Let K = Q(ζp) and let hp be its class number. Kummer showed that p divides hp if and only if p divides the numerator of some Bernoulli number. In this expository note we discuss the generalizations of this type of criterion to totally real fields and quadratic imaginary fields.

Let $f\left(x\right)$ be a cubic, monic and separable polynomial over a field of characteristic $p\ge 3$ and let $E$ be the elliptic curve given by $y^2=f\left(x\right)$. In this paper we prove that the coefficient at $x^{\frac{1}{2}p(p-1)}$ in the $p$–th division polynomial of $E$ equals the coefficient at $x^{p-1}$ in $f{\left(x\right)}^{\frac{1}{2}(p-1)}$. For elliptic curves over a finite field of characteristic $p$, the first coefficient is zero if and only if $E$ is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the zero loci...

In a recent paper we proved that there are at most finitely many complex numbers $t\ne 0,1$ such that the points $(2,\sqrt{2(2-t)})$ and $(3,\sqrt{6(3-t)})$ are both torsion on the Legendre elliptic curve defined by $y^2=x(x-1)(x-t)$. In a sequel we gave a generalization to any two points with coordinates algebraic over the field $\mathbf{Q}\left(t\right)$ and even over $\mathbf{C}\left(t\right)$. Here we reconsider the special case $(u,\sqrt{u(u-1)(u-t)})$ and $(v,\sqrt{v(v-1)(v-t)})$ with complex numbers $u$ and $v$.

We determine all modular curves X(N) (with N ≥ 7) that are hyperelliptic or bielliptic. We also give a proof that the automorphism group of X(N) is PSL₂(ℤ/Nℤ), whence it coincides with the normalizer of Γ(N) in PSL₂(ℝ) modulo ±Γ(N).

For any Eichler order $𝒪(D,N)$ of level $N$ in an indefinite quaternion algebra of discriminant $D$ there is a Fuchsian group $\Gamma (D,N)\subseteq SL(2,ℝ)$ and a Shimura curve $X(D,N)$. We associate to $𝒪(D,N)$ a set $\mathscr{H}\left(𝒪\right(D,N\left)\right)$ of binary quadratic forms which have semi-integer quadratic coefficients, and we develop a classification theory, with respect to $\Gamma (D,N)$, for primitive forms contained in $\mathscr{H}\left(𝒪\right(D,N\left)\right)$. In particular, the classification theory of primitive integral binary quadratic forms by $SL(2,\mathbb{Z})$ is recovered. Explicit fundamental domains for $\Gamma (D,N)$ allow the characterization...

Let $K=\mathbb{Q}\left(\omega \right)$, with ${\omega }^3=m$ a positive integer, be a pure cubic number field. We show that the elements $\alpha \in K^{\times }$ whose squares have the form $a-\omega$ for rational numbers $a$ form a group isomorphic to the group of rational points on the elliptic curve $E_m:y^2=x^3-m$. This result will allow us to construct unramified quadratic extensions of pure cubic number fields $K$.

Soit $F$ un polynôme en deux variables, de degré $D$ et à coefficients entiers dans $[-M,M]$ pour $M\ge 3$. Alors le nombre de zéros rationnels de $F$ est soit infini soit plus petit que $M^{2^{3^{D^2}}}$. Nous montrons aussi une version plus générale sur les corps de nombres.