We prove local results on the $ p $-adic density of elliptic curves over $\mathbb {Q} _p $ with
different reduction types, together with global results on densities of elliptic curves over …

Let $\mathfrak p $ be any point in the moduli space of genus-two curves $\mathcal M_2 $
and $ K $ its field of moduli. We provide a universal equation of a genus-two curve $\mathcal …

We study the problem of counting the number of varieties in families which have a rational
point. We give conditions on the singular fibres that force very few of the varieties in the …

The density of polynomials of degree n$n$ over Zp${\mathbb {Z}}_p$ having exactly r$r$ roots
in Qp${\mathbb {Q}}_p$ - Bhargava - 2022 - Proceedings of the London Mathematical Society …

We consider the proportion of genus one curves over ℚ of the form z 2= f (x, y) where f (x,
y)∈ ℤ [x, y] is a binary quartic form (or more generally of the form z 2+ h (x, y) z= f (x, y) …

In this paper, we compute the probability that an N× N matrix from the generalized Gaussian
Unitary Ensemble (gGUE) is positive definite, extending a previous result of Dean and …

We consider random symmetric matrices with independent entries distributed according to
the Haar measure on Z p Z_p for odd primes p and derive the distribution of their canonical …

We show that the proportion of plane cubic curves over ℚ p that have a ℚ p-rational point is
a rational function in p, where the rational function is independent of p, and we determine …

For $ n\geq 2$, we consider the map on $ M_n (\mathbb K) $ given by evaluation of a
polynomial $ f (X_1,\ldots, X_m) $ over the field $\mathbb K $. In this article, we explore the …

Let be a non-singular conic bundle over having non-split fibres and denote by the cardinality
of the fibres of Weil height at most that possess a rational point. Serre showed in 1990 that a …