Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userLiminal reciprocity and factorization statistics
185
0
0.0
(
0
)
Authors
Trevor Hyde
Ask ChatGPT about the research
No Arabic abstract
Let $M_{d,n}(q)$ denote the number of monic irreducible polynomials in $mathbb{F}_q[x_1, x_2, ldots , x_n]$ of degree $d$. We show that for a fixed degree $d$, the sequence $M_{d,n}(q)$ converges $q$-adically to an explicitly determined rational function $M_{d,infty}(q)$. Furthermore we show that the limit $M_{d,infty}(q)$ is related to the classic necklace polynomial $M_{d,1}(q)$ by an involutive functional equation, leading to a phenomenon we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of a family of symmetric group representations as a consequence of liminal reciprocity.
rate research
Read More
We announce recent results on a connection between factorization statistics of polynomials over a finite field and the structure of the cohomology of configurations in $mathbb{R}^3$ as a representation of the symmetric group. This connection parallels a result of Church, Ellenberg, and Farb relating factorization statistics of squarefree polynomials and the cohomology of configurations in $mathbb{R}^2$.
We use generating functions to relate the expected values of polynomial factorization statistics over $mathbb{F}_q$ to the cohomology of ordered configurations in $mathbb{R}^3$ as a representation of the symmetric group. Our methods lead to a new proof of the twisted Grothendieck-Lefschetz formula for squarefree polynomial factorization statistics of Church, Ellenberg, and Farb.
We obtain estimates on the number $|mathcal{A}_{boldsymbol{lambda}}|$ of elements on a linear family $mathcal{A}$ of monic polynomials of $mathbb{F}_q[T]$ of degree $n$ having factorization pattern $boldsymbol{lambda}:=1^{lambda_1}2^{lambda_2}cdots n^{lambda_n}$. We show that $|mathcal{A}_{boldsymbol{lambda}}|= mathcal{T}(boldsymbol{lambda}),q^{n-m}+mathcal{O}(q^{n-m-{1}/{2}})$, where $mathcal{T}(boldsymbol{lambda})$ is the proportion of elements of the symmetric group of $n$ elements with cycle pattern $boldsymbol{lambda}$ and $m$ is the codimension of $mathcal{A}$. Furthermore, if the family $mathcal{A}$ under consideration is sparse, then $|mathcal{A}_{boldsymbol{lambda}}|= mathcal{T}(boldsymbol{lambda}),q^{n-m}+mathcal{O}(q^{n-m-{1}})$. Our estimates hold for fields $mathbb{F}_q$ of characteristic greater than 2. We provide explicit upper bounds for the constants underlying the $mathcal{O}$--notation in terms of $boldsymbol{lambda}$ and $mathcal{A}$ with good behavior. Our approach reduces the question to estimate the number of $mathbb{F}_q$--rational points of certain families of complete intersections defined over $mathbb{F}_q$. Such complete intersections are defined by polynomials which are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning their singular locus, from which precise estimates on their number of $mathbb{F}_q$--rational points are established.
This paper has been withdrawn, as it is superseded by arXiv:0806.2122 (Bloch-Kato exponential maps for local fields with imperfect residue fields), which is a more recent version of the same paper.
We prove that the central values of additive twists of a cuspidal $L$-function define a quantum modular form in the sense of Zagier, generalizing recent results of Bettin and Drappeau. From this we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal $L$-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet $L$-functions. Furthermore we give an interpretation of quantum modularity at infinity for additive twists of $L$-functions of weight 2 cusp forms in terms of the corresponding functional equations.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
