We study the geometry of Gorenstein curve singularities of genus two, and their stable limits. These singularities come in two families, corresponding to either Weierstrass or conjugate points on a semistable tail. For every $1\le m<n$, a stability condition—using one of the markings as a reference point, and thus not ${\mathfrak{𝔖}}_n$-symmetric — defines proper Deligne–Mumford stacks $\overline{\mathcal{ℳ}}{2,n}^{(m)}$ with a dense open substack representing smooth curves.

We prove lower bounds for the discrete negative $2k$-th moment of the derivative of the Riemann zeta function for all fractional $k$. The bounds are in line with a conjecture of Gonek and Hejhal. Along the way, we prove a general formula for the discrete twisted second moment of the Riemann zeta function. This agrees with a conjecture of Conrey and Snaith.

Let $F_1=1,F_2=1,\dots$ be the Fibonacci sequence. Motivated by the identity ${\sum }_{k=0}^{\infty }1∕F_{2^k}=7-\sqrt{5}∕2$, Erdös and Graham asked whether ${\sum }_{k=1}^{\infty }1∕F_{n_k}$ is irrational for any sequence of positive integers $n_1,n_2,\dots$ with $n_{k+1}∕n_k\ge c>1$. We resolve the transcendence counterpart of their question; as a special case of our main theorem, we have that ${\sum }_{k=1}^{\infty }1∕F_{n_k}$ is transcendental when . The bound $c>2$ is best possible thanks to the identity at the beginning. This paper provides a new way to apply the subspace theorem to obtain transcendence results and extends previous nontrivial results obtainable by only Mahler’s method for special sequences of the form $n_k=d^k+r$.

We give several results concerning the connected component ${\text{Aut}}_X^0$ of the automorphism scheme of a proper variety $X$ over a field, such as its behavior with respect to birational modifications, normalization, restrictions to closed subschemes and deformations. Then, we apply our results to study the automorphism scheme of not necessarily Jacobian elliptic surfaces $f:X\to C$ over algebraically closed fields, generalizing work of Rudakov and Shafarevich, while giving counterexamples to some of their statements. We bound the dimension $h^0(X,T_X)$ of the space of global vector fields on an elliptic surface $X$ if the generic fiber of $f$ is ordinary or if $f$ admits no multiple fibers, and show that, without these assumptions, the number $h^0(X,T_X)$ can be arbitrarily large for any base curve $C$ and any field of positive characteristic. If $f$ is not isotrivial, we prove that ${\text{Aut}}_X^0\cong {\mu }_{p^n}$ and give a bound on $n$ in terms of the genus of $C$ and the multiplicity of multiple fibers of $f$. As a corollary, we reprove the nonexistence of global vector fields on K3 surfaces and calculate the connected component of the automorphism scheme of a generic supersingular Enriques surface in characteristic $2$. Finally, we present additional results on horizontal and vertical group scheme actions on elliptic surfaces which can be applied to determine ${\text{Aut}}_X^0$ explicitly in many concrete cases.

We show that every sufficiently large $x\equiv 3(4)$ can be written as the sum of three primes, each of which is a sum of a square and a prime square. The main tools are a transference version of the circle method and various sieve related ideas. In particular, a majorant of the set of primes of interest is constructed that overestimates it by a factor of less than $3$ and for which we have good control of the Fourier transform.