skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Dirac geometry II: coherent cohomology
Abstract Dirac rings are commutative algebras in the symmetric monoidal category of$$\mathbb {Z}$$-graded abelian groups with the Koszul sign in the symmetry isomorphism. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we embed this category in the larger$$\infty $$-category of Dirac stacks, which also contains formal Dirac schemes, and develop the coherent cohomology of Dirac stacks. We apply the general theory to stable homotopy theory and use Quillen’s theorem on complex cobordism and Milnor’s theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to$$\operatorname {MU}$$and$$\mathbb {F}_p$$in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves.  more » « less
Award ID(s):
1926686
PAR ID:
10535368
Author(s) / Creator(s):
;
Publisher / Repository:
Cambridge University Press
Date Published:
Journal Name:
Forum of Mathematics, Sigma
Volume:
12
ISSN:
2050-5094
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; ; (, The Bulletin of Symbolic Logic)
    Abstract For any subset$$Z \subseteq {\mathbb {Q}}$$, consider the set$$S_Z$$of subfields$$L\subseteq {\overline {\mathbb {Q}}}$$which contain a co-infinite subset$$C \subseteq L$$that is universally definable inLsuch that$$C \cap {\mathbb {Q}}=Z$$. Placing a natural topology on the set$${\operatorname {Sub}({\overline {\mathbb {Q}}})}$$of subfields of$${\overline {\mathbb {Q}}}$$, we show that ifZis not thin in$${\mathbb {Q}}$$, then$$S_Z$$is meager in$${\operatorname {Sub}({\overline {\mathbb {Q}}})}$$. Here,thinandmeagerboth mean “small”, in terms of arithmetic geometry and topology, respectively. For example, this implies that only a meager set of fieldsLhave the property that the ring of algebraic integers$$\mathcal {O}_L$$is universally definable inL. The main tools are Hilbert’s Irreducibility Theorem and a new normal form theorem for existential definitions. The normal form theorem, which may be of independent interest, says roughly that every$$\exists $$-definable subset of an algebraic extension of$${\mathbb Q}$$is a finite union of single points and projections of hypersurfaces defined by absolutely irreducible polynomials. 
    more » « less
  2. ; ; (, Journal of the Institute of Mathematics of Jussieu)
    Abstract LetKbe an imaginary quadratic field and$$p\geq 5$$a rational prime inert inK. For a$$\mathbb {Q}$$-curveEwith complex multiplication by$$\mathcal {O}_K$$and good reduction atp, K. Rubin introduced ap-adicL-function$$\mathscr {L}_{E}$$which interpolates special values ofL-functions ofEtwisted by anticyclotomic characters ofK. In this paper, we prove a formula which links certain values of$$\mathscr {L}_{E}$$outside its defining range of interpolation with rational points onE. Arithmetic consequences includep-converse to the Gross–Zagier and Kolyvagin theorem forE. A key tool of the proof is the recent resolution of Rubin’s conjecture on the structure of local units in the anticyclotomic$${\mathbb {Z}}_p$$-extension$$\Psi _\infty $$of the unramified quadratic extension of$${\mathbb {Q}}_p$$. Along the way, we present a theory of local points over$$\Psi _\infty $$of the Lubin–Tate formal group of height$$2$$for the uniformizing parameter$$-p$$. 
    more » « less
  3. ; (, Compositio Mathematica)
    We study the spaces of twisted conformal blocks attached to a$$\Gamma$$-curve$$\Sigma$$with marked$$\Gamma$$-orbits and an action of$$\Gamma$$on a simple Lie algebra$$\mathfrak {g}$$, where$$\Gamma$$is a finite group. We prove that if$$\Gamma$$stabilizes a Borel subalgebra of$$\mathfrak {g}$$, then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed$$\Gamma$$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let$$\mathscr {G}$$be the parahoric Bruhat–Tits group scheme on the quotient curve$$\Sigma /\Gamma$$obtained via the$$\Gamma$$-invariance of Weil restriction associated to$$\Sigma$$and the simply connected simple algebraic group$$G$$with Lie algebra$$\mathfrak {g}$$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic$$\mathscr {G}$$-torsors on$$\Sigma /\Gamma$$when the level$$c$$is divisible by$$|\Gamma |$$(establishing a conjecture due to Pappas and Rapoport). 
    more » « less
  4. ; ; (, Forum of Mathematics, Pi)
    Abstract Letfbe an$$L^2$$-normalized holomorphic newform of weightkon$$\Gamma _0(N) \backslash \mathbb {H}$$withNsquarefree or, more generally, on any hyperbolic surface$$\Gamma \backslash \mathbb {H}$$attached to an Eichler order of squarefree level in an indefinite quaternion algebra over$$\mathbb {Q}$$. Denote byVthe hyperbolic volume of said surface. We prove the sup-norm estimate$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$ with absolute implied constant. For a cuspidal Maaß newform$$\varphi $$of eigenvalue$$\lambda $$on such a surface, we prove that$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$ We establish analogous estimates in the setting of definite quaternion algebras. 
    more » « less
  5. (, Forum of Mathematics, Pi)
    Abstract Define theCollatz map$${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$$on the positive integers$$\mathbb {N}+1 = \{1,2,3,\dots \}$$by setting$${\operatorname {Col}}(N)$$equal to$$3N+1$$whenNis odd and$$N/2$$whenNis even, and let$${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$$denote the minimal element of the Collatz orbit$$N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $$. The infamousCollatz conjectureasserts that$${\operatorname {Col}}_{\min }(N)=1$$for all$$N \in \mathbb {N}+1$$. Previously, it was shown by Korec that for any$$\theta> \frac {\log 3}{\log 4} \approx 0.7924$$, one has$${\operatorname {Col}}_{\min }(N) \leq N^\theta $$for almost all$$N \in \mathbb {N}+1$$(in the sense of natural density). In this paper, we show that foranyfunction$$f \colon \mathbb {N}+1 \to \mathbb {R}$$with$$\lim _{N \to \infty } f(N)=+\infty $$, one has$${\operatorname {Col}}_{\min }(N) \leq f(N)$$for almost all$$N \in \mathbb {N}+1$$(in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a$$3$$-adic cyclic group$$\mathbb {Z}/3^n\mathbb {Z}$$at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email