Let $L (E /\mathbb{Q} , s)$ be the $L$-function of an elliptic curve $E$ defined over the rational field $\mathbb{Q}$. Assuming the Birch–Swinnerton-Dyer conjectures, we examine special values of the $r$th derivatives, $L^{(r)}(E , 1, \chi)$, of twists by Dirichlet characters of $L (E /\mathbb{Q} , s)$ when $L (E , 1, \chi) = • • • = L^{(r−1)} (E , 1, \chi) = 0$.

Using the machinery from the theory of Calabi–Yau differential equations, we find formulas for $1/\pi^2$ of hypergeometric and nonhypergeometric types.

We prove asymptotic formulas for the first discrete moment of the Riemann zeta function on certain vertical arithmetic progressions inside the critical strip. The results give some heuristic arguments for a stochastic periodicity that we observed in the phase portrait of the zeta function.

We study $q$-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (noncommutative) A-polynomial of a knot. Using the method of guessing, we obtain this polynomial explicitly for the $K_p = (−2, 3, 3 + 2p)$ pretzel knots for $p= −5, \dots , 5$. This is a particularly interesting family, since the pairs $(K_p,−K_{−p})$ are geometrically similar (in particular, scissors congruent) with similar character varieties. Our computation of the noncommutative $A$-polynomial complements the computation of the $A$-polynomial of the pretzel knots done by the first author and Mattman, supports the AJ conjecture for knots with reducible $A$-polynomial, and numerically computes the Kashaev invariant of pretzel knots in linear time. In a later publication, we will use the numerical computation of the Kashaev invariant to numerically verify the volume conjecture for the above-mentioned pretzel knots.

We formulate the secant conjecture, which is a generalization of the Shapiro conjecture for Grassmannians. It asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all points real if the flags defining the Schubert varieties are secant along disjoint intervals of a rational normal curve. We present theoretical evidence for this conjecture as well as computational evidence obtained in over one terahertz-year of computing, and we discuss some of the phenomena we observed in our data.

Let $\mathcal{F}_n$ be a fundamental domain of the Siegel upper half-space of degree $n$ with respect to the Siegel modular group $\operatorname{Sp}(n, \mathbb{Z})$. According to Siegel himself, $\mathcal{F}_n$ is determined by only finitely many polynomial inequalities. In case of degree $n = 2$, Gottschling determined the minimal set of inequalities. The boundary of $\mathcal{F}_2$ is of great concern in the literature not only from a homological point of view but also from the geometry of numbers. In this paper we compute the vertices of $\mathcal{F}_2$ under the condition that the defining ideal is zero-dimensional (“0-cells”). We also discuss an equivalence relation among 0-cells.

The $T$-graph of a multigraded Hilbert scheme records the zeroand one-dimensional orbits of the $T = (K^*)^n$ action on the Hilbert scheme induced from the $T$-action on $\mathbb{A}^n$. It has vertices the $T$-fixed points, and edges the one-dimensional $T$-orbits. We give a combinatorial necessary condition for the existence of an edge between two vertices in this graph. For the Hilbert scheme of points in the plane, we give an explicit combinatorial description of the equations defining the scheme parameterizing all one-dimensional torus orbits whose closures contain two given monomial ideals. For this Hilbert scheme we show that the $T$-graph depends on the ground field, resolving a question of Altmann and Sturmfels.

We examine which $p$-groups of order $\le p^6$ are Beauville. We completely classify them for groups of order $\le p^4$. We also show that the proportion of 2-generated groups of order $p^5$ that are Beauville tends to 1 as $p$ tends to infinity; this is not true, however, for groups of order $p^6$. For each prime $p$ we determine the smallest nonabelian Beauville $p$-group.

We report on some extensive computations and experiments concerning the moments of quadratic Dirichlet $L$-functions at the critical point. We computed the values of $L (1/2, \chi_d)$ for $−5 × 10^{10} \lt d \lt 1.3 × 10^{10}$ in order to numerically test conjectures concerning the moments $\sum_{|d|\lt X} L (1/2, \chi_d)^k$. Specifically, we tested the full asymptotics for the moments conjectured by Conrey, Farmer, Keating, Rubinstein, and Snaith, as well as the conjectures of Diaconu, Goldfeld, Hoffstein, and Zhang concerning additional lower-order terms in the moments. We also describe the algorithms used for this large-scale computation.