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Abstract We explicitly construct the Dirichlet series $$\begin{equation*}L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},\end{equation*}$$ where $$P_{\mathrm{Tam}}(m)$$ is the proportion of elliptic curves $$E/\mathbb{Q}$$ in short Weierstrass form with Tamagawa product m. Although there are no $$E/\mathbb{Q}$$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $$P_{\mathrm{Tam}}(1)={0.5053\dots}$$. As a corollary, we find that $$L_{\mathrm{Tam}}(-1)={1.8193\dots}$$ is the average Tamagawa product for elliptic curves over $$\mathbb{Q}$$. We give an application of these results to canonical and Weil heights. 

Abstract Hausel and Rodriguez-Villegas (2015, Astérisque 370, 113–156) recently observed that work of Göttsche, combined with a classical result of Erdös and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes $$(\mathbb {C}^{2})^{[n]}$$ on $$n$$ points, as $$n\rightarrow +\infty ,$$ is a Gumbel distribution . In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes $$((\mathbb {C}^{2})^{[n]})^{T_{\alpha ,\beta }}$$ that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdös and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer $$A\geq 2.$$ Furthermore, if $$p_{k}(A;n)$$ denotes the number of partitions of $$n$$ with exactly $$k$$ parts that are multiples of $$A$$ , then we obtain the asymptotic $$ \begin{align*} p_{k}(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^{k}}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}}, \end{align*} $$ a result which is of independent interest. 

The Eichler-Selberg trace formulas express the traces of Hecke operators on a spaces of cusp forms in terms of weighted sums of Hurwitz-Kronecker class numbers. For cusp forms on $$\text {\rm SL}_2(\mathbb{Z}),$$ Zagier proved these formulas by cleverly making use of the weight 3/2 nonholomorphic Eisenstein series he discovered in the 1970s. The holomorphic part of this form, its so-called {\it mock modular form}, is the generating function for these class numbers. In this expository note we revisit Zagier's method, and we show how to obtain such formulas for congruence subgroups, working out the details for $$\Gamma_0(2)$$ and $$\Gamma_0(4).$$ The trace formulas fall out naturally from the computation of the Rankin-Cohen brackets of Zagier's mock modular form with specific theta functions. 

Abstract The Euler–Mascheroni constant $$\gamma =0.5772\ldots \!$$ is the $$K={\mathbb Q}$$ example of an Euler–Kronecker constant $$\gamma _K$$ of a number field $K.$ In this note, we consider the size of the $$\gamma _q=\gamma _{K_q}$$ for cyclotomic fields $$K_q:={\mathbb Q}(\zeta _q).$$ Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that $$ \begin{align*} \frac{1}{Q}\sum_{Q 

In the 1980’s, Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Ap ́ery-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the 2F1 functions, the limiting distribution is semicircular (i.e. SU(2)), whereas the distribution for the 3F2 functions is the Batman distribution for the traces of the real orthogonal group O3. 

Abstract Recent works at the interface of algebraic combinatorics, algebraic geometry, number theory and topology have provided new integer-valued invariants on integer partitions. It is natural to consider the distribution of partitions when sorted by these invariants in congruence classes. We consider the prominent situations that arise from extensions of the Nekrasov–Okounkov hook product formula and from Betti numbers of various Hilbert schemes of n points on $${\mathbb {C}}^2$$ . For the Hilbert schemes, we prove that homology is equidistributed as $$n\to \infty $$ . For t -hooks, we prove distributions that are often not equidistributed. The cases where $$t\in \{2, 3\}$$ stand out, as there are congruence classes where such counts are zero. To obtain these distributions, we obtain analytic results of independent interest. We determine the asymptotics, near roots of unity, of the ubiquitous infinite products $$ \begin{align*}F_1(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi q^n\right), \ \ \ F_2(\xi; q):=\prod_{n=1}^{\infty}\left(1-(\xi q)^n\right) \ \ \ {\mathrm{and}}\ \ \ F_3(\xi; q):=\prod_{n=1}^{\infty}\left(1-\xi^{-1}(\xi q)^n\right). \end{align*} $$ 

Using predictions in mirror symmetry, Căldăraru, He, and Huang recently formulated a “Moonshine Conjecture at Landau-Ginzburg points” [arXiv:2107.12405, 2021] for Klein’s modular j j -function at j = 0 j=0 and j = 1728. j=1728. The conjecture asserts that the j j -function, when specialized at specific flat coordinates on the moduli spaces of versal deformations of the corresponding CM elliptic curves, yields simple rational functions. We prove this conjecture, and show that these rational functions arise from classical 2 F 1 _2F_1 -hypergeometric inversion formulae for the j j -function.