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).
 

Articles in Journals

• Hansheng Diao, Giovanni Rosso, Ju-Feng Wu, Perfectoid overconvergent Siegel modular forms and the overconvergent Eichler–Shimura morphism, Research in the Mathematical Sciences, 13, Peer-reviewed, 10.1007/s40687-026-00610-5, https://arxiv.org/abs/2106.00094
• Alexandros Groutides, Integrality of GL(2) x GL(2) Rankin-Selberg integrals for ramified representations, Representation Theory, To appear, Peer-reviewed, https://arxiv.org/abs/2501.16972
• Alexandros Groutides, On integral aspects of Asai periods and Euler systems for Res_{E/Q}GL(2), Forum Mathematicum, 38, Peer-reviewed, 10.1515/forum-2024-0458, https://arxiv.org/abs/2409.17888
• Alexandros Groutides, On Rankin-Selberg integral structures and Euler systems for GL2 × GL2, Journal of Number Theory, 290, Peer-reviewed, 10.1016/j.jnt.2026.03.010, https://arxiv.org/abs/2407.01377
• Zeping Hao, David Loeffler, P-adic Rankin–Selberg L-functions in universal deformation families and functional equations, Forum Mathematicum, Vol. 38 no. 3, Peer-reviewed, 10.1515/forum-2024-0497, https://arxiv.org/abs/2405.12611
• Ting-Han Huang, Ju-Feng Wu, Finite polynomial cohomology with coefficients, Rendiconti del Seminario Matematico della Università di Padova, 155, Peer-reviewed, 10.4171/rsmup/173, https://arxiv.org/abs/2209.10289
• Pak-Hin Lee, Ju-Feng Wu, On 𝑝-adic adjoint 𝐿-functions for Bianchi cuspforms: The 𝑝-split case, Transactions of the American Mathematical Society, 378, Peer-reviewed, 10.1090/tran/9388, https://arxiv.org/abs/2306.15441
• David Loeffler, On local zeta-integrals for GSp(4) and GSp(4) x GL(2), New York J Math, 30, Peer-reviewed, https://arxiv.org/abs/2011.15106
• David Loeffler, Óscar Rivero, Algebraicity of L-values for GSp4×GL2 and GSp4×GL2×GL2, Quarterly J Math, Vol. 75 no. 2, Peer-reviewed, 10.1093/qmath/haae016, https://arxiv.org/abs/2303.16114
• David Loeffler, Óscar Rivero, Eisenstein degeneration of Euler systems, Journal für die reine und angewandte Mathematik (Crelles Journal), 2024, Peer-reviewed, 10.1515/crelle-2024-0057, https://arxiv.org/abs/2201.02078
• David Loeffler, Óscar Rivero, On p-adic L-functions for GSp4×GL2, Pacific Journal of Mathematics, 335, Peer-reviewed, 10.2140/pjm.2025.335.373, https://arxiv.org/abs/2305.07707
• David Loeffler, Michael Stoll, Formalizing zeta and L-functions in Lean, Annals of Formalized Mathematics, Vol. 1, 2025, Peer-reviewed, 10.46298/afm.15328, https://arxiv.org/abs/2503.00959
• David Loeffler, Chris Williams, P-adic L-functions for GL(3), Mathematische Annalen, 394, Peer-reviewed, 10.1007/s00208-026-03377-w, https://arxiv.org/abs/2111.04535
• David Loeffler, Sarah Livia Zerbes, A universal Euler system for GSp(4), Forum Mathematicum, To appear, Peer-reviewed, 10.1515/forum-2025-0062, https://arxiv.org/abs/2411.12576
• David Loeffler, Sarah Livia Zerbes, On the Bloch-Kato conjecture for GSp(4), Cambridge J Math, To appear, Peer-reviewed, https://arxiv.org/abs/2003.05960
• David Loeffler, Sarah Livia Zerbes, Plectic structures in p-adic de Rham cohomology, Journal of Number Theory, 270, Peer-reviewed, 10.1016/j.jnt.2023.10.016, https://arxiv.org/abs/2211.12078
• David Loeffler, Sarah Livia Zerbes, Poles of p-adic Asai L-functions and Distinguished Representations, International Mathematics Research Notices, 2025, Peer-reviewed, 10.1093/imrn/rnaf363, https://arxiv.org/abs/2501.11472
• Luca Mastella, Francesco Zerman, On anticyclotomic Euler and Kolyvagin systems, Ann. Math. Québec, Published online; print version to appear, Peer-reviewed, 10.1007/s40316-026-00269-y, https://arxiv.org/abs/2505.08710
• Arshay Sheth, Control theorems for Hilbert modular varieties, Canadian Mathematical Bulletin, 68, Peer-reviewed, 10.4153/s0008439524000791, https://arxiv.org/abs/2403.04738
• Arshay Sheth, Euler Product Asymptotics for L-functions of Elliptic Curves, International Mathematics Research Notices, 2025, Peer-reviewed, 10.1093/imrn/rnaf214, http://arxiv.org/abs/2312.05236
• Arshay Sheth, Euler Products at the Centre and Applications to Chebyshev’s Bias, Mathematical Proceedings of the Cambridge Philosophical Society, 179, Peer-reviewed, 10.1017/s0305004125000234, https://arxiv.org/abs/2405.01512
• Ju-Feng Wu, On Greenberg–Benois L-invariants and Fontaine–Mazur L-invariants, Canadian Mathematical Bulletin, 67, Peer-reviewed, 10.4153/s0008439524000638, https://arxiv.org/abs/2310.11098
• Francesco Zerman, Quaternionic Kolyvagin systems and Iwasawa theory for Hida families, Math. Res. Lett., To appear, Peer-reviewed, https://arxiv.org/abs/2507.04980

Chapters in Books

• Alessio Caminata, Francesco Zerman, Hilbert-Kunz series, F-signature series, and weak p-fractals, The Landscape of Commutative Algebra (LMS Lecture Note Series), To appear, Peer-reviewed, https://arxiv.org/abs/2510.19355
• David Loeffler, Robert Rockwood, Sarah Livia Zerbes, Spherical Varieties and p-Adic Families of Cohomology Classes, Springer Proceedings in Mathematics & Statistics, Elliptic Curves and Modular Forms in Arithmetic Geometry, Vol. 527, Peer-reviewed, 10.1007/978-3-032-13123-2_11, https://arxiv.org/abs/2106.16082

Preprints

• Ufuoma Asarhasa, Rusiru Gambheera, Debanjana Kundu, Enrique Nunez Lon-wo, Arshay Sheth, On the p-ranks of class groups of certain Galois extensions, preprint, Not peer-reviewed, https://arxiv.org/abs/2408.04481
• Hansheng Diao, Giovanni Rosso, Ju-Feng Wu, Overconvergent Eichler-Shimura morphisms for GSp4, preprint, Not peer-reviewed, https://arxiv.org/abs/2506.02643
• Jean Douçot, Gabriele Rembado, Matteo Tamiozzo, Moduli spaces of untwisted wild Riemann surfaces, preprint, Not peer-reviewed, https://arxiv.org/abs/2403.18505
• Giada Grossi, David Loeffler, Sarah Livia Zerbes, Asai-Flach classes, p-adic L-functions and the Bloch-Kato conjecture for GO(4), preprint, Not peer-reviewed, https://arxiv.org/abs/2407.17055
• Giada Grossi, David Loeffler, Sarah Livia Zerbes, P-adic Asai L-functions for quadratic Hilbert eigenforms, preprint, Not peer-reviewed, http://arxiv.org/abs/2307.07004
• Alexandros Groutides, A note on toric periods in unramified families, preprint, Not peer-reviewed, https://arxiv.org/abs/2601.09418
• Alexandros Groutides, Euler systems for GSp(4) over imaginary quadratic fields, preprint, Not peer-reviewed, https://arxiv.org/abs/2511.21384
• Alexandros Groutides, Integral structures in smooth GL2(Qp)-representations and zeta integrals, preprint, Not peer-reviewed, https://arxiv.org/abs/2401.10870
• Ting-Han Huang, Ananyo Kazi, A p-adic Gross-Zagier formula for twisted triple product p-adic L-functions attached to finite slope families, preprint, Not peer-reviewed, https://arxiv.org/abs/2501.17474
• Ting-Han Huang, Ananyo Kazi, Luca Marannano, New perspectives on p-adic regulator formulae, preprint, Not peer-reviewed, https://arxiv.org/abs/2601.05406
• Ananyo Kazi, David Loeffler, P-adic Asai and twisted triple product L-functions for finite slope families, preprint, Not peer-reviewed, https://arxiv.org/abs/2504.12066
• Shin-ya Koyama, Arshay Sheth, Chebyshev's bias for modular forms, preprint, Not peer-reviewed, https://arxiv.org/abs/2509.04187
• David Loeffler, On some zeta-integrals for unramified representations of GSp(4), preprint, Not peer-reviewed, https://arxiv.org/abs/2106.16076
• David Loeffler, Arshay Sheth, The Asai--Flach Euler system in p-adic families, preprint, Not peer-reviewed, https://arxiv.org/abs/2506.19673
• David Loeffler, Sarah Livia Zerbes, An Euler system for the adjoint of a modular form, preprint, Not peer-reviewed, https://arxiv.org/abs/2312.04665
• David Loeffler, Sarah Livia Zerbes, On the Birch-Swinnerton-Dyer conjecture for modular abelian surfaces, preprint, Not peer-reviewed, https://arxiv.org/abs/2110.13102
• David Loeffler, Sarah Livia Zerbes, On the Bloch--Kato conjecture for GSp(4) x GL(2), preprint, Not peer-reviewed, https://arxiv.org/abs/2106.14511
• Arshay Sheth, Matteo Tamiozzo, Real geometric transcendence for Weierstrass elliptic functions, preprint, Not peer-reviewed, https://arxiv.org/abs/2508.14645