On Hermite's invariant for binary quintics

We use the invariant theory of binary quartics to give a new formula for the Cassels-Tate pairing on the $2$-Selmer group of an elliptic curve. Unlike earlier methods, our formula does not require us to solve any conics. An important role in our construction is played by a certain $K3$ surface defined by a $(2,2,2)$-form.

A genus one curve C of degree 5 is defined by the 4 x 4 Pfaffians of a 5 x 5 alternating matrix of linear forms on P^4. We prove a result characterising the covariants for these models in terms of their restrictions to the family of curves parametrised by the modular curve X(5). We then construct covariants describing the covering map of degree 25 from C to its Jacobian and give a practical algorithm for evaluating them.

A classical result from 1861 due to Hermite says that every separable equation of degree 5 can be transformed into an equation of the form x^5 + b x^3 + c x + d = 0. Later this was generalized to equations of degree 6 by Joubert. We show that both results can be understood as an explicit analysis of certain covariants of the symmetric groups S_5 and S_6. In case of degree 5, the classical invariant theory of binary forms of degree 5 comes into play whereas in degree 6 the exi...

In this article a complete set of invariants for ordinary quartic curves in characteristic 2 is computed.

Given an indefinite binary quaternionic Hermitian form $f$ with coefficients in a maximal order of a definite quaternion algebra over $\mathbb Q$, we give a precise asymptotic equivalent to the number of nonequivalent representations, satisfying some congruence properties, of the rational integers with absolute value at most $s$ by $f$, as $s$ tends to $+\infty$. We compute the volumes of hyperbolic 5-manifolds constructed by quaternions using Eisenstein series. In the Append...

An irreducible quintic equation is solvable by radicals if and only if its Galois group is solvable. In this work, we provide necessary and sufficient conditions for solvability, expressed in terms of invariants of the quintic.

We study the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $GF(q)$ into orbits of the group $G=PGL(2,q)$ of linear symmetries of the twisted cubic $C$. A generic line neither intersects $C$ nor lies in any of its osculating planes. While the non-generic lines have been classified into $G$-orbits in literature, it has been an open problem to classify the generic lines into $G$-orbits. For a general field $F$ of characteristic differ...

In this paper, we call a sub-scheme of dimension one in $\mathbb{P}^3$ a curve. It is well known that the arithmetic genus and the degree of an aCM curve $D$ in $\mathbb{P}^3$ is computed by the $h$-vector of $D$. However, for a given curve $D$ in $\mathbb{P}^3$, the two invariants of $D$ do not tell us whether $D$ is aCM or not. In this paper, we give a classification of curves on a smooth quintic hypersurface in $\mathbb{P}^3$ which are not aCM.

We introduce invariant rings for forms (homogeneous polynomials) and for d points on the projective space, from the point of view of representation theory. We discuss several examples, addressing some computational issues. We introduce the graphical algebra for the invariants of d points on the line. This is an expanded version of the notes for the School on Invariant Theory and Projective Geometry, Trento, September 17-22, 2012.

We determine the Waring ranks of all sextic binary forms with complex coefficients using a Geometric Invariant Theory approach. Using the five basic invariants for sextic binary forms, our results give a rapid method to determine the Waring rank of any given sextic binary form. In particular, we shed new light on a claim by E. B. Elliott at the end of the 19th century concerning the binary sextics with Waring rank 3. We show that for binary forms of arbitrary degree the cactu...