the Bernstein–Zelevinsky derivatives [Bernstein and Zelevinsky, Induced representations of reductive p-adic groups. I., Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 441–472; Aizenbud et al., Derivatives for representations of$\text{GL}(n,\mathbb{R})$and$\text{GL}(n,\mathbb{C})$, Israel J. Math. 206 (2015), 1–38]. This enables us to extend to $\text{GL}_{n}(\mathbb{R})$ and $\text{GL}_{n}(\mathbb{C})$ several further results by Moeglin and Waldspurger on the dimension of ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ and on the exactness of the generalized Whittaker functor.

For the p-adic Galois representation associated to a Hilbert modular form, Carayol has shown that, under a certain assumption, its restriction to the local Galois group at a finite place not dividing p is compatible with the local Langlands correspondence. Under the same assumption, we show that the same is true for the places dividing p, in the sense of p-adic Hodge theory, as is shown for an elliptic modular form. We also prove that the monodromy-weight conjecture holds for such representations.

Let $S$ be a Noetherian scheme of finite dimension and denote by $\unicode[STIX]{x1D70C}\in [\unicode[STIX]{x1D7D9},\mathbb{G}_{m}]_{\mathbf{SH}(S)}$ the (additive inverse of the) morphism corresponding to $-1\in {\mathcal{O}}^{\times }(S)$ . Here $\mathbf{SH}(S)$ denotes the motivic stable homotopy category. We show that the category obtained by inverting $\unicode[STIX]{x1D70C}$ in $\mathbf{SH}(S)$ is canonically equivalent to the (simplicial) local stable homotopy category of the site $S_{\text{r}\acute{\text{e}}\text{t}}$ , by which we mean the small real étale site of $S$ , comprised of étale schemes over $S$ with the real étale topology. One immediate application is that $\mathbf{SH}(\mathbb{R})[\unicode[STIX]{x1D70C}^{-1}]$ is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the $\unicode[STIX]{x1D70C}$ -local sphere (over $\mathbb{R}$ ). As further applications we show that $D_{\mathbb{A}^{1}}(k,\mathbb{Z}[1/2])^{-}\simeq \mathbf{DM}_{W}(k)[1/2]$ (improving a result of Ananyevskiy–Levine–Panin), reprove Röndigs’ result that $\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9}[1/\unicode[STIX]{x1D702},1/2])=0$ for $i=1,2$ and establish some new rigidity results.

This paper develops a general theory of canonical bases and how they arise naturally in the context of categorification. As an application, we show that Lusztig’s canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is of finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest- and highest-weight integrable representations. This generalizes past work of the author’s in the highest-weight case.

We introduce an essentially new Grothendieck topology, the Weil-étale topology, on schemes over finite fields. The cohomology groups associated with this topology should behave better than the standard étale cohomology groups. In particular there is a very natural definition of an Euler characteristic and a plausible conjecture relating the Euler characteristic of Z to the value of the zeta-function at s = 0. This conjecture is proved in certain cases.

We prove a finiteness result in the space of arcs $X_\infty$ of a singular variety X, which is an extension of the stability result of Denef and Loeser (Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math. 135 (1999), 201–232). From this follows a curve selection lemma for generically stable subsets of $X_\infty$. As a consequence, we extend to all dimensions the problem of wedges, proposed by Lejeune-Jalabert (Arcs analytiques et résolution minimale des singularités des surfaces quasi-homogénes, Lecture Notes in Mathematics, vol. 777 (Springer, Berlin, 1980), 303–336), and we obtain that an affirmative answer to this problem is equivalent to the surjectivity of the Nash map. This implies, for instance, that the Nash map is bijective for sandwiched surface singularities.

Let K be a CM number field and GK its absolute Galois group. A representation of GK is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of GK have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GLn(𝔸K), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1 . In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GLn (𝔸F) when F is a totally real number field.

This work establishes a comparison between functions on derived loop spaces (Toën and Vezzosi, Chern character, loop spaces and derived algebraic geometry, in Algebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Østvær (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. If A is a smooth commutative k-algebra and k has characteristic 0, we show that two objects, S1⊗A and ϵ(A), determine one another, functorially in A. The object S1⊗A is the S1-equivariant simplicial k-algebra obtained by tensoring A by the simplicial group S1 :=Bℤ, while the object ϵ(A) is the de Rham algebra of A, endowed with the de Rham differential, and viewed as a ϵ-dg-algebra (see the main text). We define an equivalence φ between the homotopy theory of simplicial commutative S1-equivariant k-algebras and the homotopy theory of ϵ-dg-algebras, and we show the existence of a functorial equivalence ϕ(S1 ⊗A)∼ϵ(A) . We deduce from this the comparison mentioned above, identifying the S1-equivariant functions on the derived loop space LX of a smooth k-scheme X with the algebraic de Rham cohomology of X/k. As corollaries, we obtain functorial and multiplicative versions of decomposition theorems for Hochschild homology (in the spirit of Hochschild–Kostant–Rosenberg) for arbitrary semi-separated k-schemes. By construction, these decompositions are moreover compatible with the S1-action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand.

Let $\nu _{f}(n)$ be the $n\mathrm{th}$ normalized Fourier coefficient of a Hecke–Maass cusp form $f$ for ${\rm SL }(2,\mathbb{Z})$ and let $\alpha $ be a real number. We prove strong oscillations of the argument of $\nu _{f}(n)\mu (n) \exp (2\pi i n \alpha )$ as $n$ takes consecutive integral values.

We compute the quiver of any finite monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as DO) to representation-theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of ℛ-trivial monoids, we also provide a semigroup-theoretic description of the projective indecomposable modules and compute the Cartan matrix.

We study dynamics of rational maps of degree at least 2 with coefficients in the field ${\open C}_{p}$, where p>1 is a fixed prime number. The main ingredient is to consider the action of rational maps in p-adic hyperbolic space, denoted ${\open H}_{p}$. Hyperbolic space ${\open H}_{p}$ is provided with a natural distance, for which it is connected and one dimensional (an ${\open R}$-tree). These advantages with respect to ${\open C}_{p}$ give new insight into dynamics. In this paper we prove the following results about periodic points; we give applications to the Fatou/Julia theory over ${\open C}_{p}$ and to ultrametric analysis in forthcoming papers. We prove that the existence of at least two nonrepelling periodic points implies the existence of infinitely many of them. This is in contrast with the complex setting where a rational map can have at most finitely many nonrepelling periodic points. On the other hand we prove that every rational map has a repelling fixed point, either in the projective line or in hyperbolic space. So the topological expansion of a rational map is detected by some fixed point.