Number theory

We cover a broad spectrum of number theory: rational points on varieties, arithmetic statistics, Galois representations and related topics, elliptic curves and higher dimensional abelian varieties. Our work has connections to representation theory, Lie theory, algebraic geometry and other areas.

Staff

Publications

2024

Skorobogatov, A. N., Sofos, E. (2024) Generic diagonal conic bundles revisited. Quarterly Journal of Mathematics, (doi: 10.1093/qmath/haae022)

Nicholson, J. (2024) Cancellation for (G,n)-complexes and the Swan finiteness obstruction. International Mathematics Research Notices,

Bartel, A., Johnston, H., Lenstra, H. W. (2024) Arakelov class groups of random number fields. Mathematische Annalen,

2023

Gvirtz-Chen, D., Mezzedimi, G. (2023) A Hilbert irreducibility theorem for Enriques surfaces. Transactions of the American Mathematical Society, 376, pp. 3867-3890. (doi: 10.1090/tran/8831)

Sofos, E. (2023) Gaps between prime divisors and analogues in Diophantine geometry. Glasgow Mathematical Journal, 65, pp. S129-S147. (doi: 10.1017/S0017089522000398)

Skorobogatov, A. N., Sofos, E. (2023) Schinzel Hypothesis on average and rational points. Inventiones Mathematicae, 231, pp. 673-739. (doi: 10.1007/s00222-022-01153-6)

2022

Gvirtz, D., Skorobogatov, A. N. (2022) Cohomology and the Brauer groups of diagonal surfaces. Duke Mathematical Journal, 171, pp. 1299-1347. (doi: 10.1215/00127094-2021-0029)

Blakestad, C., Gvirtz, D., Heuer, B., Shchedrina, D., Shimizu, K., Wear, P., Yao, Z. (2022) Perfectoid covers of abelian varieties. Mathematical Research Letters, 29, pp. 631-662. (doi: 10.4310/mrl.2022.v29.n3.a2)

2021

Loughran, D., Sofos, E. (2021) An Erdős–Kac law for local solubility in families of varieties. Selecta Mathematica, 27, (doi: 10.1007/s00029-021-00645-2)

Rome, N., Sofos, E. (2021) Asymptotics of the k-free diffraction measure via discretisation. Bulletin of the London Mathematical Society, 53, pp. 686-694. (doi: 10.1112/blms.12450)

Gvirtz, D., Loughran, D., Nakahara, M. (2021) Quantitative arithmetic of diagonal degree 2 K3 surfaces. Mathematische Annalen, 384, pp. 1-75. (doi: 10.1007/s00208-021-02280-w)

2020

Bartel, A., Lenstra Jr., H. W. (2020) On class groups of random number fields. Proceedings of the London Mathematical Society, 121, pp. 927-953. (doi: 10.1112/plms.12343)

Frei, C., Sofos, E. (2020) Generalised divisor sums of binary forms over number fields. Journal of the Institute of Mathematics of Jussieu, 19, pp. 137-173. (doi: 10.1017/s1474748017000469)

2019

Destagnol, K., Sofos, E. (2019) Rational points and prime values of polynomials in moderately many variables. Bulletin des Sciences Mathématiques, 156, (doi: 10.1016/j.bulsci.2019.102794)

Sofos, E. (2019) Twists of Hooley's Δ-function over number fields. Functiones et Approximatio Commentarii Mathematici, 61, pp. 57-83. (doi: 10.7169/facm/1747)

Sofos, E., Wang, Y. (2019) Finite saturation for unirational varieties. International Mathematics Research Notices, 2019, pp. 4784-4821. (doi: 10.1093/imrn/rnx318)

Browning, T.D., Sofos, E. (2019) Averages of arithmetic functions over principal ideals. International Journal of Number Theory, 15, pp. 547-567. (doi: 10.1142/s1793042119500283)

Browning, T.D., Sofos, E. (2019) Counting rational points on quartic del Pezzo surfaces with a rational conic. Mathematische Annalen, 373, pp. 977-1016. (doi: 10.1007/s00208-018-1716-6)

Bartel, A., Spencer, M. (2019) Relations between permutation representations in positive characteristic. Bulletin of the London Mathematical Society, 51, pp. 293-308. (doi: 10.1112/blms.12230)

Bartel, A., Page, A. (2019) Group representations in the homology of 3-manifolds. Commentarii Mathematici Helvetici, 94, pp. 67-88. (doi: 10.4171/CMH/455)

Schindler, D., Sofos, E. (2019) Sarnak’s saturation problem for complete intersections. Mathematika, 65, pp. 1-56. (doi: 10.1112/s002557931800030x)

2018

Frei, C., Loughran, D., Sofos, E. (2018) Rational points of bounded height on general conic bundle surfaces. Proceedings of the London Mathematical Society, 117, pp. 407-440. (doi: 10.1112/plms.12134)

2017

Bartel, A., Spencer, M. (2017) A note on Green functors with inflation. Journal of Algebra, 483, pp. 230-244. (doi: 10.1016/j.jalgebra.2017.03.031)

Bartel, A., Lenstra, H. W. (2017) Commensurability of automorphism groups. Compositio Mathematica, 153, pp. 323-346. (doi: 10.1112/S0010437X1600823X)

2016

Bartel, A., Page, A. (2016) Torsion homology and regulators of isospectral manifolds. Journal of Topology, 9, pp. 1237-1256. (doi: 10.1112/jtopol/jtw023)

Sofos, E. (2016) Serre's problem on the density of isotropic fibres in conic bundles. Proceedings of the London Mathematical Society, 113, pp. 261-288. (doi: 10.1112/plms/pdw030)

Bartel, A., Dokchitser, T. (2016) Rational representations and permutation representations of finite groups. Mathematische Annalen, 364, pp. 539-558. (doi: 10.1007/s00208-015-1223-y)

Frei, C., Sofos, E. (2016) Counting rational points on smooth cubic surfaces. Mathematical Research Letters, 23, pp. 127-143. (doi: 10.4310/mrl.2016.v23.n1.a7)

2015

Aivazidis, S., Sofos, E. (2015) On the distribution of the density of maximal order elements in general linear groups. Ramanujan Journal, 38, pp. 35-59. (doi: 10.1007/s11139-014-9619-y)

Bartel, A., Dokchitser, T. (2015) Brauer relations in finite groups. Journal of the European Mathematical Society, 17, pp. 2473-2512. (doi: 10.4171/JEMS/563)

Bartel, A. (2015) Simplicity of twists of abelian varieties. Acta Arithmetica, 171, pp. 15-22. (doi: 10.4064/aa171-1-2)

Postgraduate research students

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Number Theory - Example Research Projects

Our group has an active PhD student community, and every year we admit new PhD students. We welcome applications from across the world, and we encourage you to browse our available supervisors, and also to consult our general advice on how to navigate the application process.

Seminars

Regular seminars relevant to the group are held as part of the Algebra and Number Theory seminar series. You can find a full list of the seminars within the school on the main seminars page, where you can subscribe to their respective calendars.

Number Theory is one of the oldest disciplines of mathematics, and has connections to almost all other areas of pure mathematics. 

Our research includes:

  • rational points on curves, surfaces, and higher dimensional varieties, utilising geometric, analytic, algebraic, and probabilistic techniques;
  • classical objects of interest in algebraic number theory such as class groups, elliptic curves, higher dimensional abelian varieties;
  • analytic number theory topics, such as the Riemann zeta and Dirichlet L-functions, the distribution of prime numbers, the circle method, and the Selberg and Brun sieves;
  • diverse topics at the interface of number theory and representation theory, including (integral) Galois module structure, Galois representations, and automorphic forms, with numerous connections to representation theory of finite and of linear groups, p-adic Hodge theory, Lie theory, combinatorics, and other areas.

More information about our group, our members, our activities, and a full list of our expertise, can be found at Core Structures.

Our group has an active PhD student community, and every year we admit new PhD students.  We welcome applications from across the world, and we encourage you to browse our available supervisors, and also to consult our general advice on how to navigate the application process.