Related papers: Variations on murmurations
We introduce a new method for studying murmurations, based on random matrix theory. With this method, we exhibit murmurations or similar phenomena: assuming ratios conjectures, for elliptic curves ordered by height, quadratic twists of a…
Recently, we showed that global root numbers of modular forms are biased toward +1. Together with Pharis, we also showed an initial bias of Fourier coefficients towards the sign of the root number. First, we prove analogous results with…
We provide a brief outlook on recent developments in regularity theory for nonuniformly elliptic problems, with special emphasis on those of variational nature.
We survey on algebraically elliptic varieties in the sense of Gromov.
In this survey article, we summarise the known results towards the conjecture: elliptic curves over totally real number fields are modular. For understanding these recent results in the literature, we present some necessary background along…
We establish a case of the surprising correlation phenomenon observed in the recent works of He, Lee, Oliver, Pozdnyakov, and Sutherland between Fourier coefficients of families of modular forms and their root numbers.
Several new invariants for Lie algebroids have been discovered recently. We give an overview of these invariants and establish several relationships between them.
We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic…
We prove the murmuration phenomenon, which is a correlation between signs of functional equations and Fourier coefficients, in the case of modular forms in the weight aspect. We in particular improve the range of visibility of murmurations…
We prove the existence of "murmurations" in the family of holomorphic modular forms of level $1$ and weight $k\to\infty$, that is, correlations between their root numbers and Hecke eigenvalues at primes growing in proportion to the analytic…
We report the emergence of a striking new phenomenon in arithmetic, which we call murmurations. First observed experimentally through averages over large arithmetic datasets, murmurations can be detected and analyzed using standard…
In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.
The theory of Topological Modular Forms suggests the existence of deformation invariants for two-dimensional supersymmetric field theories that are more refined than the standard elliptic genus. In this note we give a physical definition of…
Some work in progress is announced, on the use of algebraic geometry, mostly concerning elliptic curve theory, to model turbulence. Attention is given to flows across the scales, on some convenient model space, and some current trials are…
Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F.
In this article, we study the invariant differential forms which a correspondence of curves admits. We also try to classify the correspondences of $\mathbb{P}^1$ that admits such invariant differential forms.
This note provides new closed forms evaluations of a few classes of exponential sums associated with elliptic curves and hyperelliptic curves.
We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.
The formation and evolution of nonlinear and turbulent dynamical structures in two-dimensional complex plasmas and fluids is explored by means of generalised (drift) fluid simulations. Recent numerical results on turbulence in dusty…
We present a collection of results on the evolution by curvature of networks of planar curves. We discuss in particular the existence of a solution and the analysis of singularities.