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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.
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Singular Vortex Pairs Follow Magnetic Geodesics
Abstract We consider pairs of point vortices having circulations $$\Gamma _{1}$$ and $$\Gamma _{2}$$ and confined to a two-dimensional surface $$S$$. In the limit of zero initial separation $$\varepsilon $$, we prove that they follow a magnetic geodesic in unison, if properly renormalized. Specifically, the “singular vortex pair” moves as a single-charged particle on the surface with a charge of order $$1/\varepsilon ^{2}$$ in an magnetic field $$B$$ that is everywhere normal to the surface and of strength $$|B|=\Gamma _{1} +\Gamma _{2}$$. In the case $$\Gamma _{1}=-\Gamma _{2}$$, this gives another proof of Kimura’s conjecture [11] that singular dipoles follow geodesics.
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- Award ID(s):
- 2235395
- PAR ID:
- 10529425
- Publisher / Repository:
- IMRN
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2024
- Issue:
- 14
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- 10880 to 10894
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnae106
