The Beilinson–Bloch–Kato conjecture and theta correspondence
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Peng-hao_peng-Doctorial-Mathematics-2026-thesis.pdf
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Author(s)
Peng, Hao
Advisor(s)
Zhang, Wei
Date Issued
May 2026
Publisher
Massachusetts Institute of Technology
Abstract
We study the Beilinson–Bloch–Kato conjecture in analytic rank zero for motives attached
to conjugate self-dual and self-dual automorphic representations. In the conjugate self-dual
case, we prove that the nonvanishing of the central critical value implies the vanishing of the corresponding Bloch–Kato Selmer group for relevant conjugate self-dual automorphic representations of GL(2r); in particular, this yields vanishing results for odd symmetric powers of modular elliptic curves.
The main new ingredient is an arithmetic use of local and global seesaw identities. This
allows us to relate Selmer-vanishing problems to Gan–Gross–Prasad type period problems through theta correspondence and a Burger–Sarnak type principle. In particular, for conjugate self-dual automorphic representations of GL(2r), we reduce the Selmer problem to an endoscopic Rankin–Selberg case and establish the resulting Selmer-vanishing theorem.
In the self-dual analytic rank-zero case, the same method leads to a reduction of the
Bloch–Kato conjecture to a conjectural orthogonal Gross–Prasad criterion. This provides a uniform framework that connects Selmer vanishing, theta correspondence, and Gan–Gross–Prasad periods in both the conjugate self-dual and self-dual settings.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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