The statistics of low-lying zeros of quadratic Dirichlet L-functions were conjectured by Katz and... more The statistics of low-lying zeros of quadratic Dirichlet L-functions were conjectured by Katz and Sarnak to be given by the scaling limit of eigenvalues from the unitary symplectic ensemble. The n-level densities were found to be in agreement with this in a certain neighborhood of the origin in the Fourier domain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend the neighborhood was made in the Ph.D. thesis of Peng Gao (2005), who under GRH gave the density as a complicated combinatorial factor, but it remained open whether it coincides with the Random Matrix Theory factor. For n at most 7 this was recently confirmed by Levinson and Miller. We resolve this problem for all n, not by directly doing the combinatorics, but by passing to a function field analogue, of L-functions associated to hyper-elliptic curves of given genus g over a field of q elements. We show that the answer in this case coincides with Gao's combinatorial factor up to a controlled error. We then ...
For a function field K and fixed polynomial F∈ K[x] and varying f∈ F (under certain restrictions)... more For a function field K and fixed polynomial F∈ K[x] and varying f∈ F (under certain restrictions) we give a lower bound for the degree of the greatest prime divisor of F(f) in terms of the height of f, establishing a strong result for the function field analogue of a classical problem in number theory.
We give a new derivation of an identity due to Z. Rudnick and P. Sarnak about the n-level correla... more We give a new derivation of an identity due to Z. Rudnick and P. Sarnak about the n-level correlations of eigenvalues of random unitary matrices as well as a new proof of a formula due to M. Diaconis and P. Shahshahani expressing averages of trace products over the unitary matrix ensemble. Our method uses the zero statistics of Artin-Schreier L-functions and a deep equidistribution result due to N. Katz and P. Sarnak.
We study the distribution of the zeroes of the L-functions of curves in the Artin-Schreier family... more We study the distribution of the zeroes of the L-functions of curves in the Artin-Schreier family. We consider the number of zeroes in short intervals and obtain partial results which agree with a random unitary matrix model.
For a projective curve C⊂P^n defined over F_q we study the statistics of the F_q-structure of a s... more For a projective curve C⊂P^n defined over F_q we study the statistics of the F_q-structure of a section of C by a random hyperplane defined over F_q in the q→∞ limit. We obtain a very general equidistribution result for this problem. We deduce many old and new results about decomposition statistics over finite fields in this limit. Our main tool will be the calculation of the monodromy of transversal hyperplane sections of a projective curve.
We study the minimal number of ramified primes in Galois extensions of rational function fields o... more We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric and alternating groups in many cases.
We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $... more We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $f(T_1,\ldots, T_n,X)$ is an irreducible polynomial over the field of rational functions over a finite field $\mathbb{F}_q$ of characteristic $p$, then the proportion of $n$-tuples $(t_1,\ldots, t_n)$ of monic polynomials of degree $d$ for which $f(t_1,\ldots, t_n,X)$ is reducible out of all $n$-tuples of degree $d$ monic polynomials is $O(dq^{-d/2})$.
For a polynomial $F(t,A_1,\ldots,A_n)\in\mathbf{F}_p[t,A_1,\ldots,A_n]$ ($p$ being a prime number... more For a polynomial $F(t,A_1,\ldots,A_n)\in\mathbf{F}_p[t,A_1,\ldots,A_n]$ ($p$ being a prime number) we study the factorization statistics of its specializations $$F(t,a_1,\ldots,a_n)\in\mathbf{F}_p[t]$$ with $(a_1,\ldots,a_n)\in S$, where $S\subset\mathbf{F}_p^n$ is a subset, in the limit $p\to\infty$ and $\mathrm{deg} F$ fixed. We show that for a sufficiently large and regular subset $S\subset\mathbf{F}_p^n$, e.g. a product of $n$ intervals of length $H_1,\ldots,H_n$ with $\prod_{i=1}^nH_n>p^{n-1/2+\epsilon}$, the factorization statistics is the same as for unrestricted specializations (i.e. $S=\mathbf{F}_p^n$) up to a small error. This is a generalization of the well-known P\'olya-Vinogradov estimate of the number of quadratic residues modulo $p$ in an interval.
We study the local statistics of zeros of L-functions attached to Artin-Scheier curves over finit... more We study the local statistics of zeros of L-functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier L-functions: the ordinary, polynomial (the p-rank 0 stratum) and odd-polynomial families. We compute the 1-level zero-density of the first and third families and the 2-level density of the second family for test functions with Fourier transform supported in a suitable interval. In each case we obtain agreement with a unitary or symplectic random matrix model.
Mathematical Proceedings of the Cambridge Philosophical Society
We investigate the density of square-free values of polynomials with large coefficients over the ... more We investigate the density of square-free values of polynomials with large coefficients over the rational function field 𝔽 q [t]. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial N as a sum of a k-th power of a small polynomial and a square-free polynomial.
We prove an analogue of the classical Bateman–Horn conjecture on prime values of polynomials for ... more We prove an analogue of the classical Bateman–Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$ ) polynomials $F_{1},\ldots ,F_{m}\in \mathbf{F}_{q}[t][x]$ , we show that the number of $f\in \mathbf{F}_{q}[t]$ of degree $n\geqslant \max (3,\deg _{t}F_{1},\ldots ,\deg _{t}F_{m})$ such that all $F_{i}(t,f)\in \mathbf{F}_{q}[t],1\leqslant i\leqslant m$ , are irreducible is $$\begin{eqnarray}\displaystyle \biggl(\mathop{\prod }_{i=1}^{m}\frac{\unicode[STIX]{x1D707}_{i}}{N_{i}}\biggr)q^{n+1}(1+O_{m,\,\max \deg F_{i},\,n}(q^{-1/2})), & & \displaystyle \nonumber\end{eqnarray}$$ where $N_{i}=n\deg _{x}F_{i}$ is the generic degree of $F_{i}(t,f)$ for $\deg f=n$ and $\unicode[STIX]{x1D707}_{i}$ is the number of factors into which $F_{i}$ splits over $\overline{\mathbf{F}}_{q}$ . Our proof relies on the classification of finite simple groups. We will also prove...
The statistics of low-lying zeros of quadratic Dirichlet L-functions were conjectured by Katz and... more The statistics of low-lying zeros of quadratic Dirichlet L-functions were conjectured by Katz and Sarnak to be given by the scaling limit of eigenvalues from the unitary symplectic ensemble. The n-level densities were found to be in agreement with this in a certain neighborhood of the origin in the Fourier domain by Rubinstein in his Ph.D. thesis in 1998. An attempt to extend the neighborhood was made in the Ph.D. thesis of Peng Gao (2005), who under GRH gave the density as a complicated combinatorial factor, but it remained open whether it coincides with the Random Matrix Theory factor. For n at most 7 this was recently confirmed by Levinson and Miller. We resolve this problem for all n, not by directly doing the combinatorics, but by passing to a function field analogue, of L-functions associated to hyper-elliptic curves of given genus g over a field of q elements. We show that the answer in this case coincides with Gao's combinatorial factor up to a controlled error. We then ...
For a function field K and fixed polynomial F∈ K[x] and varying f∈ F (under certain restrictions)... more For a function field K and fixed polynomial F∈ K[x] and varying f∈ F (under certain restrictions) we give a lower bound for the degree of the greatest prime divisor of F(f) in terms of the height of f, establishing a strong result for the function field analogue of a classical problem in number theory.
We give a new derivation of an identity due to Z. Rudnick and P. Sarnak about the n-level correla... more We give a new derivation of an identity due to Z. Rudnick and P. Sarnak about the n-level correlations of eigenvalues of random unitary matrices as well as a new proof of a formula due to M. Diaconis and P. Shahshahani expressing averages of trace products over the unitary matrix ensemble. Our method uses the zero statistics of Artin-Schreier L-functions and a deep equidistribution result due to N. Katz and P. Sarnak.
We study the distribution of the zeroes of the L-functions of curves in the Artin-Schreier family... more We study the distribution of the zeroes of the L-functions of curves in the Artin-Schreier family. We consider the number of zeroes in short intervals and obtain partial results which agree with a random unitary matrix model.
For a projective curve C⊂P^n defined over F_q we study the statistics of the F_q-structure of a s... more For a projective curve C⊂P^n defined over F_q we study the statistics of the F_q-structure of a section of C by a random hyperplane defined over F_q in the q→∞ limit. We obtain a very general equidistribution result for this problem. We deduce many old and new results about decomposition statistics over finite fields in this limit. Our main tool will be the calculation of the monodromy of transversal hyperplane sections of a projective curve.
We study the minimal number of ramified primes in Galois extensions of rational function fields o... more We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric and alternating groups in many cases.
We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $... more We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $f(T_1,\ldots, T_n,X)$ is an irreducible polynomial over the field of rational functions over a finite field $\mathbb{F}_q$ of characteristic $p$, then the proportion of $n$-tuples $(t_1,\ldots, t_n)$ of monic polynomials of degree $d$ for which $f(t_1,\ldots, t_n,X)$ is reducible out of all $n$-tuples of degree $d$ monic polynomials is $O(dq^{-d/2})$.
For a polynomial $F(t,A_1,\ldots,A_n)\in\mathbf{F}_p[t,A_1,\ldots,A_n]$ ($p$ being a prime number... more For a polynomial $F(t,A_1,\ldots,A_n)\in\mathbf{F}_p[t,A_1,\ldots,A_n]$ ($p$ being a prime number) we study the factorization statistics of its specializations $$F(t,a_1,\ldots,a_n)\in\mathbf{F}_p[t]$$ with $(a_1,\ldots,a_n)\in S$, where $S\subset\mathbf{F}_p^n$ is a subset, in the limit $p\to\infty$ and $\mathrm{deg} F$ fixed. We show that for a sufficiently large and regular subset $S\subset\mathbf{F}_p^n$, e.g. a product of $n$ intervals of length $H_1,\ldots,H_n$ with $\prod_{i=1}^nH_n>p^{n-1/2+\epsilon}$, the factorization statistics is the same as for unrestricted specializations (i.e. $S=\mathbf{F}_p^n$) up to a small error. This is a generalization of the well-known P\'olya-Vinogradov estimate of the number of quadratic residues modulo $p$ in an interval.
We study the local statistics of zeros of L-functions attached to Artin-Scheier curves over finit... more We study the local statistics of zeros of L-functions attached to Artin-Scheier curves over finite fields. We consider three families of Artin-Schreier L-functions: the ordinary, polynomial (the p-rank 0 stratum) and odd-polynomial families. We compute the 1-level zero-density of the first and third families and the 2-level density of the second family for test functions with Fourier transform supported in a suitable interval. In each case we obtain agreement with a unitary or symplectic random matrix model.
Mathematical Proceedings of the Cambridge Philosophical Society
We investigate the density of square-free values of polynomials with large coefficients over the ... more We investigate the density of square-free values of polynomials with large coefficients over the rational function field 𝔽 q [t]. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial N as a sum of a k-th power of a small polynomial and a square-free polynomial.
We prove an analogue of the classical Bateman–Horn conjecture on prime values of polynomials for ... more We prove an analogue of the classical Bateman–Horn conjecture on prime values of polynomials for the ring of polynomials over a large finite field. Namely, given non-associate, irreducible, separable and monic (in the variable $x$ ) polynomials $F_{1},\ldots ,F_{m}\in \mathbf{F}_{q}[t][x]$ , we show that the number of $f\in \mathbf{F}_{q}[t]$ of degree $n\geqslant \max (3,\deg _{t}F_{1},\ldots ,\deg _{t}F_{m})$ such that all $F_{i}(t,f)\in \mathbf{F}_{q}[t],1\leqslant i\leqslant m$ , are irreducible is $$\begin{eqnarray}\displaystyle \biggl(\mathop{\prod }_{i=1}^{m}\frac{\unicode[STIX]{x1D707}_{i}}{N_{i}}\biggr)q^{n+1}(1+O_{m,\,\max \deg F_{i},\,n}(q^{-1/2})), & & \displaystyle \nonumber\end{eqnarray}$$ where $N_{i}=n\deg _{x}F_{i}$ is the generic degree of $F_{i}(t,f)$ for $\deg f=n$ and $\unicode[STIX]{x1D707}_{i}$ is the number of factors into which $F_{i}$ splits over $\overline{\mathbf{F}}_{q}$ . Our proof relies on the classification of finite simple groups. We will also prove...
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