Murmurations, Mestre-Nagao sums, and convolutional neural networks for elliptic curves

Murmurations: a case study in AI-assisted mathematics
He, Y.-H., Lee, K.-H., Oliver, T. and Pozdnyakov, A. 2026. Murmurations: a case study in AI-assisted mathematics. in: Rassias, M. (ed.) Artificial Intelligence and Mathematics Springer.

Machine learning the vanishing order of rational L-functions
Oliver, T., Bieri, J., Butbaia, G., Costa, E., Deines, A., Lee, K.-H., Lowry-Duda, D., Qi, Y. and Veenstra, T. 2025. Machine learning the vanishing order of rational L-functions. Advances in Theoretical and Mathematical Physics. 29 (8), pp. 2381-2401. https://doi.org/10.4310/ATMP.260412213616

Learning Fricke signs from Maass form coefficients
Oliver, T., Bieri, J., Butbaia, G., Costa, E., Deines, A., Lee, K.-H., Lowry-Duda, D., Qi, Y. and Veenstra, T. 2025. Learning Fricke signs from Maass form coefficients. Advances in Theoretical and Mathematical Physics. 29 (8), pp. 2403-2423. https://doi.org/10.4310/ATMP.260412220615

Convergence of Kac-Moody Eisenstein series over a function field
Oliver, T., Lee, K.-H. and Liu, D. 2025. Convergence of Kac-Moody Eisenstein series over a function field. Selecta Mathematica. 31 (39). https://doi.org/10.1007/s00029-025-01031-y

Murmurations of Dirichlet characters
Oliver, T., Lee, K.-H. and Pozdnyakov, A. 2025. Murmurations of Dirichlet characters. International Mathematics Research Notices. 2025 (1) rnae277. https://doi.org/10.1093/imrn/rnae277

Murmurations of elliptic curves
Oliver, T., He, Y.-H., Lee, K.-H. and Pozdnyakov, A. 2025. Murmurations of elliptic curves. Experimental Mathematics. 34 (3), pp. 528-540. https://doi.org/10.1080/10586458.2024.2382361

Machine learning for number theory: unsupervised learning with L-functions
Oliver, T. 2024. Machine learning for number theory: unsupervised learning with L-functions. 8th International Congress on Mathematical Software (ICMS2024). Durham University 22 - 25 Jul 2024 Springer. https://doi.org/10.1007/978-3-031-64529-7_21

PCA, arithmetic, and murmurations
Oliver, T. 2024. PCA, arithmetic, and murmurations. Murmurations in Arithmetic. ICERM (Brown University) 06 - 08 Jul 2023 World Scientific Publishing. https://doi.org/10.1142/S2810939224400021

Preprint: Murmurations of Dirichlet Characters
Lee, K.-H., Oliver, T. and Pozdnyakov, A. 2023. Preprint: Murmurations of Dirichlet Characters. arXiv. https://doi.org/10.48550/arXiv.2307.00256

Machine Learning Class Numbers of Real Quadratic Fields
Oliver, T., Amir, M., He, Y.-H., Lee, K.-H. and Sultanow, E. 2023. Machine Learning Class Numbers of Real Quadratic Fields. International Journal of Data Science in the Mathematical Sciences. 1 (2), pp. 107-134. https://doi.org/10.1142/s2810939223500016

Counting points on elliptic curves
Oliver, T. and Wuthrich, C. 2023. Counting points on elliptic curves. LMS newsletter. 509, pp. 31-35. https://doi.org/10.1112/NLMS

Supervised learning of arithmetic invariants
Oliver, T. 2023. Supervised learning of arithmetic invariants. in: Yang-Hui He (ed.) Machine Learning in Pure Mathematics and Theoretical Physics World Scientific Publishing. pp. 331-363

Machine learning invariants of arithmetic curves
He, Y.-H., Lee, K.-H. and Oliver, T. 2023. Machine learning invariants of arithmetic curves. Journal of Symbolic Computation. 115, pp. 478-491. https://doi.org/10.1016/j.jsc.2022.08.017

Ratios of Artin L-functions
Hochfilzer, L. and Oliver, T. 2022. Ratios of Artin L-functions. Journal of Number Theory. 236, pp. 1-40. https://doi.org/10.1016/j.jnt.2021.07.007

Preprint: Murmurations of Elliptic Curves
He, Y.-H., Lee, K.-H., Oliver, T. and Pozdnyakov, A. 2022. Preprint: Murmurations of Elliptic Curves. arXiv. https://doi.org/10.48550/arxiv.2204.10140

Preprint: Machine Learning Class Numbers of Real Quadratic Fields
Amir, M., He, Y.-H., Lee, K.-H., Oliver, T. and Sultanow, E. 2022. Preprint: Machine Learning Class Numbers of Real Quadratic Fields. arXiv. https://doi.org/10.48550/arxiv.2209.09283

Preprint: Convergance of Kac-Moody Eisenstein Series over a Function Field
Lee, K.-H., Liu, D. and Oliver, T. 2022. Preprint: Convergance of Kac-Moody Eisenstein Series over a Function Field. arXiv. https://doi.org/10.48550/arxiv.2203.08628

Machine-learning the Sato-Tate conjecture
He, Y.-H., Lee, K.-H. and Oliver, T. 2022. Machine-learning the Sato-Tate conjecture. Journal of Symbolic Computation. 111, pp. 61-72. https://doi.org/10.1016/j.jsc.2021.11.002

Ratios of Artin L-functions
Hochfilzer, L. and Oliver, T. 2022. Ratios of Artin L-functions. Journal of Number Theory. 236, pp. 1-40. https://doi.org/10.1016/j.jnt.2021.07.007

Machine-learning number fields
He, Y.-H., Lee, K.-H. and Oliver, T. 2022. Machine-learning number fields. Mathematics, Computation and Geometry of Data. 2 (1), pp. 49-66. https://doi.org/10.4310/MCGD.2022.v2.n1.a2

Character expansion of Kac–Moody correction factors
Lee, K.-H., Liu, D and Oliver, T. 2021. Character expansion of Kac–Moody correction factors. Pacific Journal of Mathematics. 313 (1), pp. 159-183. https://doi.org/10.2140/pjm.2021.313.159

Preprint: Twisting moduli for gl(2)
Bedert, B., Cooper, G., Oliver, T. and Zhang, P. 2020. Preprint: Twisting moduli for gl(2). arXiv. https://doi.org/10.48550/arxiv.2003.02557

Preprint: Machine-Learning the Sato-Tate Conjecture
He, Y.-H., Lee, K.-H. and Oliver, T. 2020. Preprint: Machine-Learning the Sato-Tate Conjecture. arXiv. https://doi.org/10.48550/arxiv.2010.01213

Preprint: Machine-learning number fields
He, Y.-H., Lee, K.-H. and Oliver, T. 2020. Preprint: Machine-learning number fields. arXiv. https://doi.org/10.48550/arxiv.2011.08958

Preprint: Machine-learning arithmetic curves
He, Y.-H., Lee, K.-H. and Oliver, T. 2020. Preprint: Machine-learning arithmetic curves. arXiv. https://doi.org/10.48550/arxiv.2012.04084

Twisting moduli for GL(2)
Bedert, B., Cooper, G., Oliver, T. and Zhang, P. 2020. Twisting moduli for GL(2). Journal of Number Theory. 217, pp. 142-162. https://doi.org/10.1016/j.jnt.2020.04.008

Weil's converse theorem for Maass forms and cancellation of zeros
Oliver, T. and Neururer, M. 2020. Weil's converse theorem for Maass forms and cancellation of zeros. Acta Arithmetica. 196, pp. 387-422. https://doi.org/10.4064/aa190811-3-2

Preprint: Ratios of Artin L-functions
Hochfilzer, L. and Oliver, T. 2019. Preprint: Ratios of Artin L-functions. arXiv. https://doi.org/10.48550/arxiv.1910.02821

Preprint: Weil's Converse Theorem for Maass Forms and Cancellation of Zeros
Neururer, M. and Oliver, T. 2018. Preprint: Weil's Converse Theorem for Maass Forms and Cancellation of Zeros. arXiv. https://doi.org/10.48550/arxiv.1809.06586

A conjectural extension of Hecke’s converse theorem
Bettin, S., Bober, J., Booker, A., Conrey, B., Lee, M., Molteni, G., Oliver, T., Platt, D. and Steiner, R. 2018. A conjectural extension of Hecke’s converse theorem. The Ramanujan Journal. 47, pp. 659-684. https://doi.org/10.1007/s11139-017-9953-y

A conjectural extension of hecke's converse theorem
Bettin, S., Bober, J.W., Booker, A.R., Conrey, B., Lee, M., Molteni, G., Oliver, T., Platt, D.J. and Steiner, R.S. 2017. A conjectural extension of hecke's converse theorem. arXiv. https://doi.org/10.48550/arxiv.1704.02570

Notes on low degree L-data
Oliver, T. 2017. Notes on low degree L-data. Analytic Number Theory and Related Areas. Research Institute for Mathematical Sciences, Kyoto University 04 - 06 Nov 2015

Automorphicity and mean-periodicity
Oliver, T. 2017. Automorphicity and mean-periodicity. Journal of the Mathematical Society of Japan. 69 (1), pp. 25-51. https://doi.org/10.2969/jmsj/06910025

Zeta integrals on arithmetic surfaces
Oliver, T. 2016. Zeta integrals on arithmetic surfaces. St Petersburg Math. J.. 27, pp. 1003-1028. https://doi.org/10.1090/spmj/1432

Higher Dimensional Adeles, Mean-Periodicity and Zeta Functions of Arithmetic Surfaces
Oliver, T. 2014. Higher Dimensional Adeles, Mean-Periodicity and Zeta Functions of Arithmetic Surfaces. PhD thesis University of Nottingham School of Mathematical Sciences