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1. The K-theory of perfectoid rings

   https://doi.org/10.4171/DM/X22  

   Antieau, Ben; Mathew, Akhil; Morrow, Matthew (January 2022, Documenta Mathematica)

   We establish various properties of thep-adic algebraic\text{K}-theory of smooth algebras over perfectoid rings living over perfectoid valuation rings. In particular, thep-adic\text{K}-theory of such rings is homotopy invariant, and coincides with thep-adic\text{K}-theory of thep-adic generic fibre in high degrees. In the case of smooth algebras over perfectoid valuation rings of mixed characteristic the latter isomorphism holds in all degrees and generalises a result of Nizioł. 

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2. Hypergeometric L-functions in average polynomial time, II

   https://doi.org/10.1007/s40993-024-00593-8  

   Costa, Edgar; Kedlaya, Kiran S; Roe, David (January 2025, Research in Number Theory)

   Abstract We describe an algorithm for computing, for all primes$$p \le X$$ $p\le X$, the trace of Frobenius atpof a hypergeometric motive over$$\mathbb {Q}$$ $Q$in time quasilinear inX. This involves computing the trace modulo$$p^e$$ $p^e$for suitablee; as in our previous work treating the case$$e=1$$ $e=1$, we combine the Beukers–Cohen–Mellit trace formula with average polynomial time techniques of Harvey and Harvey–Sutherland. The key new ingredient for$$e>1$$ $e>1$is an expanded version of Harvey’s “generic prime” construction, making it possible to incorporate certainp-adic transcendental functions into the computation; one of these is thep-adic Gamma function, whose average polynomial time computation is an intermediate step which may be of independent interest. We also provide an implementation in Sage and discuss the remaining computational issues around tabulating hypergeometricL-series. 

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3. Geometric $$L$$-packets of Howe-unramified toral supercuspidal representations

   https://doi.org/10.4171/JEMS/1396  

   Chan, Charlotte; Oi, Masao (March 2025, Journal of the European Mathematical Society)

   We show thatL-packets of toral supercuspidal representations arising from unramified maximal tori ofp-adic groups are realized by Deligne–Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra’s characterization of real discrete series representations by their characters on regular elements of compact maximal tori, a characterization which Langlands relied on in his construction ofL-packets of these representations. In parallel to the real case, we characterize the members of Kaletha’s toralL-packets by their characters on sufficiently regular elements of elliptic maximal tori. 

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4. Drinfeld's lemma for F -isocrystals, II: Tannakian approach

   https://doi.org/10.1112/S0010437X23007571  

   Kedlaya, Kiran S; Xu, Daxin (January 2024, Compositio Mathematica)

   We prove a Tannakian form of Drinfeld's lemma for isocrystals on a variety over a finite field, equipped with actions of partial Frobenius operators. This provides an intermediate step towards transferring V. Lafforgue's work on the Langlands correspondence over function fields from$$\ell$$-adic to$$p$$-adic coefficients. We also discuss a motivic variant and a local variant of Drinfeld's lemma. 

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5. p -ADIC L -FUNCTIONS AND RATIONAL POINTS ON CM ELLIPTIC CURVES AT INERT PRIMES

   https://doi.org/10.1017/S147474802300021X  

   Burungale, Ashay A; Kobayashi, Shinichi; Ota, Kazuto (May 2024, Journal of the Institute of Mathematics of Jussieu)

   Abstract LetKbe an imaginary quadratic field and$$p\geq 5$$a rational prime inert inK. For a$$\mathbb {Q}$$-curveEwith complex multiplication by$$\mathcal {O}_K$$and good reduction atp, K. Rubin introduced ap-adicL-function$$\mathscr {L}_{E}$$which interpolates special values ofL-functions ofEtwisted by anticyclotomic characters ofK. In this paper, we prove a formula which links certain values of$$\mathscr {L}_{E}$$outside its defining range of interpolation with rational points onE. Arithmetic consequences includep-converse to the Gross–Zagier and Kolyvagin theorem forE. A key tool of the proof is the recent resolution of Rubin’s conjecture on the structure of local units in the anticyclotomic$${\mathbb {Z}}_p$$-extension$$\Psi _\infty $$of the unramified quadratic extension of$${\mathbb {Q}}_p$$. Along the way, we present a theory of local points over$$\Psi _\infty $$of the Lubin–Tate formal group of height$$2$$for the uniformizing parameter$$-p$$. 

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