Regularity of quotients of Drinfeld modular schemes

Let $A$ be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal $I \subset A$, Drinfeld defined the notion of structure of level $I$ on a Drinfeld module. We extend this to that of level $N$, where $N$ is a finitely generated torsion $A$-module. The case where $N=(I^{-1}/A)^d$, where $d$ is the rank of the Drinfeld module,coincides with the structure of level $I$. The moduli functor is representable by a regular affine scheme. The automorphism group $\mathrm{Aut}_{A}(N)$ acts on the moduli space. Our theorem gives a class of subgroups for which the quotient of the moduli scheme is regular. Examples include generalizations of $\Gamma_0$ and of $\Gamma_1$. We also show that parabolic subgroups appearing in the definition of Hecke correspondences are such subgroups

Unsupervised Domain Adaptation for MRI Volume Segmentation and Classification Using Image-to-Image Translation

Unsupervised domain adaptation is a type of domain adaptation and exploits labeled data from the source domain and unlabeled data from the target one. In the Cross-Modality Domain Adaptation for Medical Image Segmenta-tion challenge (crossMoDA2022), contrast enhanced T1 MRI volumes for brain are provided as the source domain data, and high-resolution T2 MRI volumes are provided as the target domain data. The crossMoDA2022 challenge contains two tasks, segmentation of vestibular schwannoma (VS) and cochlea, and clas-sification of VS with Koos grade. In this report, we presented our solution for the crossMoDA2022 challenge. We employ an image-to-image translation method for unsupervised domain adaptation and residual U-Net the segmenta-tion task. We use SVM for the classification task. The experimental results show that the mean DSC and ASSD are 0.614 and 2.936 for the segmentation task and MA-MAE is 0.84 for the classification task

String-theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication

It is known that the L-function of an elliptic curve defined over Q is given by the Mellin transform of a modular form of weight 2. Does that modular form have anything to do with string theory? In this article, we address a question along this line for elliptic curves that have complex multiplication defined over number fields. So long as we use diagonal rational N=(2,2) superconformal field theories for the string-theory realizations of the elliptic curves, the weight-2 modular form turns out to be the Boltzmann-weighted (q^{L_0-c/24}-weighted) sum of U(1) charges with F e^{ \pi i F} insertion computed in the Ramond sector.Comment: 48 pages; minor corrections and improvements in v