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This paper is the first in a series of two dedicated to the study of period relations of the type L ( 1 2 + k , Π ) ∈ ( 2 π i ) d ⋅ k Ω ( − 1 ) k \bf Q ( Π ) , 1 2 + k critical , \begin{equation*} L\Big (\frac {1}{2}+k,\Pi \Big )\;\in \;(2\pi i)^{d\cdot k}\Omega _{(-1)^k}\textrm {\bf Q}(\Pi ),\quad \frac {1}{2}+k\;\text {critical}, \end{equation*} for certain automorphic representations Π \Pi of a reductive group G . G. In this paper we discuss the case G = G L ( n + 1 ) × G L ( n ) . G=\mathrm {GL}(n+1)\times \mathrm {GL}(n). The case G = G L ( 2 n ) G=\mathrm {GL}(2n) is discussed in part two. Our method is representation theoretic and relies on the author’s recent results on global rational structures on automorphic representations. We show that the above period relations are intimately related to the field of definition of the global representation Π \Pi under consideration. The new period relations we prove are in accordance with Deligne’s Conjecture on special values of L L -functions, and the author expects this method to apply to other cases as well.

In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is contained in the singular set. Combining this result with the results of the first paper in this series, the author obtains asymptotic results on the occurrence of tempered representations in induction and restriction problems for real, reductive algebraic groups., Accepted to Transactions of the American Mathematical Society

In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $\mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.

In this paper, we consider Sjamaar's holomorphic slice theorem, the birational equivalence theorem of Guillemin and Sternberg, and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the K\"ahler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, the theory extends if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. However, we also believe that the setting we work in is intrinsically interesting, and elucidates the key role played by the following fact: the moment image of $e^t \cdot x$ increases as $t \in \mathbb{R}$ increases., Comment: 25 pages

In this paper we discuss a simple relation, which was previously missed, between the high co-dimensional isoperimetric problem of finding a filling with small volume to a given cycle and extinction estimates for singular, high co-dimensional, mean curvature flow. The utility of this viewpoint is first exemplified by two results which, once casted in the light of this relation, are almost self-evident. The first is a genuine, 5-line proof, for the isoperimetric inequality for k k -cycles in R n \mathbb {R}^n , with a constant differing from the optimal constant by a factor of only k \sqrt {k} , as opposed to a factor of k k k^k produced by all of the other soft methods. The second is a 3-line proof of a lower bound for extinction for arbitrary co-dimensional, singular, mean curvature flows starting from cycles, generalizing the main result of Giga and Yama-uchi (1993). We then turn to use the above-mentioned relation to prove a bound on the parabolic Hausdorff measure of the space-time track of high co-dimensional, singular, mean curvature flow starting from a cycle, in terms of the mass of that cycle. This bound is also reminiscent of a Michael-Simon isoperimetric inequality. To prove it, we are led to study the geometric measure theory of Euclidean rectifiable sets in parabolic space and prove a co-area formula in that setting. This formula, the proof of which occupies most of this paper, may be of independent interest.

We study the transverse Lusternik-Schnirelmann category theory of a Riemannian foliation F on a closed manifold M. The essential transverse category cat e (M, F) is introduced in this paper, and we prove that cat e (M, F) is always finite for a Riemannian foliation. Necessary and sufficient conditions are derived for when the usual transverse category cat (M, F) is finite, and thus cat e (M, F) = cat(M, F) holds. A fundamental point of this paper is to use properties of Riemannian submersions and the Molino Structure Theory for Riemannian foliations to transform the calculation of cat e (M, F) into a standard problem about O(q)-equivariant LS category theory. A main result, Theorem 1.6, states that for an associated O(q)-manifold W, we have that cat e (M, F) = cat O(q) (Ŵ). Hence, the traditional techniques developed for the study of smooth compact Lie group actions can be effectively employed for the study of the LS category of Riemannian foliations. A generalization of the Lusternik-Schnirelmann theorem is derived: given a C 1 -function f: M → R which is constant along the leaves of a Riemannian foliation F, the essential transverse category cat e (M, F) is a lower bound for the number of critical leaf closures of f.

Let F ^ \hat {F} be a free pro- p p non-abelian group, and let Δ \Delta be a commutative Noetherian complete local ring with a maximal ideal I I such that c h a r ( Δ / I ) = p > 0 \mathrm {char}(\Delta /I)=p>0 . Zubkov [Sibirsk. Mat. Zh. 28 (1987), pp. 64–69] showed that when p ≠ 2 p\neq 2 , the pro- p p congruence subgroup \[ G L 2 1 ( Δ ) = ker ⁡ ( G L 2 ( Δ ) ⟶ Δ → Δ / I G L 2 ( Δ / I ) ) GL_{2}^{1}(\Delta )=\ker (GL_{2}(\Delta )\overset {\Delta \to \Delta /I}{\longrightarrow }GL_{2}(\Delta /I)) \] admits a pro- p p identity, i.e., there exists an element 1 ≠ w ∈ F ^ 1\neq w\in \hat {F} that vanishes under any continuous homomorphism F ^ → G L 2 1 ( Δ ) \hat {F}\to GL_{2}^{1}(\Delta ) . In this paper we investigate the case p = 2 p=2 . The main result is that when c h a r ( Δ ) = 2 \mathrm {char}(\Delta )=2 , the pro- 2 2 group G L 2 1 ( Δ ) GL_{2}^{1}(\Delta ) admits a pro- 2 2 identity. This result was obtained by the use of trace identities that originate in PI-theory.

We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. “Point” means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of a solution, blow-up occurrence, and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e., we show that the global solution scatters as t → ± ∞ t\to \pm \infty in the L 2 L^2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

Let G G be a connected, linear, real reductive Lie group with compact centre. Let K > G K>G be compact. Under a condition on K K , which holds in particular if K K is maximal compact, we give a geometric expression for the multiplicities of the K K -types of any tempered representation (in fact, any standard representation) π \pi of G G . This expression is in the spirit of Kirillov’s orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of π | K \pi |_K obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of SU ( p , 1 ) \textrm {SU}(p,1) , SO 0 ( p , 1 ) \textrm {SO}_0(p,1) , and SO 0 ( 2 , 2 ) \textrm {SO}_0(2,2) restrict multiplicity freely to maximal compact subgroups.

Let f f and g g be algebraically independent entire functions. We first give an estimate of the Nevanlinna counting function for the common zeros of f n − 1 f^n-1 and g n − 1 g^n-1 for sufficiently large n n . We then apply this estimate to study divisible sequences in the sense that f n − 1 f^n-1 is divisible by g n − 1 g^n-1 for infinitely many n n . For the first part of establishing our gcd estimate, we need to formulate a truncated second main theorem for effective divisors by modifying a theorem from a paper by Hussein and Ru and explicitly computing the constants involved for a blowup of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 along a point with its canonical divisor and the pull-back of vertical and horizontal divisors of P 1 × P 1 \mathbb {P}^1\times \mathbb {P}^1 .

In this paper we engage in a general study of the asymptoticexpansion of the Witten–Reshetikhin–Turaev invariants of mapping tori ofsurface mapping class group elements. We use the geometric constructionof the Witten–Reshetikhin–Turaev topological quantum field theory via thegeometric quantization of moduli spaces of flat connections on surfaces. Weidentify assumptions on the mapping class group elements that allow us toprovide a full asymptotic expansion. In particular, we show that our resultsapply to all pseudo-Anosov mapping classes on a punctured torus and show byexample that our assumptions on the mapping class group elements are strictlyweaker than hitherto successfully considered assumptions in this context. Theproof of our main theorem relies on our new results regarding asymptoticexpansions of oscillatory integrals, which allows us to go significantly beyondthe standard transversely cut out assumption on the fixed point set. Thismakes use of the Picard–Lefschetz theory for Laplace integrals.

In this paper, we consider representations of the absolute Galois group Gal ( Q ¯ / Q ) \text {Gal}(\overline {\mathbb Q}/\mathbb Q) attached to modular forms for noncongruence subgroups of SL 2 ( Z ) \text {SL}_2(\mathbb Z) . When the underlying modular curves have a model over Q \mathbb {Q} , these representations are constructed by Scholl in [Invent. Math. 99 (1985), pp. 49–77] and are referred to as Scholl representations, which form a large class of motivic Galois representations. In particular, by a result of Belyi, Scholl representations include the Galois actions on the Jacobian varieties of algebraic curves defined over Q \mathbb Q . As Scholl representations are motivic, they are expected to correspond to automorphic representations according to the Langlands philosophy. Using recent developments on automorphy lifting theorem, we obtain various automorphy and potential automorphy results for potentially G L 2 \mathrm {GL}_2 -type Galois representations associated to noncongruence modular forms. Our results are applied to various kinds of examples. In particular, we obtain potential automorphy results for Galois representations attached to an infinite family of spaces of weight 3 noncongruence cusp forms of arbitrarily large dimensions.

In this paper we generalize a result of Galloway and Mendes in two different situations: in the first case for marginally outer trapped surfaces (MOTSs) of genus greater than 1 1 and, in the second case, for MOTSs of high dimension with negative σ \sigma -constant. In both cases we obtain a splitting result for the ambient manifold when it contains a stable closed MOTS which saturates a lower bound for the area (in dimension 2 2 ) or for the volume (in dimension ≥ 3 \ge 3 ). These results are extensions of theorems of Nunes and Moraru to general (non-time-symmetric) initial data sets.

In this paper we extend a recent breakthrough of Chen and Cheng on the existence of a constant scalar Kähler metric on a compact Kähler manifold to Calabi’s extremal metric. There are no new a priori estimates needed, but rather there are necessary modifications adapted to the extremal case. We prove that there exists an extremal metric with extremal vector V V if and only if the modified Mabuchi energy is proper, modulo the action of the subgroup in the identity component of the automorphism group which commutes with the flow of V V . We introduce two essentially equivalent notions, called reductive properness and reduced properness. We observe that one can test reductive properness/reduced properness only for invariant metrics. We prove that existence of an extremal metric is equivalent to reductive properness/reduced properness of the modified Mabuchi energy.

In this paper we give a structure theorem for the module of vector valued modular forms in the case of a three dimensional ball with the action of the Picard modular group Γ 3 [ − 3 ] \Gamma _3 [\sqrt {-3}] . The corresponding modular variety of dimension 3 3 is a copy of the Segre cubic.

A scalar sequence ξ \xi is said to be admissible for a positive operator A A if A = ∑ ξ j P j A= \sum \xi _jP_j for some rank-one projections P j P_j , or, equivalently, if diag ξ \xi is the diagonal of V A V ∗ VAV^* for some partial isometry V V having as domain the closure of the range of A A . The main result of this paper is that if A A is the sum of infinitely many projections (converging in the strong operator topology) and ξ \xi is a nonsummable sequence in [ 0 , 1 ] [0,1] that satisfies the Kadison condition that requires that either ∑ { ξ i ∣ ξ i ≤ 1 2 } + ∑ { ( 1 − ξ i ) ∣ ξ i > 1 2 } = ∞ \sum \{\xi _i \mid \xi _i\le \frac {1}{2}\}+ \sum \{(1-\xi _i) \mid \xi _i> \frac {1}{2}\} = \infty or the difference ∑ { ξ i ∣ ξ i ≤ 1 2 } − ∑ { ( 1 − ξ i ) ∣ ξ i > 1 2 } \sum \{\xi _i \mid \xi _i\le \frac {1}{2}\}- \sum \{(1-\xi _i) \mid \xi _i> \frac {1}{2}\} is an integer, then ξ \xi is admissible for A A . This result extends Kadison’s carpenter’s theorem and provides an independent proof of it.

In recent papers Masser and Zannier have proved various results of “relative Manin–Mumford” type for various families of abelian varieties, some with field of definition restricted to the algebraic numbers. Typically these imply the finiteness of the set of torsion points on a curve in the family. After Bertrand, Masser, and Zannier discovered some surprising counterexamples for multiplicative extensions of elliptic families, the three authors together with Pillay settled completely the situation for this case over the algebraic numbers. Here we treat the last remaining case of surfaces, that of additive extensions of elliptic families, and even over the field of all complex numbers. In particular analogous counterexamples do not exist. There are finiteness consequences for Pell’s equation over polynomial rings and integration in elementary terms. Our work can be made effective (as opposed to most of that preceding), mainly because we use counting results only for analytic curves.

In this paper we study the singular set of Dirichlet-minimizing $Q$-valued maps from $\mathbb{R}^m$ into a smooth compact manifold $\mathcal{N}$ without boundary. Similarly to what happens in the case of single valued minimizing harmonic maps, we show that this set is always $(m-3)$-rectifiable with uniform Minkowski bounds. Moreover, as opposed to the single valued case, we prove that the target $\mathcal{N}$ being non-positively curved but not simply connected does not imply continuity of the map., Comment: 43 pages

For certain quasi-split reductive groups $G$ over a general field $F$, we construct an automorphism $\iota_G$ of $G$ over $F$, well-defined as an element of ${\rm Aut}(G)(F)/jG(F)$ where $j:G(F) \rightarrow {\rm Aut}(G)(F)$ is the inner-conjugation action of $G(F)$ on $G$. The automorphism $\iota_G$ generalizes (although only for quasi-split groups) an involution due to Moeglin-Vigneras-Waldspurger in [MVW] for classical groups which takes any irreducible admissible representation $\pi$ of $G(F)$ for $G$ a classical group and $F$ a local field, to its contragredient $\pi^\vee$. The paper also formulates a conjecture on the contragredient of an irreducible admissible representation of $G(F)$ for $G$ a reductive algebraic group over a local field $F$ in terms of the (enhanced) Langlands parameter of the representation., Comment: arXiv admin note: substantial text overlap with arXiv:1512.04347

In this paper we prove the rectifiability of and measure bounds on the singular set of the free boundary for minimizers of a functional first considered by Alt-Caffarelli. Our main tools are the Quantitative Stratification and Rectifiable-Reifenberg framework of Naber-Valtorta, which allow us to do a type of "effective dimension-reduction." The arguments are sufficiently robust that they apply to a broad class of related free boundary problems as well., Comment: This version has minor corrections and additional references

We use the progenerator constructed in our previous paper to give a necessary condition for a simple module of a finite reductive group to be cuspidal, or more generally to obtain information on which Harish-Chandra series it can lie in. As a first application we show the irreducibility of the smallest unipotent character in any Harish-Chandra series. Secondly, we determine a unitriangular approximation to part of the unipotent decomposition matrix of finite orthogonal groups and prove a gap result on certain Brauer character degrees., Comment: arXiv admin note: text overlap with arXiv:1611.07373

In this paper we prove undecidability of finite systems of equations in free Lie algebras of rank at least three over an arbitrary field. We show that the ring of integers $\mathbb{Z}$ is interpretable by positive existential formulas in such free Lie algebras over a field of characteristic zero., Comment: arXiv admin note: text overlap with arXiv:1606.03617

In this paper, we use Borcherds lifting and the big CM value formula of Bruinier, Kudla, and Yang to give an explicit factorization formula for the norm of $\Psi(\frac{d_1+\sqrt{d_1}}2) -\Psi(\frac{d_2+\sqrt{d_2}}2)$, where $\Psi$ is the $j$-invariant or the Weber invariant $\omega_2$. The $j$-invariant case gives another proof of the well-known Gross-Zagier factorization formula of singular moduli, while the Weber invariant case gives a proof of the Yui-Zagier conjecture for $\omega_2$. The method used here could be extended to deal with other modular functions on a genus zero modular curve., Comment: accepted to appear in Trans. AMS

In this paper, we consider the problem of solving the Cauchy–Riemann equation with prescribed support in a domain of a complex manifold for forms or currents. We are especially interested in the case when the domain is a Hartogs triangle in C 2 \mathbb {C}^2 or C P 2 \mathbb {C}\mathbb {P}^2 . In particular, we show that the strong L 2 L^2 Dolbeault cohomology group on the Hartogs triangle in C P 2 \mathbb {C}\mathbb {P}^2 is infinitely dimensional.

Magnus proved that, given two elements $x$ and $y$ of a finitely generated free group $F$ with equal normal closures $\langle x\rangle^F=\langle y\rangle^F$, then $x$ is conjugated either to $y$ or $y^{-1}$. More recently, this property, called the Magnus property, has been generalized to oriented surface groups. In this paper, we consider an analogue property for profinite surface groups. While Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for ${\mathscr S}$ a class of finite groups, we prove that, if $x$ and $y$ are \emph{algebraically simple} elements of the pro-${\mathscr S}$ completion $\hat{\Pi}^{\mathscr S}$ of an orientable surface group $\Pi$, such that, for all $n\in{\mathbb N}$, there holds $\langle x^n\rangle^{\hat{\Pi}^{\mathscr S}}=\langle y^n\rangle^{\hat{\Pi}^{\mathscr S}}$, then $x$ is conjugated to $y^s$ for some $s\in(\hat{\mathbb Z}^{\mathscr S})^\ast$. As a matter of fact, a much more general property is proved and further extended to a wider class of profinite completions. The most important application of the theory above is a generalization of the description of centralizers of profinite Dehn twists to profinite Dehn multitwists., Comment: 27 pages. Final version, to appear on Transactions of the American Mathematical Society

We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multiplicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two type of forms which does not exist in lower dimensions. Finally, we produce some counter-examples concerning Carleson measures on the infinite polydisc., Comment: This paper has been accepted for publication in Transactions of the AMS

Given two cyclic A ∞ A_\infty -algebras A A and B B , in this paper we prove that there exists a cyclic A ∞ A_\infty -algebra structure on their tensor product A ⊗ B A\otimes B which is unique up to a cyclic A ∞ A_\infty -quasi-isomorphism. Furthermore, the Kontsevich class of A ⊗ B A\otimes B is equal to the cup product of the Kontsevich classes of A A and B B on the moduli space of curves.

Let X X be a space of homogeneous type and let L \mathfrak {L} be a nonnegative self-adjoint operator on L 2 ( X ) L^2(X) enjoying Gaussian estimates. The main aim of this paper is twofold. Firstly, we prove (local) nontangential and radial maximal function characterizations for the local Hardy spaces associated to L \mathfrak {L} . This gives the maximal function characterization for local Hardy spaces in the sense of Coifman and Weiss provided that L \mathfrak {L} satisfies certain extra conditions. Secondly we introduce local Hardy spaces associated with a critical function ρ \rho which are motivated by the theory of Hardy spaces related to Schrödinger operators and of which include the local Hardy spaces of Coifman and Weiss as a special case. We then prove that these local Hardy spaces can be characterized by (local) nontangential and radial maximal functions related to L \mathfrak {L} and ρ \rho , and by global maximal functions associated to ‘perturbations’ of L \mathfrak {L} . We apply our theory to obtain a number of new results on maximal characterizations for the local Hardy type spaces in various settings ranging from Schrödinger operators on manifolds to Schrödinger operators on connected and simply connected nilpotent Lie groups.

In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex. One consequence of our general result is the following: for any dimension there exists some ϵ > 0 \epsilon > 0 so that if the squeezing function on a smoothly bounded convex domain is greater than 1 − ϵ 1-\epsilon outside a compact set, then the domain is strongly pseudoconvex (and hence the squeezing function limits to one on the boundary). Another consequence is the following: for any dimension d d there exists some ϵ > 0 \epsilon > 0 so that if the holomorphic sectional curvature of the Bergman metric on a smoothly bounded convex domain is within ϵ \epsilon of − 4 / ( d + 1 ) -4/(d+1) outside a compact set, then the domain is strongly pseudoconvex (and hence the holomorphic sectional curvature limits to − 4 / ( d + 1 ) -4/(d+1) on the boundary).

At the 1987 Ramanujan Centenary meeting Dyson asked for a coherent group-theoretical structure for Ramanujan's mock theta functions analogous to Hecke's theory of modular forms. Many of Ramanujan's mock theta functions can be written in terms of $R(\zeta,q)$, where $R(z,q)$ is the two-variable generating function of Dyson's rank function and $\zeta$ is a root of unity. Building on earlier work of Watson, Zwegers, Gordon and McIntosh, and motivated by Dyson's question, Bringmann, Ono and Rhoades studied transformation properties of $R(\zeta,q)$. In this paper we strengthen and extend the results of Bringmann, Rhoades and Ono, and the later work of Ahlgren and Treneer. As an application we give a new proof of Dyson's rank conjecture and show that Ramanujan's Dyson rank identity modulo $5$ from the Lost Notebook has an analogue for all primes greater than $3$. The proof of this analogue was inspired by recent work of Jennings-Shaffer on overpartition rank differences mod $7$.

This paper examines a general method for producing twists of a comodule algebra by tensoring it with a torsor then taking co-invariants. We examine the properties that pass from the original algebra to the twisted algebra and vice versa. We then examine the special case where the algebra is a 4-dimensional Sklyanin algebra viewed as a comodule algebra over the Hopf algebra of functions on the non-cyclic group of order 4 with the torsor being the 2x2 matrix algebra. The twisted algebra is an "exotic elliptic algebra". We show that the twisted algebra has many of the good properties that the Sklyanin algebra has, and that it has some new properties that make it quite unusual by comparison.

In this paper, we will introduce notions of relative version of imprimitivity bimodules and relative version of strong Morita equivalence for pairs of $C^*$-algebras $(\mathcal{A}, \mathcal{D})$ such that $\mathcal{D}$ is a $C^*$-subalgebra of $\mathcal{A}$ with certain conditions. We will then prove that two pairs $(\mathcal{A}_1, \mathcal{D}_1)$ and $(\mathcal{A}_2, \mathcal{D}_2)$ are relatively Morita equivalent if and only if their relative stabilizations are isomorphic. In particularly, for two pairs $(\mathcal{O}_A, \mathcal{D}_A)$ and $(\mathcal{O}_B, \mathcal{D}_B)$ of Cuntz--Krieger algebras with their canonical masas, they are relatively Morita equivalent if and only if their underlying two-sided topological Markov shifts $(\bar{X}_A,\bar{\sigma}_A)$ and $(\bar{X}_B,\bar{\sigma}_B)$ are flow equivalent. We also introduce a relative version of the Picard group ${\operatorname{Pic}}(\mathcal{A}, \mathcal{D})$ for the pair $(\mathcal{A}, \mathcal{D})$ of $C^*$-algebras and study them for the Cuntz--Krieger pair $(\mathcal{O}_A, \mathcal{D}_A)$., Comment: 38 pages

In [J. Inst. Math. Jussieu 14 (2015), 149–184] and [Int. Math. Res. Not. IMRN 7 (2017), 2014–2099] a family of Rankin-Selberg integrals was shown to represent the twisted standard L \mathcal {L} -function L ( s , π , χ , s t ) \mathcal {L}(s,\pi ,\chi ,\mathfrak {st}) of a cuspidal representation π \pi of the exceptional group of type G 2 G_2 . These integral representations bind the analytic behavior of this L \mathcal {L} -function with that of a family of degenerate Eisenstein series for quasi-split forms of S p i n 8 Spin_8 associated to an induction from a character on the Heisenberg parabolic subgroup. This paper is divided into two parts. In Part 1 we study the poles of these degenerate Eisenstein series in the right half-plane R e ( s ) > 0 \mathfrak {Re}(s)>0 . In Part 2 we use the results of Part 1 to prove the conjecture, made by J. Hundley and D. Ginzburg in [Israel J. Math. 207 (2015), 835–879], for stable poles and also to give a criterion for π \pi to be a CAP representation with respect to the Borel subgroup of G 2 G_2 in terms of the analytic behavior of L ( s , π , χ , s t ) \mathcal {L}(s,\pi ,\chi ,\mathfrak {st}) at s = 3 2 s=\frac {3}{2} .

In this paper, we study A 1 \mathbb {A}^1 -equivalence classes of zero cycles on open algebraic surfaces. We prove the logarithmic version of Mumford’s theorem on zero cycles. We also prove that the log Bloch conjecture holds for surfaces with log Kodaira dimension − ∞ -\infty .

We present an algorithm which given a presentation of a group $G$ without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant ${\kappa}$, outputs a collection of tracks in an appropriate presentation complex. We give two applications: the first is an algorithm which decides if $G$ admits an essential free decomposition, the second is an algorithm which; if $G$ is relatively hyperbolic; decides if it admits an essential elementary splitting., Comment: This is a rewritten version of the paper "Finding tracks in 2-complexes". The statements of the main theorems are unchanged and the proof is essentially the same, but the presentation has been substantially improved. To appear in Transactions of the American Mathematical Society

In this paper, we prove the irreducibility of global descents for even unitary groups. More generally, through Fourier-Jacobi coefficients of automorphic forms, we give a bijection between a certain set of irreducible cuspidal automorphic representations of U ( n , n ) ( A ) \mathrm {U}(n,n)(\mathbb {A}) and a certain set of irreducible square-integrable automorphic representations of U ( 2 n , 2 n ) ( A ) \mathrm {U}(2n, 2n)(\mathbb {A}) . We also give three applications of the irreducibility of global descents. As a global application, we prove a rigidity theorem for irreducible generic cuspidal automorphic representations of U ( n , n ) \mathrm {U}(n,n) . Moreover, as a local application, we prove the irreducibility of explicit local descents for a couple of supercuspidal representations and a local converse theorem for generic representations in the case of U ( n , n ) \mathrm {U}(n,n) .

In this paper we discuss the existence of certain classes of cuspidal automorphic representations having non-zero Fourier coefficients for a general semisimple algebraic group $ G$ defined over a number field $ k$ such that its Archimedean group $ G_\infty $ is not compact. When $ G$ is quasi-split over $ k$, we obtain a result on existence of generic cuspidal automorphic representations which generalize results of Vignéras, Henniart, and Shahidi. We also discuss: (i) the existence of cuspidal automorphic forms with non-zero Fourier coefficients for congruence of subgroups of $ G_\infty $, and (ii) applications related to the work of Bushnell and Henniart on generalized Whittaker models.

Let S S be an orientable surface with negative Euler characteristic. For k ∈ N k \in \mathbb {N} , let C k ( S ) \mathcal {C}_{k}(S) denote the k-curve graph, whose vertices are isotopy classes of essential simple closed curves on S S and whose edges correspond to pairs of curves that can be realized to intersect at most k k times. The theme of this paper is that the geometry of Teichmüller space and of the mapping class group captures local combinatorial properties of C k ( S ) \mathcal {C}_{k}(S) , for large k k . Using techniques for measuring distance in Teichmüller space, we obtain upper bounds on the following three quantities for large k k : the clique number of C k ( S ) \mathcal {C}_{k}(S) (exponential in k k , which improves on previous bounds of Juvan, Malnič, and Mobar and Przytycki); the maximum size of the intersection, whenever it is finite, of a pair of links in C k \mathcal {C}_{k} (quasi-polynomial in k k ); and the diameter in C 0 ( S ) \mathcal {C}_{0}(S) of a large clique of C k ( S ) \mathcal {C}_{k}(S) (uniformly bounded). As an application, we obtain quasi-polynomial upper bounds, depending only on the topology of S S , on the number of short simple closed geodesics on any unit-square tiled surface homeomorphic to S S .

In this paper, we analyze the asymptotic behavior of $\kappa$-noncollapsed and positively curved steady Ricci solitons and prove that any $n$-dimensional $\kappa$-noncollapsed steady K\"ahler-Ricci soliton with non-negative sectional curvature must be flat., Comment: This is a final version. We added some details in the proofs of Proposition 4.3 and Corollary 4.5

A Cayley octad is a set of 8 points in P3 which are the base locus of a net of quadrics. Blowing up the points of the octad gives a morphism to P2 defined by the net; the fibres of this morphism are intersections of two quadrics in the net, hence curves of genus 1. The generic fibre therefore has a group structure, and the action of this group on itself extends to a birational action on the whole variety. In particular, if the generic fibre has a large group of rational points, the birational automorphism group, and hence the birational geometry, of the variety must be complicated. It is natural to ask whether the converse is true: if the generic fibre has only a small group of rational points, is the birational geometry of the variety correspondingly simple? In this paper we study a special Cayley octad with the property that the generic fibre has finitely many rational points. In Section 1 we find that such an octad only exists in characteristic 2, and is unique up to projective transformations. Our main results then show that the simplicity of the generic fibre is indeed reected in the simplicity of the birational geometry of our blowup. In Section 2 we show that the cones of nef and movable divisors are rational polyhedral, as predicted by the Morrison{Kawamata conjecture. Finally, in Section 3 we prove that our blowup has the \best possible" birational geometric properties: it is a Mori dream space.

We work over an algebraically closed field of characteristic zero. The purpose of this paper is to characterize a nondegenerate projective variety X X with a linear projection which induces a nonbirational map to its image. As an application, for smooth X X of degree d d and codimension e e , we prove the “semiampleness” of the ( d − e + 1 ) (d-e+1) th twist of the ideal sheaf. This improves a linear bound of the regularity of smooth projective varieties by Bayer–Mumford–Bertram–Ein–Lazarsfeld, and gives an asymptotic regularity bound.

In this paper, we study the Heegner points on more general modular curves other than $X_0(N)$, which generalizes Gross' work "Heegner points on $X_0(N)$". The explicit Gross-Zagier formula and the Euler system property are stated in this case. Using such kind of Heegner points, we construct certain families of quadratic twists of $X_0(36)$, with the ranks of Mordell-Weil groups being zero and one respectively, and show that the $2$-part of their BSD conjectures hold., comments are welcome

The purpose of this paper is to lay the foundations of a theory of invariants in \'etale cohomology for smooth Artin stacks. We compute the invariants in the case of the stack of elliptic curves, and we use the theory we developed to get some results regarding Brauer groups of algebraic spaces., Comment: final version

In this paper we prove that the space of flat metrics (nonpositively curved Euclidean cone metrics) on a closed, oriented surface is marked-length-spectrally rigid. In other words, two flat metrics assigning the same lengths to all closed curves differ by an isometry isotopic to the identity. The novel proof suggests a stronger rigidity conjecture for this class of metrics.

We construct several new spaces of quantum sequences and their quantum families of maps in the sense of Sołtan. The noncommutative distributional symmetries associated with these quantum maps are noncommutative versions of spreadability and partial exchangeability. Then, we study simple relations between these symmetries. We will focus on studying two kinds of noncommutative distributional symmetries: monotone spreadability and boolean spreadability. We provide an example of a spreadable sequence of random variables for which the usual unilateral shift is an unbounded map. As a result, it is natural to study bilateral sequences of random objects, which are indexed by integers, rather than unilateral sequences. At the end of the paper, we will show Ryll-Nardzewski type theorems for monotone independence and boolean independence: Roughly speaking, an infinite bilateral sequence of random variables is monotonically (boolean) spreadable if and only if the variables are identically distributed and monotone (boolean) with respect to the conditional expectation onto its tail algebra. For an infinite sequence of noncommutative random variables, boolean spreadability is equivalent to boolean exchangeability.

This note identifies errors in the joint paper [Trans. Amer. Math. Soc. 355 (2003), 813–836] by the authors and T. Constantinescu. The Main Theorem requires a stronger hypothesis. The applications to interpolation problems are affected by this change; some survive, but others do not and are withdrawn.

In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic $n$-orbifold, $n\ge 4$. Many of the results are more general and apply to locally symmetric spaces associated to arithmetic lattices in $\mathbb{R}$-simple Lie groups of type $B_n$ and $D_n$. We use a combination of techniques from algebraic groups and quadratic forms to prove several results about these spaces.

In this paper we investigate strictly convex central configurations of the planar five-body problem, and prove some necessary conditions for such configurations. In particular, given such a central configuration with multiplier λ \lambda and total mass M M , we show that all exterior edges are less than r 0 = ( M / λ ) 1 / 3 r_0=(M/\lambda )^{1/3} , at most two interior edges are less than or equal to r 0 r_0 , and its subsystem with four masses cannot be a central configuration. We also obtain some other necessary conditions for strictly convex central configurations with five bodies, and show numerical examples of strictly convex central configurations with five bodies that have either one or two interior edges less than or equal to r 0 r_0 . Our work develops some formulae in a classic work by W. L. Williams in 1938 and we rectify some unsupported assumptions there.

In this paper we prove the dry version of the Ten Martini problem: Cantor spectrum with all gaps open, for the extended Harper’s model in the non-self-dual region for Diophantine frequencies.

We further develop the general theory of the "mixed modular derived category" introduced by the authors in a previous paper in this series. We then use it to study positivity and Q-Koszulity phenomena on flag varieties., 33 pages