Related papers: Class Numbers via 3-Isogenies and Elliptic Surface…

We investigate exponential sums over singular binary quartic forms, proving an explicit formula for the finite field Fourier transform of this set. Our formula shares much in common with analogous formulas proved previously for other vector…

A formula for the sum of quadratic residues modulus a prime $p=4n-1$ is studied. We relate some terms on this formula with roots of quadratics and provide an exhaustive analysis of new concepts based on these roots. A number of formulas for…

Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other,…

It is a well known result that the number of points over a finite field on the Legendre family of elliptic curves can be written in terms of a hypergeometric function modulo $p$. In this paper, we extend this result, due to Igusa, to a…

The theory of the isoptic curves is widely studied in the Euclidean plane $\bE^2$ (see \cite{CMM91} and \cite{Wi} and the references given there). The analogous question was investigated by the authors in the hyperbolic $\bH^2$ and elliptic…

For any odd prime $p,$ we construct an infinite family of pairs of imaginary quadratic fields $\mathbb{Q}(\sqrt{d}),\mathbb{Q}(\sqrt{d+1})$ whose class numbers are both divisible by $p.$ One of our theorems settles Iizuka's conjecture for…

We give explicit formulae for the logarithmic class group pairing on an elliptic curve defined over a number field. Then we relate it to the descent relative to a suitable cyclic isogeny. This allows us to connect the resulting Selmer group…

We discuss the computation of coefficients of the L-series associated to a hyperelliptic curve over Q of genus at most 3, using point counting, generic group algorithms, and p-adic methods.

Let $h(-n)$ be the class number of the imaginary quadratic field with discriminant $-n$. We establish an asymtotic formula for correlations involving $h(-n)$ and $h(-n-l)$, over fundamental discriminants that avoid the congruence class…

We propose an algorithm that calculates isogenies between elliptic curves defined over an extension $K$ of $\mathbb{Q}_2$. It consists in efficiently solving with a logarithmic loss of $2$-adic precision the first order differential…

Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p odd and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity…

By combining tools from different areas of mathematics, we obtain 3D visualizations of elliptic curves over different fields that faithfully capture the underlying algebra and geometry.

Let $p \geq 5$ be a prime and for $a, b \in \mathbb{F}_{p}$, let $E_{a,b}$ denote the elliptic curve over $\mathbb{F}_{p}$ with equation $y^2=x^3+a\,x + b$. As usual define the trace of Frobenius $a_{p,\,a,\,b}$ by \begin{equation*}…

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice…

Let $E$ be a non CM elliptic curve defined over $\Q$. There is an isogeny-torsion graph associated to $E$ and there is also a Galois representation $\rho_{E,l^{\infty}} \colon \Gal(\Qbar/\Q) \to \GL_2(\ZZ_{\ell})$ associated to $E$ for…

It is well known that there is a one-to-one correspondence between supersingular $j$-invariants up to the action of $\text{Gal}(\mathbb{F}_{p^2}/\mathbb{F}_p)$ and type classes of maximal orders in $B_{p,\infty}$ by Deuring's theorem.…

We classify birational maps into elliptic fibrations of a general quasismooth hypersurface in $\mathbb{P}(1,a_{1},a_{2},a_{3},a_{4})$ of degree $\sum_{i=1}^{4}a_{i}$ that has terminal singularities.

We prove the generalised Tate conjecture for H^3 of products of elliptic curves over finite fields, by slightly modifying an argument of M. Spiess concerning the Tate conjecture. We prove it fully if the elliptic curves run among at most 3…

Using an explicit family of plane quartic curves, we prove the existence of a genus 3 curve over any finite field of characteristic 3 whose number of rational points stays within a fixed distance from the Hasse-Weil-Serre upper bound. We…

In this paper, we are going to prove the relation between rank of elliptic curves and the non-triviality of class groups of infinitely many real quadratic fields.