Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes, then\[\mathrm{rank}_{\mathbb{Z}} E(\mathbb{Q}) =1\quad \mathrm{and}\quad \#\Sha(E) \lt \infty X(E)\implies \mathrm{ord}_{s=1}L(E,s) = 1.\] We also prove the corresponding result for the abelian variety associated with a weight 2 newform $f$ of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria for $f$ and $H^1_f(\mathbb{Q},V)$, where $V$ is the $p$-adic Galois representation associated with $f$, that ensure that $\mathrm{ord}_{s=1}L(f,s) = 1$. The main theorem is proved using the Iwasawa theory of $V$ over an imaginary quadratic field to show that the $p$-adic logarithm of a suitable Heegner point is non-zero.

If $k$ is a sufficiently large positive integer, we show that the Diophantine equation\[n(n+d)\cdots(n+(k-1)d) = y^\ell\]has at most finitely many solutions in positive integers $n$, $d$, $y$ and $\ell$, with $\mathrm{gcd}(n,d)=1$ and $\ell \ge 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.

In this paper we give an example of a triangulated category, linear over a field of characteristic zero, which does not carry a DG-enhancement. The only previous examples of triangulated categories without a model have been constructed by Muro, Schwede and Strickland. These examples are however not linear over a field.

We show that any two birational projective Calabi-Yau manifolds have isomorphic small quantum cohomology algebras after a certain change of Novikov rings. The key tool used is a version of an algebra called symplectic cohomology, which is constructed using Hamiltonian Floer cohomology. Morally, the idea of the proof is to show that both small quantum products are identical deformations of symplectic cohomology of some common open affine subspace.

A knot is said to be slice if it bounds a smooth properly embedded disk in $B^4$. We demonstrate that the Conway knot is not slice. This completes the classification of slice knots under $13$ crossings and gives the first example of a non-slice knot which is both topologically slice and a positive mutant of a slice knot.

For each $n$, let $M_n$ be an $n\times n$ random matrix with independent $\pm 1$ entries. We show that $\mathbb{P}\{M_n \mathrm{is\ singular}\} = (1/2 + o_n(1))^n$, which settles an old problem. Some generalizations are considered.

We prove among other things the existence of Hodge generic abelian varieties defined over the algebraic numbers and not isogenous to any Jacobian. Actually, we also show that in various interpretations these abelian varieties make up the majority, and we give certain uniform bounds on the possible degree of the fields of definition. In particular, this yields a new answer (in strong form) to a question of Katz and Oort, compared to previous work of Chai and Oort (2012, conditional on the André-Oort Conjecture) and by Tsimerman (2012 unconditionally); their constructions provided abelian varieties with complex multiplication (so not ``generic"). Our methods are completely different, and they also answer a related question posed by Chai and Oort in their paper.