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1. On symmetric representations of 𝑆𝐿₂(ℤ)

   https://doi.org/10.1090/proc/16205  

   Ng, Siu-Hung ; Wang, Yilong ; Wilson, Samuel ( January 2023 , Proceedings of the American Mathematical Society)

   We introduce the notions of symmetric and symmetrizable representations of${\operatorname {SL}_2(\mathbb {Z})}$. The linear representations of${\operatorname {SL}_2(\mathbb {Z})}$arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of${\operatorname {SL}_2(\mathbb {Z})}$. By investigating a$\mathbb {Z}/2\mathbb {Z}$-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of${\operatorname {SL}_2(\mathbb {Z})}$are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of${\operatorname {SL}_2(\mathbb {Z})}$that are subrepresentations of a symmetric one.

    

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2. The stable exotic Cuntz algebras are higher-rank graph algebras

   https://doi.org/10.1090/bproc/180  

   Boersema, Jeffrey ; Browne, Sarah ; Gillaspy, Elizabeth ( March 2024 , Proceedings of the American Mathematical Society, Series B)

   For each odd integer$n \geq 3$, we construct a rank-3 graph$\Lambda _n$with involution$\gamma _n$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)$is stably isomorphic to the exotic Cuntz algebra$\mathcal E_n$. This construction is optimal, as we prove that a rank-2 graph with involution$(\Lambda ,\gamma )$can never satisfy$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n$, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math.10(2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution$(\Lambda , \gamma )$whose real$C^*$-algebra$C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )$is stably isomorphic to the suspension$S \mathbb {R}$. In the Appendix, we show that the$i$-fold suspension$S^i \mathbb {R}$is stably isomorphic to a graph algebra iff$-2 \leq i \leq 1$.

    

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3. The reduced ring of the 𝑅𝑂(𝐶₂)-graded 𝐶₂-equivariant stable stems

   https://doi.org/10.1090/bproc/203  

   Belmont, Eva ; Xu, Zhouli ; Zhang, Shangjie ( January 2024 , Proceedings of the American Mathematical Society, Series B)

   We describe in terms of generators and relations the ring structure of the$RO(C_2)$-graded$C_2$-equivariant stable stems$\pi _\star ^{C_2}$modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational$C_2$-equivariant sphere$\pi _\star ^{C_2}(\mathbb {S}_\mathbb {Q})$.

    

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4. Cyclic isogenies of elliptic curves over fixed quadratic fields

   https://doi.org/10.1090/mcom/3894  

   Banwait, Barinder ; Najman, Filip ; Padurariu, Oana ( March 2024 , Mathematics of Computation)

   Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over$\mathbb {Q}$. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised.

   In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields$\mathbb {Q}(\sqrt {d})$with$|d| > 10^4$we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over$19$quadratic fields, including$\mathbb {Q}(\sqrt {213})$and$\mathbb {Q}(\sqrt {-2289})$. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves$X_0(125)$and$X_0(169)$, which may be of independent interest.

    

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5. A modular construction of unramified 𝑝-extensions of ℚ(ℕ^{1/𝕡})

   https://doi.org/10.1090/bproc/141  

   Lang, Jaclyn ; Wake, Preston ( October 2022 , Proceedings of the American Mathematical Society, Series B)

   We show that for primes$N, p \geq 5$with$N \equiv -1 \bmod p$, the class number of$\mathbb {Q}(N^{1/p})$is divisible by$p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when$N \equiv -1 \bmod p$, there is always a cusp form of weight$2$and level$\Gamma _0(N^2)$whose$\ell$th Fourier coefficient is congruent to$\ell + 1$modulo a prime above$p$, for all primes$\ell$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree-$p$extension of$\mathbb {Q}(N^{1/p})$.

    

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