An official website of the United States government
Abstract Let f : ℙ 1 → ℙ 1 {f:\mathbb{P}^{1}\to\mathbb{P}^{1}} be a map of degree > 1 {>1} defined over a function field k = K ( X ) {k=K(X)} , where K is a number field and X is a projective curve over K . For each point a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} satisfying a dynamical stability condition, we prove that the Call–Silverman canonical height for specialization f t {f_{t}} at point a t {a_{t}} , for t ∈ X ( ℚ ¯ ) {t\in X(\overline{\mathbb{Q}})} outside a finite set, induces a Weil height on the curve X ; i.e., we prove the existence of a ℚ {\mathbb{Q}} -divisor D = D f , a {D=D_{f,a}} on X so that the function t ↦ h ^ f t ( a t ) - h D ( t ) {t\mapsto\hat{h}_{f_{t}}(a_{t})-h_{D}(t)} is bounded on X ( ℚ ¯ ) {X(\overline{\mathbb{Q}})} for any choice of Weil height associated to D . We also prove a local version, that the local canonical heights t ↦ λ ^ f t , v ( a t ) {t\mapsto\hat{\lambda}_{f_{t},v}(a_{t})} differ from a Weil function for D by a continuous function on X ( ℂ v ) {X(\mathbb{C}_{v})} , at each place v of the number field K . These results were known for polynomial maps f and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} without the stability hypothesis,[21, 14],and for maps f that are quotients of endomorphisms of elliptic curves E over k and all points a ∈ ℙ 1 ( k ) {a\in\mathbb{P}^{1}(k)} . [32, 29].Finally, we characterize our stability condition in terms of the geometry of the induced map f ~ : X × ℙ 1 ⇢ X × ℙ 1 {\tilde{f}:X\times\mathbb{P}^{1}\dashrightarrow X\times\mathbb{P}^{1}} over K ; and we prove the existence of relative Néron models for the pair ( f , a ) {(f,a)} , when a is a Fatou point at a place γ of k , where the local canonical height λ ^ f , γ ( a ) {\hat{\lambda}_{f,\gamma}(a)} can be computed as an intersection number.
more »
« less
Faltings height and Néron–Tate height of a theta divisor
We prove a formula, which, given a principally polarized abelian variety $$(A,\lambda )$$ over the field of algebraic numbers, relates the stable Faltings height of $$A$$ with the Néron–Tate height of a symmetric theta divisor on $$A$$ . Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of $$(A,\lambda )$$ .
more »
« less
- Award ID(s):
- 2044564
- PAR ID:
- 10336343
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 158
- Issue:
- 1
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 1 to 32
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We report a new, to the best of our knowledge, lensless microscopy configuration by integrating the concepts of transverse translational ptychography and defocus multi-height phase retrieval. In this approach, we place a tilted image sensor under the specimen for introducing linearly increasing phase modulation along one lateral direction. Similar to the operation of ptychography, we laterally translate the specimen and acquire the diffraction images for reconstruction. Since the axial distance between the specimen and the sensor varies at different lateral positions, laterally translating the specimen effectively introduces defocus multi-height measurements while eliminating axial scanning. Lateral translation further introduces sub-pixel shift for pixel super-resolution imaging and naturally expands the field of view for rapid whole slide imaging. We show that the equivalent height variation can be precisely estimated from the lateral shift of the specimen, thereby addressing the challenge of precise axial positioning in conventional multi-height phase retrieval. Using a sensor with 1.67 µm pixel size, our low-cost and field-portable prototype can resolve the 690 nm linewidth on the resolution target. We show that a whole slide image of a blood smear with a field of view can be acquired in 18 s. We also demonstrate accurate automatic white blood cell counting from the recovered image. The reported approach may provide a turnkey solution for addressing point-of-care and telemedicine-related challenges.more » « less
-
On a smooth, compact, Riemannian manifold without boundary(M,g), let\Delta_{g}be the Laplace–Beltrami operator. We define the orthogonal projection operator \Pi_{I_\lambda}\colon L^{2}(M)\to \bigoplus_{\mathclap{\lambda_j\in I_\lambda}}\ker(\Delta_{g}+\lambda_{j}^{2}) for an intervalI_{\lambda}centered around\lambda\in\Rof a small, fixed length. The Schwartz kernel,\Pi_{I_\lambda}(x,y), of this operator plays a key role in the analysis of monochromatic random waves, a model for high energy eigenfunctions. It is expected that\Pi_{I_\lambda}(x,y)has universal asymptotics as\lambda \to \inftyin a shrinking neighborhood of the diagonal inM\times M(providedI_{\lambda}is chosen appropriately) and hence that certain statistics for monochromatic random waves have universal behavior. These asymptotics are well known for the torus and the round sphere, and were recently proved to hold near points inMwith few geodesic loops by Canzani–Hanin. In this article, we prove that the same universal asymptotics hold in the opposite case of Zoll manifolds (manifolds all of whose geodesics are closed with a common period) under an assumption on the volume of loops with length incommensurable with the minimal common period.more » « less
-
Abstract Accretion disks around both stellar-mass and supermassive black holes (BHs) are likely often warped. Whenever a disk is warped, its scale height varies with azimuth. Sufficiently strong warps cause extreme compressions of the scale height, which fluid parcels “bounce” off of twice per orbit to high latitudes. We study the dynamics of strong warps using (i) the nearly analytic “ring theory” of Fairbairn & Ogilvie, which we generalize to the Kerr metric, and (ii) three-dimensional general-relativistic hydrodynamic simulations of tori (“rings”) around BHs, using theH-AMRcode. We initialize a ring with a warp and study its evolution on tens of orbital periods. The simulations agree excellently with the ring theory until the warp amplitude,ψ, reaches a critical valueψc. Whenψ > ψc, the rings enter the bouncing regime. We analytically derive (and numerically validate) that in the non-Keplerian regime, whererg = GM/c2is the gravitational radius, andMis the mass of the central object. Whenever the scale height bounces, the vertical velocity becomes supersonic, leading to “nozzle shocks” as gas collides at the scale height minima. Nozzle shocks damp the warp within ≈10–20 orbits, which is not captured by the ring theory. Nozzle shock dissipation leads to inflow timescales 1–2 orders of magnitude shorter than unwarpedαdisks, which may result in rapid variability, such as in changing-look active galactic nuclei or in the soft state of X-ray binaries. We propose that steady disks with strong warps may self-regulate to have amplitudes nearψc.more » « less
-
We study the essential minimum of the (stable) Faltings' height on the moduli space of elliptic curves. We prove that, in contrast to the Weil height on a projective space and the Néron-Tate height of an abelian variety, Faltings' height takes at least two values that are smaller than its essential minimum. We also provide upper and lower bounds for this quantity that allow us to compute it up to five decimal places. In addition, we give numerical evidence that there are at least four isolated values before the essential minimum. One of the main ingredients in our analysis is a good approximation of the hyperbolic Green function associated to the cusp of the modular curve of level one. To establish this approximation, we make an intensive use of distortion theorems for univalent functions. Our results have been motivated and guided by numerical experiments that are described in detail in the companion files.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1112/S0010437X21007661
