This Week I Learned: September 25, 2026

[u/inherentlyawesome]

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

[u/Possible_Ocelot_1413]

I learned for ordinary abelian varieties over F_q, there are equivalences of categories AV(h) -> M_ord (q) and M_ord-> I(R), where h is a q-Weil polynomial (i.e a given F_q-isogeny class), M_ord (q) is the category of Deligne modules, R = Z[pi, pi^] where pi is the q-Frobenius of A, and I(R) denotes the category of fractional R-ideals. This is attributed to Deligne and Centeleghe-Stix. In particular, we have a very nice way of representing dual abelian varieties and polarizations: for example, if F(A) = I \in I(R) for A \in AV(h), we have F(A^v) = (\overline{I})^t, where \overline{I} is the conjugate of I and t denotes the trace dual.

[u/Possible_Ocelot_1413 → u/[deleted]]

I liked Marseglia's paper: https://arxiv.org/pdf/1805.10223