Szpiro’s conjecture asserts the existence of an absolute constant $K \gt 6$ such that if $E$ is an elliptic curve over $\mathbb{Q}$, the minimal discriminant $\Delta(E)$ of $E$ is bounded above in modulus by the $K$th power of the conductor $N(E)$ of $E$ . An immediate consequence of this is the existence of an absolute upper bound on $\min\{v_p(\Delta(E )) : p |\Delta(E )\}$. In this paper, we will prove this local version of Szpiro’s conjecture under the (admittedly strong) additional hypotheses that $N(E)$ is divisible by a “large” prime $p$ and that $E$ possesses a nontrivial rational isogeny. We will also formulate a related conjecture that if true, we prove to be sharp. Our construction of families of curves for which $\min{v_p(\Delta(E)) : p | \Delta(E )} \ge 6$ provides an alternative proof of a result of Masser on the sharpness of Szpiro’s conjecture.We close the paper by reporting on recent computations of examples of curves with large Szpiro ratio.

Harmonic weak Maass forms of half-integral weight have been the subject of much recent work. They are closely related to Ramanujan’s mock theta functions, and their theta lifts give rise to Arakelov Green functions, and their coefficients are often related to central values and derivatives of Hecke L-functions. We present an algorithm to compute harmonic weak Maass forms numerically, based on the automorphymethod due to Hejhal and Stark. As explicit examples we consider harmonic weak Maass forms of weight 1/2 associated to the elliptic curves 11a1, 37a1, 37b1. We have made extensive numerical computations, and the data we obtained are presented in this paper. We expect that experiments based on our data will lead to a better understanding of the arithmetic properties of the Fourier coefficients of harmonic weak Maass forms of half-integral weight.

Let $\zeta (s)$ be the Riemann zeta function. In this paper, we study repeated values of $\zeta (s)$ on the critical line, and we give evidence to support our conjecture that for every nonzero complex number $z$, the equation $\zeta (1/2 + i t) = z$ has at most two solutions $t \in R$. We prove a number of related results, some of which are unconditional, and some of which depend on the truth of the Riemann hypothesis. We also propose some related conjectures that are implied by Montgomery’s pair correlation conjecture.

We reduce the computation of Poisson traces on quotients of symplectic vector spaces by finite subgroups of symplectic automorphisms to a finite one by proving several results that bound the degrees of such traces as well as the dimension in each degree. This applies more generally to traces on all polynomial functions that are invariant under invariant Hamiltonian flow.We implement these approaches by computer together with direct computation for infinite families of groups, focusing on complex reflection and abelian subgroups of ${\rm GL}_2(\mathbb{C}) \lt {\rm Sp}_4(\mathbb{C})$, Coxeter groups of rank $\le 3$ and types $A_4$, $B_4 = C_4$, and $D_4$, and subgroups of ${\rm SL}_2(\mathbb{C})$.

We develop a semistability algorithm for vector bundles that are given as a kernel of a surjective morphism between splitting bundles on the projective space $\mathbb{P}^N$ over an algebraically closed field $K$ . This class of bundles is a generalization of syzygy bundles. We show how to implement this algorithm in a computer algebra system. Further, we give applications, mainly concerning the computation of Tannaka dual groups of stable vector bundles of degree 0 on $\mathbb{P}^N$ and on certain smooth complete intersection curves. We also use our algorithm to close a case left open in a recent work of L. Costa, P. Macias Marques, and R. M. Miró- Roig regarding the stability of the syzygy bundle of general forms. Finally, we apply our algorithm to provide a computational approach to tight closure. All algorithms are implemented in the computer algebra system CoCoA.

We study a multivariate Markov chain on the symmetric group with remarkable enumerative properties. We conjecture that the stationary distribution of this Markov chain can be expressed in terms of positive sums of Schubert polynomials. This Markov chain is a multivariate generalization of a Markov chain introduced by the first author in the study of random affine Weyl group elements.

It was conjectured in Yukihiko Namikawa, “On the Canonical Holomorphic Map from the Moduli Space of Stable Curves to the Igusa Monoidal Transform,” that the Torelli map $M_g \to A_g$ associating to a curve its Jacobian extends to a regular map from the Deligne–Mumford moduli space of stable curves $\bar{M}_g$ to the (normalization of the) Igusa blowup $\bar{A}^{\rm cent}_g$. A counterexample in genus $g = 9$ was found in Valery Alexeev and Adrian Brunyate, “Extending Torelli Map to Toroidal Compactifications of Siegel Space.” Here, we prove that the extended map is regular for all $g \le 8$, thus completely solving the problem in every genus.

We investigate the frequency of positive squareful numbers $x, y, z \le B$ for which $x + y = z$ and present a conjecture concerning its asymptotic behavior.