Related papers: Zhang's Conjecture and the Effective Bogomolov Con…

We investigate versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings, for other classes of varieties. We first obtain analogues for certain Fano threefolds. We use these results to prove the…

Study of the level curve for the real part of $\eta(s)=0$ with $\eta(s)=\pi^{-s/2}\Gamma(s/2)\zeta^\prime(s)$ gives a new classification of the zeros of $\zeta(s)$ and of $\zeta^\prime(s)$. We conjecture that for type 2 zeros, $\liminf…

Let $f: S\longrightarrow B$ be a non-trivial fibration from a complex projective smooth surface $S$ to a smooth curve $B$ of genus $b$. Let $c_f$ the Clifford index of the generic fibre $F$ of $f$. In [arXiv:1401.7502v4] it is proved that…

In this paper we study the degeneration behavior of the norm of the Riemann $\theta$-function in a family of principally polarized abelian varieties over the punctured complex unit disc in terms of the associated polarized real torus. As an…

We give results on the asymptotic in Waring's problem over function fields that are stronger than the results obtained over the integers using the main conjecture in Vinogradov's mean value theorem. Similar estimates apply to Manin's…

In this paper we establish some foundations regarding sheaves of vector spaces on graphs and their invariants, such as homology groups and their limits. We then use these ideas to prove the Hanna Neumann Conjecture of the 1950's; in fact,…

We present a family of conjectural relations in the tautological cohomology of the moduli spaces of stable algebraic curves of genus $g$ with $n$ marked points. A large part of these relations has a surprisingly simple form: the…

This is an introduction to the topics of the title, from the 2017 Grenoble Summer school on Arakelov geometry and arithmetic applications. We review Arithmetic intersection numbers, explain the definition of the height of a variety and its…

A conjecture of Batyrev and Manin predicts the asymptotic behaviour of rational points of bounded height on smooth projective varieties over number fields. We prove some new cases of this conjecture for conic bundle surfaces equipped with…

The statement of the Riemann hypothesis makes sense for all global fields, not just the rational numbers. For function fields, it has a natural restatement in terms of the associated curve. Weil's work on the Riemann hypothesis for curves…

On an abelian scheme over a smooth curve over $\overline{\mathbb Q}$ a symmetric relatively ample line bundle defines a fiberwise N\'eon-Tate height. If the base curve is inside a projective space, we also have a height on its…

These brief notes record our puzzles and findings surrounding Givental's recent conjecture which expresses higher genus Gromov-Witten invariants in terms of the genus-0 data. We limit our considerations to the case of a projective line,…

Towards confirming Sun's conjecture on the strict log-concavity of combinatorial sequence involving the n$th$ Bernoulli number, Chen, Guo and Wang proposed a conjecture about the log-concavity of the function…

We state a conjecture about the zeta function of crepant resolutions of Berglund--H\"ubsch orbifold hypersurfaces over a finite field. In addition to numerical evidence, we show that our conjectural zeta function satisfies the Weil…

We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement to the normal form. Using this result, we prove the Herman invariant tori…

We prove new cases of Vojta's conjectures for surfaces in the context of function fields, with truncation equal to one and providing an effective explicit description of the exceptional set. We also prove a general and explicit result…

De Bruijn and Newman introduced a deformation of the completed Riemann zeta function $\zeta$, and proved there is a real constant $\Lambda$ which encodes the movement of the nontrivial zeros of $\zeta$ under the deformation. The Riemann…

Several invariants of polarized metrized graphs and their applications in Arithmetic Geometry are studied recently. In this paper, we explicitly calculated these admissible invariants for all curves of genus $3$. We find the sharp lower…

The Graded Classification Conjecture (GCC) states that the pointed $K_0^{\operatorname{gr}}$-group is a complete invariant of the Leavitt path algebras of finite graphs when these algebras are considered with their natural grading by…

We explicitly bound the Faltings height of a curve over Q polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the…