In this paper, we prove an asymptotic formula with a power saving error term for traces of weight zero weakly holomorphic modular forms of level N along Galois orbits of Heegner points on the modular curve X0(N). We use this result to study the distribution of partition ranks modulo 2. In particular, we give an asymptotic formula with a power saving error term for the number of partitions of a positive integer n with even (respectively, odd) rank. We use these results to deduce a strong quantitative form of equidistribution of partition ranks modulo 2.

In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function ${\rm\Gamma}$ and an analogous function ${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.

Let $q$ be a prime and $- D\lt - 4$ be an odd fundamental discriminant such that $q$ splits in $ \mathbb{Q} ( \sqrt{- D} )$ . For $f$ a weight-zero Hecke–Maass newform of level $q$ and ${\Theta }_{\chi } $ the weight-one theta series of level $D$ corresponding to an ideal class group character $\chi $ of $ \mathbb{Q} ( \sqrt{- D} )$ , we establish a hybrid subconvexity bound for $L(f\times {\Theta }_{\chi } , s)$ at $s= 1/ 2$ when $q\asymp {D}^{\eta } $ for $0\lt \eta \lt 1$ . 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\rightarrow \infty $ for $q\leq {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 $q$ 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.

Abstract. Let $H$ be the Hilbert class field of a $\text{CM}$ number field $K$ with maximal totally real subfield $F$ of degree $n$ over $\mathbb{Q}$ . We evaluate the second term in the Taylor expansion at $s\,=\,0$ of the Galois-equivariant $L$ -function ${{\Theta }_{{{S}_{\infty }}\,}}\left( s \right)$ associated to the unramified abelian characters of $\text{Gal}\left( H/K \right)$ . This is an identity in the group ring $\mathbb{C}\left[ \text{Gal}\left( H/K \right) \right]$ expressing $\Theta _{{{S}_{\infty }}}^{(n)}\,\left( 0 \right)$ as essentially a linear combination of logarithms of special values $\left\{ \Psi ({{z}_{\sigma }}) \right\}$ , where $\Psi :\,{{\mathbb{H}}^{n}}\,\to \,\mathbb{R}$ is a Hilbert modular function for a congruence subgroup of $S{{L}_{2}}\left( {{\mathcal{O}}_{F}} \right)$ and $\left\{ {{z}_{\sigma }}\,:\,\sigma \,\in \,\text{Gal}\left( H/K \right) \right\}$ are $\text{CM}$ points on a universal Hilbert modular variety. We apply this result to express the relative class number ${{h}_{H}}/{{h}_{K}}$ as a rational multiple of the determinant of an $\left( {{h}_{K}}\,-\,1 \right)\,\times \,\left( {{h}_{K}}\,-\,1 \right)$ matrix of logarithms of ratios of special values $\Psi ({{z}_{\sigma }})$ , thus giving rise to candidates for higher analogs of elliptic units. Finally, we obtain a product formula for $\Psi ({{z}_{\sigma }})$ in terms of exponentials of special values of $L$ -functions.

We use a variant of a method of Goncharov, Kontsevich and Zhao [5, 16] to meromorphically continue the multiple Hurwitz zeta function to , to locate the hyperplanes containing its possible poles and to compute the residues at the poles. We explain how to use the residues to locate trivial zeros of .