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We prove Malle's conjecture for $G \times A$, with $G=S_3, S_4, S_5$ and $A$ an abelian group. This builds upon work of the fourth author, who proved this result with restrictions on the primes dividing $A$.

In this paper we prove a sparse equidistribution theorem for Gross points over the rational function field $\mathbb{F}_q(t)$. We apply this result to study the reduction map from CM Drinfeld modules to supersingular Drinfeld modules. Our proofs rely crucially on a period formula due to M. Papikian and F.-T. Wei/J. Yu, and a Lindel\"of-type bound for central values of Rankin-Selberg $L$-functions associated to twists of automorphic forms of Drinfeld-type by ideal class group characters., Comment: 20 pages

Let $G$ be a transitive subgroup of $S_d$ and $E$ be a CM field of degree $2d$ with a maximal totally real $G$-field. If the Galois group of the Galois closure of $E$ is isomorphic to the wreath product of $C_2$ and $G$, then we say that $E$ is a $G$-Weyl CM field. Let $N_{2d}^{\textrm{Weyl}}(X,G)$ count the $G$-Weyl CM fields $E$ of degree $2d$ with discriminant $|d_E| \leq X$ and define \begin{align*} N_{2d}^{\textrm{Weyl}}(X):=\sum_{G \leq S_d}N_{2d}^{\textrm{Weyl}}(X,G). \end{align*} Further, let $N_{2d}^{\textrm{cm}}(X)$ count the CM fields $E$ of degree $2d$ with discriminant $|d_E| \leq X$. Assuming a weak form of the upper bound in Malle's conjecture which is known to be true in many cases, we build upon an approach of Kl\"uners to prove that \begin{align*} \frac{N_{2d}^{\textrm{Weyl}}(X,G)}{N_{2d}^{\textrm{cm}}(X)} = C(d, G) + O(X^{-\alpha(d,G)}) \end{align*} and \begin{align} \frac{N_{2d}^{\textrm{Weyl}}(X)}{N_{2d}^{\textrm{cm}}(X)} = 1 + O(X^{-\beta(d)}) \qquad \qquad (0.1) \end{align} for some explicit positive constants $C(d,G), \alpha(d,G)$, and $\beta(d)$. We then apply these distribution results to study the Colmez conjecture. Using the recently proved averaged Colmez conjecture, we deduce that the Colmez conjecture is true for $G$-Weyl CM fields. Combined with (0.1), we conclude that the Colmez conjecture is true for an asymptotic density of 100% of CM fields of degree $2d$; in other words, the Colmez conjecture is true for a random CM field., Comment: The introduction has been reorganized and Section 3 has been expanded. 21 pages, 3 tables

In this paper, we establish an asymptotic formula with an effective bound on the error term for the Andrews smallest parts function $\mathrm{spt}(n)$. We use this formula to prove recent conjectures of Chen concerning inequalities which involve the partition function $p(n)$ and $\mathrm{spt}(n)$. Further, we strengthen one of the conjectures, and prove that for every $\epsilon>0$ there is an effectively computable constant $N(\epsilon) > 0$ such that for all $n\geq N(\epsilon)$, we have \begin{equation*} \frac{\sqrt{6}}{\pi}\sqrt{n}\,p(n)<\mathrm{spt}(n)<\left(\frac{\sqrt{6}}{\pi}+\epsilon\right) \sqrt{n}\,p(n). \end{equation*} Due to the conditional convergence of the Rademacher-type formula for $\mathrm{spt}(n)$, we must employ methods which are completely different from those used by Lehmer to give effective error bounds for $p(n)$. Instead, our approach relies on the fact that $p(n)$ and $\mathrm{spt}(n)$ can be expressed as traces of singular moduli., Comment: Changed the title. Added more details and simplified some arguments in Section 3

The Colmez conjecture relates the Faltings height of an abelian variety with complex multiplication by the ring of integers of a CM field $E$ to logarithmic derivatives of certain Artin $L$--functions at $s=0$. In this paper, we prove that if $F$ is any fixed totally real number field of degree $[F:\mathbb{Q}] \geq 3$, then there are infinitely many CM extensions $E/F$ such that $E/\mathbb{Q}$ is $\textit{non-abelian}$ and the Colmez conjecture is true for $E$. Moreover, these CM extensions are explicitly constructed to be ramified at "arbitrary" prescribed sets of prime ideals of $F$. We also prove that the Colmez conjecture is true for a generic class of non-abelian CM fields called Weyl CM fields, and use this to develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. We illustrate these results by evaluating the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field in terms of the Barnes double Gamma function at algebraic arguments. This can be seen as an explicit non-abelian Chowla-Selberg formula. A crucial input to the proofs is an averaged version of the Colmez conjecture which was recently proved independently by Andreatta-Goren-Howard-Madapusi Pera and Yuan-Zhang., Comment: The introduction has been expanded to include an explicit evaluation of the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field. We also develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. (35 pages, 2 figures, submitted for publication)

Let q be a prime and -D < -4 be an odd fundamental discriminant such that q splits in Q(\sqrt{-D}). For f a weight zero Hecke-Maass newform of level q and h the weight one theta series of level D corresponding to an ideal class group character of Q(\sqrt{-D}), we establish a hybrid subconvexity bound for L(f \times h,s) at the central point when q = D^{\eta} for 0 < \eta < 1. With this circle of ideas, we show that the Heegner points of level q and discriminant D become equidistributed, in a natural sense, as q, D become large with q < D^{1/20-\varepsilon}. Our approach to these problems is connected to estimating the L^2-restriction norm of a Maass form of large level when restricted to the collection of Heegner points. We furthermore establish bounds for quadratic twists of Hecke-Maass L-functions with simultaneously large level and large quadratic twist, and hybrid bounds for quadratic Dirichlet L-functions in certain ranges., Comment: 24 pages, submitted for publication

In this paper we develop the analytic theory of a multiple zeta function in d independent complex variables defined over a global function field. This is the function field analog of the Euler-Zagier multiple zeta function of depth d.