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1. Galois automorphisms on Harish-Chandra series and Navarro’s self-normalizing Sylow $$2$$-subgroup conjecture

   https://doi.org/10.1090/tran/7590  

   Schaeffer Fry, A. A. (July 2019, Transactions of the American Mathematical Society)

   Navarro has conjectured a necessary and sufficient condition for a finite group G to have a self-normalizing Sylow 2-subgroup, which is given in terms of the ordinary irreducible characters of G. In a previous article, Schaeffer Fry has reduced the proof of this conjecture to showing that certain related statements hold for simple groups. In this article, we describe the action of Galois automorphisms on the Howlett-Lehrer parametrization of Harish-Chandra induced characters. We use this to complete the proof of the conjecture by showing that the remaining simple groups satisfy the required conditions. 

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2. Algebraic stability of meromorphic maps descended from Thurston’s pullback maps

   https://doi.org/10.1090/tran/8221  

   Ramadas, Rohini (January 2021, Transactions of the American Mathematical Society)

   null (Ed.)

   Let ϕ : S 2 → S 2 \phi :S^2 \to S^2 be an orientation-preserving branched covering whose post-critical set has finite cardinality n n . If ϕ \phi has a fully ramified periodic point p ∞ p_{\infty } and satisfies certain additional conditions, then, by work of Koch, ϕ \phi induces a meromorphic self-map R ϕ R_{\phi } on the moduli space M 0 , n \mathcal {M}_{0,n} ; R ϕ R_{\phi } descends from Thurston’s pullback map on Teichmüller space. Here, we relate the dynamics of R ϕ R_{\phi } on M 0 , n \mathcal {M}_{0,n} to the dynamics of ϕ \phi on S 2 S^2 . Let ℓ \ell be the length of the periodic cycle in which the fully ramified point p ∞ p_{\infty } lies; we show that R ϕ R_{\phi } is algebraically stable on the heavy-light Hassett space corresponding to ℓ \ell heavy marked points and ( n − ℓ ) (n-\ell ) light points. 

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3. Hodge classes and the Jacquet–Langlands correspondence

   https://doi.org/10.1017/fmp.2023.20  

   Ichino, Atsushi; Prasanna, Kartik (September 2023, Forum of Mathematics, Pi)

   We prove that the Jacquet–Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in ℓ-adic cohomology is Galois invariant for all ℓ. 

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4. Vanishing of quartic and sextic twists of L-functions

   https://doi.org/10.1007/s40993-023-00499-x  

   Berg, Jennifer; Ryan, Nathan C.; Young, Matthew P. (February 2024, Research in Number Theory)

   Abstract LetEbe an elliptic curve over$${{\mathbb {Q}}}$$ $Q$. We conjecture asymptotic estimates for the number of vanishings of$$L(E,1,\chi )$$ $L(E,1,\chi )$as$$\chi $$ $\chi$varies over all primitive Dirichlet characters of orders 4 and 6, subject to a mild hypothesis onE. Our conjectures about these families come from conjectures about random unitary matrices as predicted by the philosophy of Katz-Sarnak. We support our conjectures with numerical evidence. Compared to earlier work by David, Fearnley and Kisilevsky that formulated analogous conjectures for characters of any odd prime order, in the composite order case, we need to justify our use of random matrix theory heuristics by analyzing the equidistribution of the squares of normalized Gauss sums. To do this, we introduce the notion of totally order$$\ell $$ $\ell$characters to quantify how quickly the quartic and sextic Gauss sums become equidistributed. Surprisingly, the rate of equidistribution in the full family of quartic (resp., sextic) characters is much slower than in the sub-family of totally quartic (resp., sextic) characters. We provide a conceptual explanation for this phenomenon by observing that the full family of order$$\ell $$ $\ell$twisted elliptic curveL-functions, with$$\ell $$ $\ell$even and composite, is a mixed family with both unitary and orthogonal aspects. 

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5. The fields of values of characters of degree not divisible by p

   https://doi.org/10.1017/fmp.2021.1  

   Navarro, Gabriel; Tiep, Pham Huu (January 2021, Forum of Mathematics, Pi)

   Abstract We study the fields of values of the irreducible characters of a finite group of degree not divisible by a prime p . In the case where $p=2$ , we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups. 

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