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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 → ∞ ) 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 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.

Mordell-Weil ranks of families of elliptic curves associated to Pythagorean triples

Bartosz Naskręcki (2013)

Acta Arithmetica

We study the family of elliptic curves y² = x(x-a²)(x-b²) parametrized by Pythagorean triples (a,b,c). We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over ℚ is 1, and for some explicitly given infinite family the rank is 2. To each family we attach an elliptic surface fibered over the projective line. We show that the lower bounds for the rank are optimal, in the sense that for each generic fiber of such an elliptic surface its corresponding Mordell-Weil...

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