Shimura varieties and the Birch–Swinnerton-Dyer conjecture

One of the most famous open problems in mathematics is the Birch–Swinnerton-Dyer (BSD) conjecture, which predicts that the size of the set of rational points on an elliptic curve is determined by the order of vanishing at s = 1 of its Hasse–Weil L-function. Building a crucial breakthrough due to Kolyvagin in the 1990's—the discovery of the first example of an ""Euler system""—the BSD conjecture has now been proved for a wide class of elliptic curves over the rationals: those where the order of vanishing of the L-function (the ""analytic rank"") is 0 or 1, which conjecturally accounts for 100% of elliptic curves.

However, the case of elliptic curves over the rationals is only the tip of an iceberg. Versions of the BSD conjecture are also expected to hold for elliptic curves over number fields, and more generally for abelian varieties of any dimension (with elliptic curves being the case of dimension 1). Even more generally, the Bloch–Kato conjecture predicts that for any L-function arising from geometry, its order of vanishing at any integer point encodes geometric information. However, these conjectures are far beyond the reach of Kolyvagin's Euler system.

The aim of my proposal is to prove new cases of the BSD conjecture and the Bloch–Kato conjecture, using new Euler systems arising from the geometry of unitary and symplectic Shimura varieties. In particular, I will prove the rank 0 case of the BSD conjecture for abelian surfaces over the rationals, elliptic curves over imaginary quadratic fields, and abelian three-folds with complex multiplication, assuming appropriate modularity results hold for these objects (which are known in many cases).
 

Research ouputs

• A universal Euler system for GSp(4), David Loeffler, Sarah Livia Zerbes, Forum Mathematicum, To appear, Yes, 10.1515/forum-2025-0062, https://arxiv.org/abs/2411.12576
• Algebraicity of L-values for GSp4×GL2 and GSp4×GL2×GL2, David Loeffler, Óscar Rivero, Quarterly J Math, Vol. 75 no. 2, Yes, 10.1093/qmath/haae016, https://arxiv.org/abs/2303.16114
• Control theorems for Hilbert modular varieties, Arshay Sheth, Canadian Mathematical Bulletin, 68, Yes, 10.4153/s0008439524000791, https://arxiv.org/abs/2403.04738
• Eisenstein degeneration of Euler systems, David Loeffler, Óscar Rivero, Journal für die reine und angewandte Mathematik (Crelles Journal), 2024, Yes, 10.1515/crelle-2024-0057, https://arxiv.org/abs/2201.02078
• Euler Product Asymptotics for L-functions of Elliptic Curves, Arshay Sheth, International Mathematics Research Notices, 2025, Yes, 10.1093/imrn/rnaf214, http://arxiv.org/abs/2312.05236
• Euler Products at the Centre and Applications to Chebyshev’s Bias, Arshay Sheth, Mathematical Proceedings of the Cambridge Philosophical Society, 179, Yes, 10.1017/s0305004125000234, https://arxiv.org/abs/2405.01512
• Finite polynomial cohomology with coefficients, Ting-Han Huang, Ju-Feng Wu, Rendiconti del Seminario Matematico della Università di Padova, 155, Yes, 10.4171/rsmup/173, https://arxiv.org/abs/2209.10289
• On anticyclotomic Euler and Kolyvagin systems, Luca Mastella, Francesco Zerman, Ann. Math. Québec, To appear, Yes, https://arxiv.org/abs/2505.08710
• On Greenberg–Benois L-invariants and Fontaine–Mazur L-invariants, Ju-Feng Wu, Canadian Mathematical Bulletin, 67, Yes, 10.4153/s0008439524000638, https://arxiv.org/abs/2310.11098
• On integral aspects of Asai periods and Euler systems for Res_{E/Q}GL(2), Alexandros Groutides, Forum Mathematicum, 38, Yes, 10.1515/forum-2024-0458, https://arxiv.org/abs/2409.17888
• On local zeta-integrals for GSp(4) and GSp(4) x GL(2), David Loeffler, New York J Math, 30, Yes, https://arxiv.org/abs/2011.15106
• On 𝑝-adic adjoint 𝐿-functions for Bianchi cuspforms: The 𝑝-split case, Pak-Hin Lee, Ju-Feng Wu, Transactions of the American Mathematical Society, 378, Yes, 10.1090/tran/9388, https://arxiv.org/abs/2306.15441
• On p-adic L-functions for GSp4×GL2, David Loeffler, Óscar Rivero, Pacific Journal of Mathematics, 335, Yes, 10.2140/pjm.2025.335.373, https://arxiv.org/abs/2305.07707
• On Rankin-Selberg integral structures and Euler systems for GL2 × GL2, Alexandros Groutides, Journal of Number Theory, 290, Yes, 10.1016/j.jnt.2026.03.010, https://arxiv.org/abs/2407.01377
• On the Bloch-Kato conjecture for GSp(4), David Loeffler, Sarah Livia Zerbes, Cambridge J Math, To appear, Yes, https://arxiv.org/abs/2003.05960
• P-adic L-functions for GL(3), David Loeffler, Chris Williams, Mathematische Annalen, 394, Yes, 10.1007/s00208-026-03377-w, https://arxiv.org/abs/2111.04535
• P-adic Rankin–Selberg L-functions in universal deformation families and functional equations, Zeping Hao, David Loeffler, Forum Mathematicum, Vol. 38 no. 3, Yes, 10.1515/forum-2024-0497, https://arxiv.org/abs/2405.12611
• Perfectoid overconvergent Siegel modular forms and the overconvergent Eichler–Shimura morphism, Hansheng Diao, Giovanni Rosso, Ju-Feng Wu, Research in the Mathematical Sciences, 13, Yes, 10.1007/s40687-026-00610-5, https://arxiv.org/abs/2106.00094
• Plectic structures in p-adic de Rham cohomology, David Loeffler, Sarah Livia Zerbes, Journal of Number Theory, 270, Yes, 10.1016/j.jnt.2023.10.016, https://arxiv.org/abs/2211.12078
• Poles of p-adic Asai L-functions and Distinguished Representations, David Loeffler, Sarah Livia Zerbes, International Mathematics Research Notices, 2025, Yes, 10.1093/imrn/rnaf363, https://arxiv.org/abs/2501.11472
• Quaternionic Kolyvagin systems and Iwasawa theory for Hida families, Francesco Zerman, Math. Res. Lett., To appear, Yes, https://arxiv.org/abs/2507.04980
• Hilbert-Kunz series, F-signature series, and weak p-fractals, Alessio Caminata, Francesco Zerman, The Landscape of Commutative Algebra (LMS Lecture Note Series), To appear, Yes, https://arxiv.org/abs/2510.19355
• Spherical Varieties and p-Adic Families of Cohomology Classes, David Loeffler, Robert Rockwood, Sarah Livia Zerbes, Springer Proceedings in Mathematics & Statistics, Elliptic Curves and Modular Forms in Arithmetic Geometry, Vol. 527, Yes, 10.1007/978-3-032-13123-2_11, https://arxiv.org/abs/2106.16082
• A note on toric periods in unramified families, Alexandros Groutides, [PREPRINT], No, https://arxiv.org/abs/2601.09418
• A p-adic Gross-Zagier formula for twisted triple product p-adic L-functions attached to finite slope families, Ting-Han Huang, Ananyo Kazi, [PREPRINT], No, https://arxiv.org/abs/2501.17474
• An Euler system for the adjoint of a modular form, David Loeffler, Sarah Livia Zerbes, [PREPRINT], No, https://arxiv.org/abs/2312.04665
• Asai-Flach classes, p-adic L-functions and the Bloch-Kato conjecture for GO(4), Giada Grossi, David Loeffler, Sarah Livia Zerbes, [PREPRINT], No, https://arxiv.org/abs/2407.17055
• Chebyshev's bias for modular forms, Shin-ya Koyama, Arshay Sheth, [PREPRINT], No, https://arxiv.org/abs/2509.04187
• Euler systems for GSp(4) over imaginary quadratic fields, Alexandros Groutides, [PREPRINT], No, https://arxiv.org/abs/2511.21384
• Integral structures in smooth GL2(Qp)-representations and zeta integrals, Alexandros Groutides, [PREPRINT], No, https://arxiv.org/abs/2401.10870
• Integrality of GL(2) x GL(2) Rankin-Selberg integrals for ramified representations, Groutides, Alexandros, [PREPRINT], No, http://arxiv.org/abs/2501.16972
• Moduli spaces of untwisted wild Riemann surfaces, Jean Douçot, Gabriele Rembado, Matteo Tamiozzo, [PREPRINT], No, https://arxiv.org/abs/2403.18505
• New perspectives on p-adic regulator formulae, Ting-Han Huang, Ananyo Kazi, Luca Marannano, [PREPRINT], No, https://arxiv.org/abs/2601.05406
• On some zeta-integrals for unramified representations of GSp(4), David Loeffler, [PREPRINT], No, https://arxiv.org/abs/2106.16076
• On the Birch-Swinnerton-Dyer conjecture for modular abelian surfaces, Loeffler, David, Zerbes, Sarah Livia, [PREPRINT], No, https://arxiv.org/abs/2110.13102
• On the Bloch--Kato conjecture for GSp(4) x GL(2), Loeffler, David, Zerbes, Sarah Livia, [PREPRINT], No, https://arxiv.org/abs/2106.14511
• On the p-ranks of class groups of certain Galois extensions, Ufuoma Asarhasa, Rusiru Gambheera, Debanjana Kundu, Enrique Nunez Lon-wo, Arshay Sheth, [PREPRINT], No, https://arxiv.org/abs/2408.04481
• Overconvergent Eichler-Shimura morphisms for GSp4, Hansheng Diao, Giovanni Rosso, Ju-Feng Wu, [PREPRINT], No, https://arxiv.org/abs/2506.02643
• P-adic Asai and twisted triple product L-functions for finite slope families, Ananyo Kazi, David Loeffler, [PREPRINT], No, https://arxiv.org/abs/2504.12066
• P-adic Asai L-functions for quadratic Hilbert eigenforms, Grossi, Giada, Loeffler, David, Zerbes, Sarah Livia, [PREPRINT], No, http://arxiv.org/abs/2307.07004
• Real geometric transcendence for Weierstrass elliptic functions, Arshay Sheth, Matteo Tamiozzo, [PREPRINT], No, https://arxiv.org/abs/2508.14645
• The Asai--Flach Euler system in p-adic families, David Loeffler, Arshay Sheth, [PREPRINT], No, https://arxiv.org/abs/2506.19673