Genus two curves with everywhere good reduction over quadratic fields

We address the question of existence of absolutely simple abelian varieties of dimension 2 with everywhere good reduction over quadratic fields. The emphasis will be given to the construction of pairs $(K,C)$, where $K$ is a quadratic number field and $C$ is a genus $2$ curve with everywhere good reduction over $K$. We provide the first infinite sequence of pairs $(K,C)$ where $K$ is a real (complex) quadratic field and $C$ has everywhere good reduction over $K$. Moreover, we show that the Jacobian of $C$ is an absolutely simple abelian variety

Arithmetic Aspects of Bianchi Groups

We discuss several arithmetic aspects of Bianchi groups, especially from a computational point of view. In particular, we consider computing the homology of Bianchi groups together with the Hecke action, connections with automorphic forms, abelian varieties, Galois representations and the torsion in the homology of Bianchi groups. Along the way, we list several open problems and conjectures, survey the related literature, presenting concrete examples and numerical data.Comment: 35 pages, 171 references, 3 tables, 2 figure

Quadratic Fields Admitting Elliptic Curves with Rational jj-Invariant and Good Reduction Everywhere

Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg x\log^{-1/2}(x)$. In this paper, we assume the $abc$-conjecture to show the sharp asymptotic $\sim cx\log^{-1/2}(x)$ for this number, obtaining formulae for $c$ in both the real and imaginary cases. Our method has three ingredients: (1) We make progress towards a conjecture of Granville: Given a fixed elliptic curve $E/\mathbb{Q}$ with short Weierstrass equation $y^2 = f(x)$ for reducible $f \in \mathbb{Z}[x]$, we show that the number of integers $d$, $|d| \leq D$, for which the quadratic twist $dy^2 = f(x)$ has an integral non-$2$-torsion point is at most $D^{2/3+o(1)}$, assuming the $abc$-conjecture. (2) We apply the Selberg--Delange method to obtain a Tauberian theorem which allows us to count integers satisfying certain congruences while also being divisible only by certain primes. (3) We show that for a polynomially sparse subset of the natural numbers, the number of pairs of elements with least common multiple at most $x$ is $O(x^{1-\epsilon})$ for some $\epsilon > 0$. We also exhibit a matching lower bound. If instead of the $abc$-conjecture we assume a particular tail bound, we can prove all the aforementioned results and that the coefficient $c$ above is greater in the real quadratic case than in the imaginary quadratic case, in agreement with an experimentally observed bias.Comment: 35 pages, 1 figur