More Like this

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. 

   more » « less
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. 

   more » « less
3. Moduli Spaces of Quadratic Maps: Arithmetic and Geometry

   https://doi.org/10.1093/imrn/rnae126  

   Ramadas, Rohini ( June 2024 , International Mathematics Research Notices)

   We establish an implication between two long-standing open problems in complex dynamics. The roots of the $n$th Gleason polynomial $G_{n}\in{\mathbb{Q}}[c]$ comprise the $0$-dimensional moduli space of quadratic polynomials with an $n$-periodic critical point. $\operatorname{Per}_{n}(0)$ is the $1$-dimensional moduli space of quadratic rational maps on ${\mathbb{P}}^{1}$ with an $n$-periodic critical point. We show that if $G_{n}$ is irreducible over ${\mathbb{Q}}$, then $\operatorname{Per}_{n}(0)$ is irreducible over ${\mathbb{C}}$. To do this, we exhibit a ${\mathbb{Q}}$-rational smooth point on a projective completion of $\operatorname{Per}_{n}(0)$, using the admissible covers completion of a Hurwitz space. In contrast, the Uniform Boundedness Conjecture in arithmetic dynamics would imply that for sufficiently large $n$, $\operatorname{Per}_{n}(0)$ itself has no ${\mathbb{Q}}$-rational points.

    

   more » « less
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. 

   more » « less
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 . 

   more » « less