数论讨论班

摘要：Let L be a finite extension of Q_p and G_K be its absolute Galois group. Given two crystalline Z_p representations of G_K, we are interested in the relation between congruences of the two representations and congruences of their Bloch-Kato Selmer groups. Under certain assumptions, we do not obtain enough information from the congruences of the representations, and will consider instead the reductions of the prismatic F-crystals attached to these crystalline representations and explain how these reductions induce congruences of their first syntomic cohomology groups, which can be regarded as refinements of local Bloch-Kato Selmer groups.

摘要：Kudla—Rapoport conjecture predicts a precise identity between the arithmetic intersection number of special cycles on unitary Rapoport-Zink spaces and the derivative of local representation densities of hermitian forms. This result was proven by C. Li and W. Zhang in 2022 at the hyperspecial level. In this talk, I will discuss and motivate the conjecture and its global counterpart. If time permits, I will present a new proof of the conjecture using a global method.

摘要：Tracing back to Ribet's celebrated level-raising theorem for modular forms, presented at the 1983 ICM, a series of works by recent mathematicians—including Lu-Tian, Liu-Tian-Xiao-Zhang, Zhu, Wang, and others—have shown that one can obtain certain level-raising results for automorphic representations by establishing the surjectivity of the cycle class map for Shimura varieties. More precisely, under some generic conditions, cycles arising from the supersingular locus of the mod p fiber generate the non-Eisenstein parts of the cohomology of Shimura varieties. Inspired by the work of Rong Zhou, the above philosophy also extends to higher Chow cycles. In this talk, I will focus on the mod p fiber of \(U(2r,1)\) Shimura varieties, where r is an odd inner prime. We exhibit higher Chow cycles on the supersingular locus and prove the surjectivity of the cycle class map for the higher Chow group under some generic conditions. A key ingredient of the proof is to establish a strong form of Ihara's lemma for unitary groups. This work is joint with Hao Fu.

摘要：In 1989, Adolphson and Sperber established that for non-degenerate exponential sums, the associated L-function is a polynomial whose precise degree is determined by the volume of its Newton polyhedron. Building on this work, Haessig and Sperber (2014) developed a framework to analyze arbitrary algebraic families of lower order deformations of non-degenerate toric exponential sums, computing their relative cohomology and establishing sharp estimates for their relative Frobenius map. This talk explores the associated symmetric power L-functions within this framework, detailing their construction and the p-adic cohomological methods that establish the main structural theorems for these families.