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1. A unipotent circle action on 𝑝-adic modular forms

   https://doi.org/10.1090/btran/52  

   Howe, Sean (November 2020, Transactions of the American Mathematical Society, Series B)

   null (Ed.)

   Following a suggestion of Peter Scholze, we construct an action of G m ^ \widehat {\mathbb {G}_m} on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the p p -adic modular curve whose ring of functions is Serre’s space of p p -adic modular functions. This action is a local, p p -adic analog of a global, archimedean action of the circle group S 1 S^1 on the lattice-unstable locus of the modular curve over C \mathbb {C} . To construct the G m ^ \widehat {\mathbb {G}_m} -action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates q q ; along the way we also prove a natural generalization of Dwork’s equation τ = log ⁡ q \tau =\log q for extensions of Q p / Z p \mathbb {Q}_p/\mathbb {Z}_p by μ p ∞ \mu _{p^\infty } valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of G m ^ \widehat {\mathbb {G}_m} integrates the differential operator θ \theta coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and p p -adic L L -functions. 

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2. Character bounds for regular semisimple elements and asymptotic results on Thompson’s conjecture

   https://doi.org/10.1007/s00209-022-03193-3  

   Larsen, Michael; Taylor, Jay; Tiep, Pham Huu (February 2023, Mathematische Zeitschrift)

   Abstract For every integer k there exists a bound $$B=B(k)$$ B = B ( k ) such that if the characteristic polynomial of $$g\in \textrm{SL}_n(q)$$ g ∈ SL n ( q ) is the product of $$\le k$$ ≤ k pairwise distinct monic irreducible polynomials over $$\mathbb {F}_q$$ F q , then every element x of $$\textrm{SL}_n(q)$$ SL n ( q ) of support at least B is the product of two conjugates of g . We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions ( p ,  q ), in the special case that $$n=p$$ n = p is prime, if g has order $$\frac{q^p-1}{q-1}$$ q p - 1 q - 1 , then every non-scalar element $$x \in \textrm{SL}_p(q)$$ x ∈ SL p ( q ) is the product of two conjugates of g . The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups. 

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3. Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields

   https://doi.org/10.1007/s40993-024-00584-9  

   Marseglia, Stefano (March 2025, Research in Number Theory)

   Abstract We give an algorithm to compute representatives of the conjugacy classes of semisimple square integral matrices with given minimal and characteristic polynomials. We also give an algorithm to compute the $$\mathbb {F}_q$$ $F_q$-isomorphism classes of abelian varieties over a finite field $$\mathbb {F}_q$$ $F_q$which belong to an isogeny class determined by a characteristic polynomial hof Frobenius when his ordinary, or qis prime and hhas no real roots. 

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4. 𝚤Hall algebra of the projective line and 𝑞-Onsager algebra

   https://doi.org/10.1090/tran/8798  

   Lu, Ming; Ruan, Shiquan; Wang, Weiqiang (April 2023, Transactions of the American Mathematical Society)

   The ı \imath Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of 1 1 -periodic complexes of coherent sheaves on the projective line. This ı \imath Hall algebra is shown to realize the universal q q -Onsager algebra (i.e., ı \imath quantum group of split affine A 1 A_1 type) in its Drinfeld type presentation. The ı \imath Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two ı \imath Hall algebras, explaining the isomorphism of the q q -Onsager algebra under the two presentations. 

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5. Rationality of twists of the Siegel modular variety of genus 2 and level 3

   https://doi.org/10.1090/proc/15854  

   Calegari, Frank; Chidambaram, Shiva (February 2022, Proceedings of the American Mathematical Society)

   Let  ρ ¯ : G Q → GSp 4 ⁡ ( F 3 ) \overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3) be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider ρ ¯ \overline {\rho } to be the Galois representation associated to the  3 3 -torsion of a principally polarized abelian surface  A / Q A/\mathbf {Q} . We prove that the moduli space  A 2 ( ρ ¯ ) \mathcal {A}_2(\overline {\rho }) of principally polarized abelian surfaces  B / Q B/\mathbf {Q} admitting a symplectic isomorphism  B [ 3 ] ≃ ρ ¯ B[3] \simeq \overline {\rho } of Galois representations is never rational over  Q \mathbf {Q} when  ρ ¯ \overline {\rho } is surjective, even though it is both rational over  C \mathbf {C} and unirational over  Q \mathbf {Q} via a map of degree  6 6 . 

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