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We show that the Shimura varieties of level one parametrizing QM-abelian varieties have rarely rational points.

We study the characters induced by suitable level structures of abelian varieties with quaternionic multiplication following the methods of Mazur, Momose, who studied the characters induced by elliptic curves, and Arai--Momose, who studied the characters induced by abelian surfaces with quaternionic multiplication. Also using results on such characters we show that Shimura varieties parametrizing them have rarely rational points.

Federated learning is a distributed machine learning approach in which a single server and multiple clients collaboratively build machine learning models without sharing datasets on clients. A challenging issue of federated learning is data heterogeneity (i.e., data distributions may differ across clients). To cope with this issue, numerous federated learning methods aim at personalized federated learning and build optimized models for clients. Whereas existing studies empirically evaluated their own methods, the experimental settings (e.g., comparison methods, datasets, and client setting) in these studies differ from each other, and it is unclear which personalized federate learning method achieves the best performance and how much progress can be made by using these methods instead of standard (i.e., non-personalized) federated learning. In this paper, we benchmark the performance of existing personalized federated learning through comprehensive experiments to evaluate the characteristics of each method. Our experimental study shows that (1) there are no champion methods, (2) large data heterogeneity often leads to high accurate predictions, and (3) standard federated learning methods (e.g. FedAvg) with fine-tuning often outperform personalized federated learning methods. We open our benchmark tool FedBench for researchers to conduct experimental studies with various experimental settings.

We compute the Mordell-Weil groups of the modular Jacobian varieties of hyperelliptic modular curves $X_1(M, MN)$ over every number field which is the composition of quadratic fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.

In this paper, we determine all modular Jacobian varieties $J_1(M,MN)$ over the number field $\mathbb{Q}(\zeta_M)$ with the Mordell-Weil rank zero following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown.

Federated learning is a distributed machine learning method in which a single server and multiple clients collaboratively build machine learning models without sharing datasets on clients. Numerous methods have been proposed to cope with the data heterogeneity issue in federated learning. Existing solutions require a model architecture tuned by the central server, yet a major technical challenge is that it is difficult to tune the model architecture due to the absence of local data on the central server. In this paper, we propose Federated learning via Model exchange (FedMe), which personalizes models with automatic model architecture tuning during the learning process. The novelty of FedMe lies in its learning process: clients exchange their models for model architecture tuning and model training. First, to optimize the model architectures for local data, clients tune their own personalized models by comparing to exchanged models and picking the one that yields the best performance. Second, clients train both personalized models and exchanged models by using deep mutual learning, in spite of different model architectures across the clients. We perform experiments on three real datasets and show that FedMe outperforms state-of-the-art federated learning methods while tuning model architectures automatically.