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1. Negative Moments of L –Functions With Small Shifts Over Function Fields

   https://doi.org/10.1093/imrn/rnad118  

   Florea, Alexandra (June 2023, International Mathematics Research Notices)

   Abstract We consider negative moments of quadratic Dirichlet $$L$$–functions over function fields. Summing over monic square-free polynomials of degree $2g+1$ in $$\mathbb{F}_{q}[x]$$, we obtain an asymptotic formula for the $$k^{\textrm{th}}$$ shifted negative moment of $$L(1/2+\beta ,\chi _{D})$$, in certain ranges of $$\beta $$ (e.g., when roughly $$\beta \gg \log g/g $$ and $k<1$). We also obtain non-trivial upper bounds for the $$k^{\textrm{th}}$$ shifted negative moment when $$\log (1/\beta ) \ll \log g$$. Previously, almost sharp upper bounds were obtained in [ 3] in the range $$\beta \gg g^{-\frac{1}{2k}+\epsilon }$$. 

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2. Calibrated representations of the double Dyck path algebra

   https://doi.org/10.1007/s00208-024-02937-2  

   González, Nicolle; Gorsky, Eugene; Simental, José (August 2024, Mathematische Annalen)

   Abstract The double Dyck path algebra$$\mathbb {A}_{q,t}$$ $A_{q,t}$and its polynomial representation first arose as a key figure in the proof of the celebrated Shuffle Theorem of Carlsson and Mellit. A geometric formulation for an equivalent algebra$$\mathbb {B}_{q,t}$$ $B_{q,t}$was then given by the second author and Carlsson and Mellit using the K-theory of parabolic flag Hilbert schemes. In this article, we initiate the systematic study of the representation theory of the double Dyck path algebra$$\mathbb {B}_{q,t}$$ $B_{q,t}$. We define a natural extension of this algebra and study its calibrated representations. We show that the polynomial representation is calibrated, and place it into a large family of calibrated representations constructed from posets satisfying certain conditions. We also define tensor products and duals of these representations, thus proving (under suitable conditions) the category of calibrated representations is generically monoidal. As an application, we prove that tensor powers of the polynomial representation can be constructed from the equivariant K-theory of parabolic Gieseker moduli spaces. 

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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. Asymptotic second moment of Dirichlet L -functions along a thin coset

   https://doi.org/10.1017/fms.2025.44  

   Garcia, Bradford; Young, Matthew P (January 2025, Forum of Mathematics, Sigma)

   Abstract We prove an asymptotic formula for the second moment of central values of DirichletL-functions restricted to a coset. More specifically, consider a coset of the subgroup of characters modulodinside the full group of characters moduloq. Suppose that$$\nu _p(d) \geq \nu _p(q)/2$$for all primespdividingq. In this range, we obtain an asymptotic formula with a power-saving error term; curiously, there is a secondary main term of rough size$$q^{1/2}$$here which is not predicted by the integral moments conjecture of Conrey, Farmer, Keating, Rubinstein, and Snaith. The lower-order main term does not appear in the second moment of the Riemann zeta function, so this feature is not anticipated from the analogous archimedean moment problem. We also obtain an asymptotic result for smallerd, with$$\nu _p(q)/3 \leq \nu _p(d) \leq \nu _p(q)/2$$, with a power-saving error term fordlarger than$$q^{2/5}$$. In this more difficult range, the secondary main term somewhat changes its form and may have size roughlyd, which is only slightly smaller than the diagonal main term. 

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5. 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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