Related papers: Geometric non-vanishing

We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related…

Let $\pi$ be an irreducible, cuspidal automorphic representation of $GL_n(\mathbb{A}_\mathbb{Q})$ ($n\geq 3$), which is tempered only for $n=3$. Let $s$ be a complex number such that $\Re(s)\notin \left[1/n, 1-1/n\right]$ if $n\neq 4$;…

We prove an implicit function theorem for functions on infinite-dimensional Banach manifolds, invariant under the (local) action of a finite dimensional Lie group. Motivated by some geometric variational problems, we consider group actions…

Given elliptic modular forms f and g satisfying certain conditions on their weights and levels, we prove (a quantitative version of the statement) that there exist infinitely many imaginary quadratic fields K and characters chi of the ideal…

We study the low-lying zeros of certain Artin $L$-functions associated with $D_4$-quartic function fields. Specifically, we prove that when ordered by conductor, at least $77\%$ of these $L$-functions are non-vanishing at the central point.…

We calculate certain "wide moments" of central values of Rankin--Selberg $L$-functions $L(\pi\otimes \Omega, 1/2)$ where $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_2$ over $\mathbb{Q}$ and $\Omega$ is a Hecke character…

Let $K/F$ be a finite Galois extension of number fields. It is well known that the Tchebotarev density theorem implies that an irreducible, finitely ramified $p$-adic representation $\rho$ of the absolute Galois group of $K$ is determined…

In this article, we prove non-vanishing results for $L$-functions associated to holomorphic cusp forms of half-integral weight on average (over an orthogonal basis of Hecke eigenforms). This extends a result of W. Kohnen to forms of…

We give a complete description of the arboreal Galois representation of a certain postcritically finite cubic polynomial over a large class of number fields and for a large class of basepoints. This is the first such example that is not…

We show how non-vanishing of p-adic L functions controls the dimensions of Selmer varieties associated to the complement of the origin in an elliptic curve with CM. As a corollary, one obtains a \pi_1-proof of the theorem of Siegel for such…

We define an axiomatic class of L-functions extending the Selberg class. We show in particular that one can recast the traditional conditions of an Euler product, analytic continuation and functional equation in terms of distributional…

This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the…

We propose a new heuristic approach to integral moments of L-functions over function fields, which we demonstrate in the case of Dirichlet characters ramified at one place (the function field analogue of the moments of the Riemann zeta…

We define a quantitative invariant of Liouville cobordisms with monotone filling through an action-completed symplectic cohomology theory. We illustrate the non-trivial nature of this invariant by computing it for annulus subbundles of the…

We prove a strengthening of Mui\'c's integral non-vanishing criterion for Poincar\'e series on unimodular locally compact Hausdorff groups and use it to prove a result on non-vanishing of L-functions associated to cusp forms of…

We study the behaviour of automorphic L-Invariants associated to cuspidal representations of GL(2) of cohomological weight 0 under abelian base change and Jacquet-Langlands lifts to totally definite quaternion algebras. Under a standard…

Let $\omega$ be a primitive cubic root of unity. We study the non-vanishing problem for the family of Hecke $L$-functions associated to primitive cubic characters defined over the Eisenstein quadratic number field $\mathbb{Q}(\omega)$. We…

The Riemann Zeta-Function is the most studied L-function; it's zeroes give information about the prime numbers. We can associate L-functions to a wide array of objects, and in general, the zeroes of these L-functions give information about…

We consider equations of the form Bf=g, where B is a Galois connection between lattices of functions. This includes the case where B is the Legendre-Fenchel transform, or more generally a Moreau conjugacy. We characterise the existence and…

Let $G$ be a reductive group over a local field $F$ of characteristic zero, Archimedean or not. Let $X$ be a $G$-space. In this paper we study the existence of generalized Whittaker quotients for the space of Schwartz functions on $X$,…