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Manin’s and Peyre’s conjectures on rational points and adelic mixing

Alex Gorodnik, François Maucourant, Hee Oh (2008)

Annales scientifiques de l'École Normale Supérieure

Let $X$ be the wonderful compactification of a connected adjoint semisimple group $G$ defined over a number field $K$ . We prove Manin’s conjecture on the asymptotic (as $T \to \infty$ ) of the number of $K$ -rational points of $X$ of height less than $T$ , and give an explicit construction of a measure on $X (𝔸)$ , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points $𝐆 (K)$ on $X (𝔸)$ . Our approach is based on the mixing property of $L^{2} (𝐆 (K) ∖ 𝐆 (𝔸))$ which we obtain with a rate of convergence.

Manin’s conjecture for a quartic del Pezzo surface with $A_{4}$ singularity

Tim D. Browning, Ulrich Derenthal (2009)

Annales de l’institut Fourier

The Manin conjecture is established for a split singular del Pezzo surface of degree four, with singularity type $A_{4}$ .

Manin’s conjecture for a singular sextic del Pezzo surface

Daniel Loughran (2010)

Journal de Théorie des Nombres de Bordeaux

We prove Manin’s conjecture for a del Pezzo surface of degree six which has one singularity of type $A_{2}$ . Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.

We generalize L. J. Mordell’s construction of cubic surfaces for which the Hasse principle fails.

Motivic measures

Eduard Looijenga (1999/2000)

Séminaire Bourbaki

Displaying 1 – 13 of 13

Manin’s and Peyre’s conjectures on rational points and adelic mixing

Manin’s conjecture for a quartic del Pezzo surface with $A_{4}$ singularity

Manin’s conjecture for a singular sextic del Pezzo surface

Manin's conjecture for two quartic del Pezzo surfaces with 3A₁ and A₁ + A₂ singularity types

Manin's conjecture on a nonsingular quartic del Pezzo surface

Méthodes algébriques et démonstrations de transcendance locales.

Modular curves and the Eisenstein ideal

Modular equations of hyperelliptic X₀(N) and an application

Modulare Einheiten für Funktionenkörper.

Mordell's conjecture over function fields.

Mordell-Weil rank of the jacobians of the curves defined by $y^{p} = f (x)$

More cubic surfaces violating the Hasse principle

Motivic measures

Currently displaying 1 – 13 of 13