On Incidences of φ\varphi and σ\sigma in the Function Field Setting

Erd\H{o}s first conjectured that infinitely often we have $\varphi(n) = \sigma(m)$, where $\varphi$ is the Euler totient function and $\sigma$ is the sum of divisor function. This was proven true by Ford, Luca and Pomerance in 2010. We ask the analogous question of whether infinitely often we have $\varphi(F) = \sigma(G)$ where $F$ and $G$ are polynomials over some finite field $\mathbb{F}_q$. We find that when $q\not=2$ or $3$, then this can only trivially happen when $F=G=1$. Moreover, we give a complete characterisation of the solutions in the case $q=2$ or $3$. In particular, we show that $\varphi(F) = \sigma(G)$ infinitely often when $q=2$ or $3$

Expected Values of LL-functions Away from the Central Point

We compute the expected value of Dirichlet $L$-functions defined over $\mathbb{F}_q[T]$ attached to cubic characters evaluated at an arbitrary $s \in (0,1)$. We find a transition term at the point $s=\frac{1}{3}$, reminiscent of the transition at the point $s=\frac{1}{2}$ of the bound for the size of an $L$-function implied by the Lindel\"of hypothesis. We show that at $s=\frac{1}{3}$, the expected value matches corresponding statistics of the group of unitary matrices multiplied by a weight function.Comment: 31 page

Low-lying zeros in families of elliptic curve L-functions over function fields

We investigate the low-lying zeros in families of L-functions attached to quadratic and cubic twists of elliptic curves defined over Fq(T). In particular, we present precise expressions for the expected values of traces of high powers of the Frobenius class in these families with a focus on the lower order behavior. As an application we obtain results on one-level densities and we verify that these elliptic curve families have orthogonal symmetry type. In the quadratic twist families our results refine previous work of Comeau-Lapointe. Moreover, in this case we find a lower order term in the one-level density reminiscent of the deviation term found by Rudnick in the hyperelliptic ensemble. On the other hand, our investigation is the first to treat these questions in families of cubic twists of elliptic curves and in this case it turns out to be more complicated to isolate lower order terms due to a larger degree of cancellation among lower order contributions

The Distribution of the Number of Points on Abelian Curves over Finite Fields

Classical results due to Katz and Sarnak show that if the genus is fixed and q tends to infinity, then the number of points on a family of curves over a finite field of q elements is distributed as the trace of a random matrix in the monodromy group associated to the family. Every smooth projective curve C corresponds to a finite Galois extension of the field of polynomials with coefficients in the finite field. Therefore, some natural families to consider are the curves that correspond to a extensions with a fixed Galois group. This thesis involves determining the distribution of the families with fixed abelian Galois group, G, when q is fixed and the genus tends to infinity. Several authors determined that the distribution for the family of prime-cyclic curves as well as for the family of n-quadratic curves is that of a sum of q+1 random variables. This thesis shows that if we fix any abelian group, the distribution will be that of q+1 random variables. The above results deal only with the distribution for the coarse irreducible moduli space of the families. It has been shown that if you look at the whole (coarse) moduli space, the distribution is the same in the case of prime-cyclic curves. We are able to show that the distribution is the same for the coarse moduli space of curves with G=(Z/QZ)^n, Q a prime. Some work is done towards proving this true for all abelian groups