数学前沿

Let $G$ be a simple group over a global function field $K$, and let $\pi$ be a cuspidal automorphic representation of $G$. Suppose $K$ has two places $u$\nonbreakingspace and\nonbreakingspace $v$ (satisfying a mild restriction on the residue field cardinality), at which the group $G$ is quasi-split, such that $\pi _u$ is tempered and $\pi _v$ is unramified and generic. We prove that $\pi _w$ is tempered at all unramified places\nonbreakingspace $K_w$ at which $G$ is unramified quasi-split. \par More generally, the set of unitary spherical representations is partitioned according to nilpotent conjugacy classes in the Lie algebra of $G$. We show that if $\pi _v$ is in the set corresponding to the nilpotent class $N$, and if $\pi _u$ satisfies an analogous hypothesis, then $\pi _w$ belongs to the same class $N$, where $w$ is as above. These results are consistent with conjectures of Shahidi and Arthur. \par The proofs use the Galois parametrization of cuspidal representations due to V. Lafforgue to relate the local Satake parameters of $\pi$ to Deligne’s theory of Frobenius weights. The main observation is that, in view of the classification of unitary spherical representations, due to Barbasch and the first-named author, the theory of weights excludes almost all complementary series as possible local components of $\pi$. This in turn determines the local Frobenius weights at all unramified places. In order to apply this observation in practice, we need a result of the second-named author with Gan and Sawin on the weights of discrete series representations.

We describe an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl $m$-function at the point. We show that bulk universality of the Christoffel–Darboux kernel holds for any point where the imaginary part of the $m$-function has a positive finite nontangential limit. This approach is based on studying a matrix version of the Christoffel–Darboux kernel and the realization that bulk universality for this kernel at a point is equivalent to the fact that the corresponding $m$-function has normal limits at the same point. Our approach automatically applies to other self-adjoint systems with $2\times 2$ transfer matrices such as continuum Schrö and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle.

We consider Schrödinger operators $H=-\Delta+V({\mathbf x})$ in ${\mathbb{R}}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis $[\lambda_*,+\infty)$. We also construct a family of eigenfunctions of the absolutely continuous spectrum; these eigenfunctions are small perturbations of the exponentials. The proof is based on a version of the multi-scale analysis in the momentum space with several new ideas introduced along the way.

We develop a theory of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence, which also includes the possibility of dealing with several systems associated with sufficiently independent matrix transformations. Our main results go far beyond the existing literature, also surpassing those of two unpublished preprints the authors made available on the arXiv in 2018. The main new feature is that \Highlight [Maureen]our results apply now without any restriction on the matrices defining the corresponding Mahler systems. As a consequence, we settle several problems concerning expansions of numbers in multiplicatively independent bases. For instance, we prove that no irrational real number can be automatic in two multiplicatively independent integer bases, and we give a new proof and a broad algebraic generalization of Cobham’s theorem in automata theory. We also provide a new proof and a multivariate generalization of Nishioka’s theorem, a landmark result in Mahler’s method.

We prove that if $\Sigma$ is a closed surface of genus at least $3$ and $G$ is a split real semisimple Lie group of rank of at least $3$ acting faithfully by isometries on a Riemannian symmetric space $N$, then there exists a Hitchin representation $\rho\colon \pi_1(\Sigma)\to G$ and a $\rho$-equivarient unstable minimal map from the universal cover of $\Sigma$ to $N$. This follows from a new lower bound on the index of high energy minimal maps into an arbitrary symmetric space of non-compact type. Taking $G=\mathrm{PSL}(n,\mathbb{R})$, $n\ge 4$, this disproves the Labourie Conjecture

We show that a complete, two-sided stable minimal hypersurface in $\mathbf{R}^5$ is flat.

In this work, we prove the $L^3$-based strong Onsager conjecture for the three-dimensional Euler equations. Our main theorem states that there exist weak solutions which dissipate the total kinetic energy, satisfy the local energy inequality, and belong to $C^0_t (W^{\frac{1}{3}-, 3} \cap L^{\infty-})$. More precisely, for every $\beta\lt \frac{1}{3}$, we can construct such solutions in the space $C^0_t (B^{\beta}_{3,\infty} \cap L^{\frac{1}{1-3\beta}} )$.

We establish a fibre sequence relating the classical Grothendieck-Witt theory of a ring $R$ to the homotopy $\mathrm{C}_2$-orbits of its K-theory and Ranicki’s original (non-periodic) symmetric L-theory. We use this fibre sequence to remove the assumption that $2$ is a unit in $R$ from various results about Grothendieck-Witt groups. For instance, we solve the homotopy limit problem for Dedekind rings whose fraction field is a number field, calculate the various flavours of Grothendieck-Witt groups of $\mathbb{Z}$, show that the Grothendieck-Witt groups of rings of integers in number fields are finitely generated and that the comparison map from quadratic to symmetric Grothendieck-Witt theory of coherent rings of global dimension $d$ is an equivalence in degrees $\geq d+3$. As an important tool, we establish the hermitian analogue of Quillen’s localisation-dévissage sequence for Dedekind rings and use it to solve a conjecture of Berrick-Karoubi.

We introduce a `tremor’ deformation on strata of translation surfaces. Using it, we give new examples of behaviors of horocycle flow orbits $U_q$ in strata of translation surfaces. In the genus 2 stratum $\mathcal{H}(1,1)$ we find orbits $U_q$ which are generic for a measure whose support is strictly contained in $\overline{U_q}$ and find orbits which are not generic for any measure. We also describe a horocycle orbit-closure whose Hausdorff dimension is not an integer.

Let $F$ be a totally real field. We prove the existence of all symmetric power liftings of those cuspidal automorphic representations of $\mathrm {GL}_2(\mathbf{A}_F)$ associated to Hilbert modular forms of regular weight.

This is a continuation of our previous work on the locally analytic vectors of the completed cohomology of modular curves. We construct differential operators on modular curves with infinite level at $p$ in both “holomorphic” and “anti-holomorphic” directions. As applications, we reprove a classicality result of Emerton which says that every absolutely irreducible two dimensional Galois representation that is regular de Rham at $p$ and appears in the completed cohomology of modular curves comes from an eigenform. Moreover, we give a geometric description of the locally analytic representations of $\mathrm{GL}_2(\mathbb{Q}_p)$ attached to such a Galois representation in the completed cohomology.

In this paper, we construct the admissible canonical bundle of a quasi-projective family of curves, prove that it is a big adelic line bundle when the family has maximal variation, and apply it to prove a uniform Bogomolov-type theorem for curves over global fields of all characteristics. This gives a different approach to the uniform Mordell–Lang type of result of Vojta, Dimitrov–Gao–Habegger and Kuhne. Our treatment is based on the theory of adelic line bundles on quasi-projective varieties recently introduced by Yuan–Zhang.

Let $E$ be a CM elliptic curve defined over $\mathbb{Q}$ and $p$ a prime. We show that $\text{corank}_{\mathbb{Z}_{p}}\mathrm{Sel}_{p^\infty}(E_{/\mathbb{Q}}) = 0 \implies \mathrm{ord}_{s=1}L(s,E_{/\mathbb{Q}}) = 0$ and the for the $p^\infty$-Selmer group $\mathrm{Sel}_{p^\infty}(E_{/\mathbb{Q}})$ and the complex $L$-function $L(s,E_{/\mathbb{Q}})$. Along with Smith’s work on the distribution of $2^\infty$-Selmer groups, this leads to the first instance of the even parity Goldfeld conjecture: For 50% of the positive square-free integers $n$, we have $\mathrm{ord}_{s=1}L(s,E^{(n)}_{/\mathbb{Q}}) = 0$, where $E^{(n)}: ny^2 = x^3 -x$ is a quadratic twist of the congruent number elliptic curve $E: y^2 = x^3 – x$.

Let $G$ be a higher rank semisimple real Lie group or the product of at least two automorphism groups of regular trees. We prove all probability measure preserving actions of lattices in such groups have cost one, answering Gaboriau’s fixed price question for this class of groups. We prove the minimal number of generators of a torsion-free lattice in $G$ is sublinear in its co-volume, settling a conjecture of Abért–Gelander–Nikolov. As a consequence, we derive new estimates on the growth of first mod-$p$ homology groups of higher rank locally symmetric spaces. Our method of proof is novel, using low intensity Poisson point processes on $G$ and the geometry of their associated Voronoi tesselations. We prove, as the intensity limits to zero, these tessellations partition the space into “horoball-like” cells such that any two share an unbounded border. We use this new phenomenon to construct low cost graphings for orbit equivalence relations of higher rank lattices.

The Gopakumar–Vafa conjecture predicts that the BPS invariants of a symplectic $6$-manifold, defined in terms of the Gromov–Witten invariants, are integers and all but finitely many vanish in every homology class. The integrality part of this conjecture was proved earlier by Ionel and Parker. This article proves the finiteness part. The proof relies on a modification of Ionel and Parker’s cluster formalism using results from geometric measure theory.

We analyze stabilization with respect to $\mathbb{P}^1$ in the Morel–Voevodsky unstable motivic homotopy theory. We introduce a refined notion of cellularity (a.k.a., biconnectivity) in various motivic homotopy categories taking into account both the simplicial and Tate circles. Under suitable cellularity hypotheses, we refine the Whitehead theorem by showing that a map of nilpotent motivic spaces can be seen to be an equivalence if it so after taking (Voevodsky) motives. We then establish a version of the Freudenthal suspension theorem for $\mathbb{P}^1$-suspension, again under suitable cellularity hypotheses. As applications, we resolve Murthy’s conjecture on splitting of corank $1$ vector bundles on smooth affine algebras over algebraically closed fields having characteristic $0$ and compute new unstable motivic homotopy of motivic spheres.

We prove the Scholze–Weinstein conjecture on the existence and uniqueness of local models for local Shimura varieties, as well as the test function conjecture of Haines–Kottwitz in this framework. To this end, we establish a specialization principle for well-behaved $p$-adic kimberlites, show that these include the v-sheaf local models, determine their special fibers using hyperbolic localization for the étale cohomology of small v-stacks, and analytic the resulting specialization morphism using convolution.

In 1987, Segal gave a functorial definition of Conformal Field Theory (CFT) that was designed to capture the mathematical essence of the Conformal Bootstrap formalism pioneered in physics by Belavin, Polyakov and Zamolodchikov. In Segal’s formulation, the basic objects of CFT, the correlation functions of conformal primary fields, are viewed as functions on the moduli space of Riemann surfaces with marked points which behave naturally under gluing of surfaces. In this paper we give a probabilistic realisation of Segal’s axioms in Liouville Conformal Field Theory (LCFT), a CFT that plays a fundamental role in the theory of random surfaces and two-dimensional quantum gravity. Namely, to a Riemann surface $\Sigma$ with marked points and boundary given by a union of parameterised circles, we associate a Hilbert-Schmidt operator $\mathcal{A}_\Sigma$, called the _amplitude_ of $\Sigma$, which acts on a tensor product of Hilbert spaces assigned to the boundary circles. We show that this correspondence is functorial: gluing of surfaces along boundary circles maps to a composition of the corresponding operators. Correlation functions of LCFT, constructed probabilistically in earlier works by the authors and F. David, can then be expressed as compositions of the amplitudes of simple building blocks where $\Sigma$ is a sphere with $b\in {1,2,3}$ disks removed (hence $b$ boundary circles) and $3-b$ marked points. These amplitudes in turn are shown to be determined by basic objects of LCFT: its _spectrum_ and its _structure constants_ determined in earlier works by the authors. As a consequence, we obtain a formula for the correlation functions as multiple integrals over the spectrum of LCFT, the structure of these integrals being related to a pant decomposition of the surface. The integrand is the square modulus of a function called conformal block: its structure is encoded by the commutation relations of an algebra of operators called the Virasoro algebra and it depends holomorphically on the moduli of the surface with marked points. The integration measure involves a product of structure constants, which have an explicit expression, the so-called DOZZ formula. Such a holomorphic factorisation of correlation functions has been conjectured in physics since the 80’s and we give here its first rigorous derivation for a non-trivial CFT.