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1. MULTIPLICITY ONE AT FULL CONGRUENCE LEVEL

   https://doi.org/10.1017/S1474748020000225  

   Le, Daniel; Morra, Stefano; Schraen, Benjamin (May 2020, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Let $$F$$ be a totally real field in which $$p$$ is unramified. Let $$\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $$v$$ above $$p$$ . Let $$\mathfrak{m}$$ be the corresponding Hecke eigensystem. We describe the $$\mathfrak{m}$$ -torsion in the $$\text{mod}\,p$$ cohomology of Shimura curves with full congruence level at $$v$$ as a $$\text{GL}_{2}(k_{v})$$ -representation. In particular, it only depends on $$\overline{r}|_{I_{F_{v}}}$$ and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $$\text{GL}_{2}(\mathbf{F}_{q})$$ -projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math.   200 (1) (2015), 1–96]. 

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2. G -valued crystalline deformation rings in the Fontaine–Laffaille range

   https://doi.org/10.1112/S0010437X23007297  

   Booher, Jeremy; Levin, Brandon (August 2023, Compositio Mathematica)

   Let$$G$$be a split reductive group over the ring of integers in a$$p$$-adic field with residue field$$\mathbf {F}$$. Fix a representation$$\overline {\rho }$$of the absolute Galois group of an unramified extension of$$\mathbf {Q}_p$$, valued in$$G(\mathbf {F})$$. We study the crystalline deformation ring for$$\overline {\rho }$$with a fixed$$p$$-adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for$$G$$-valued representations. In particular, we give a root theoretic condition on the$$p$$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups. 

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3. ON THE EXISTENCE OF ADMISSIBLE SUPERSINGULAR REPRESENTATIONS OF -ADIC REDUCTIVE GROUPS

   https://doi.org/10.1017/fms.2019.50  

   HERZIG, FLORIAN; KOZIOŁ, KAROL; VIGNÉRAS, MARIE-FRANCE (January 2020, Forum of Mathematics, Sigma)

   Suppose that $$\mathbf{G}$$ is a connected reductive group over a finite extension $$F/\mathbb{Q}_{p}$$ and that $$C$$ is a field of characteristic  $$p$$ . We prove that the group $$\mathbf{G}(F)$$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over  $$C$$ . 

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4. Every finite abelian group is the group of rational points of an ordinary abelian variety over 𝔽₂, 𝔽₃ and 𝔽₅

   https://doi.org/10.1090/proc/16127  

   Marseglia, Stefano; Springer, Caleb (February 2023, Proceedings of the American Mathematical Society)

   We show that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over ${\mathbb{F}}_2$, ${\mathbb{F}}_3$and ${\mathbb{F}}_5$. We produce partial results for abelian varieties over a general finite field  ${\mathbb{F}}_q$. In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over ${\mathbb{F}}_q$when $q$is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over  ${\mathbb{F}}_2$. 

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5. An inverse Jacobian algorithm for Picard curves

   https://doi.org/10.1007/s40993-021-00253-1  

   Lario, Joan-C.; Somoza, Anna; Vincent, Christelle (June 2021, Research in Number Theory)

   null (Ed.)

   Abstract We study the inverse Jacobian problem for the case of Picard curves over $${\mathbb {C}}$$ C . More precisely, we elaborate on an algorithm that, given a small period matrix $$\varOmega \in {\mathbb {C}}^{3\times 3}$$ Ω ∈ C 3 × 3 corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre–Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike–Weng (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most $$4$$ 4 . In particular, we obtain the complete list of maximal CM Picard curves defined over $${\mathbb {Q}}$$ Q . In the appendix, Vincent gives a correction to the generalization of Takase’s formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan–Ionica–Lauter–Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016]. 

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