Related papers: 2-Selmer Groups over Multiquadratic Extensions

We investigate degree bounds for fields of rational invariants of representations of finite groups. We prove many cases of a bound for $\mathbb{Z}/p\mathbb{Z}$ conjectured by Blum-Smith, Garcia, Hidalgo, and Rodriguez. For arbitrary groups,…

We consider the following question : given a family over abelian varieties $\mathcal{A}$ over a curve $B$ defined over a number field $k$, how does the rank of the Mordell-Weil group of the fibres $\mathcal{A}_t(k)$ vary? A specialisation…

We study the Siegel-Weil formula in the second term range ($n+1\leq m\leq n+r$) for unitary groups of hermitian forms over a skew-field $D$ with involution of the second kind.

We study Kloosterman sums on the orthogonal groups $SO_{3,3}$ and $SO_{4,2}$, associated to short elements of their respective Weyl groups. An explicit description for these sums is obtained in terms of multi-dimensional exponential sums.…

We classify elliptic curves over the rationals whose N\'eron model over the integers is semi-abelian, with good reduction at p=2, and whose Mordell--Weil group contains an element of order two that stays non-trivial at p=2. Furthermore, we…

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…

We consider the degree-diameter problem for Cayley graphs of dihedral groups. We find upper and lower bounds on the maximum number of vertices of such a graph with diameter 2 and degree $d$. We completely determine the asymptotic behaviour…

In this paper, we bring the terminology of the Kunz coordinates of numerical semigroups to gapsets and we generalize this concept to $m$-extensions. It allows us to identify gapsets and, in general, $m$-extensions with tilings of boards. As…

Let $G$ be a finite group and $K$ a number field. We construct a $G$-extension $E/F$, with $F$ of transcendence degree $2$ over $K$, that specializes to all $G$-extensions of $K_\mathfrak{p}$, where $\mathfrak{p}$ runs over all but finitely…

Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for…

It follows from the Grothendieck-Ogg-Shafarevich formula that the rank of an abelian variety (with trivial trace) defined over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on bad…

Using the Riemann Hypothesis over finite fields and bounds for the size of spherical codes, we give explicit upper bounds, of polynomial size with respect to the size of the field, for the number of geometric isomorphism classes of…

For any k<2n we construct a complete system of invariants in the problem of classifying singularities of immersed k-dimensional submanifolds of a symplectic 2n-manifold at a generic double point.

We present several upper and lower bounds on the Kronecker coefficients of the symmetric group. We prove $k$-stability of the Kronecker coefficients generalizing the (usual) stability, and giving a new upper bound. We prove a lower bound…

Given a finite set of points $S\subset\mathbb{R}^d$, a $k$-set of $S$ is a subset $A \subset S$ of size $k$ which can be strictly separated from $S \setminus A $ by a hyperplane. Similarly, a $k$-facet of a point set $S$ in general position…

We explicitly bound the Faltings height of a curve over Q polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings' delta invariant and the self-intersection of the…

Let $E$ be an elliptic curve over $\mathbb{Q}$. Then, we show that the average analytic rank of $E$ over cyclic extensions of degree $l$ over $\mathbb{Q}$ with $l$ a prime not equal to $2$, is at most $2+r_{\mathbb{Q}}(E)$, where…

Let $k$ denote an algebraically closed field. We revisit a construction of the author of families of elliptic curves over the rational function field $k(t)$. Combining a combinatorial analysis with a rank formula of Ulmer we prove that, for…

Let C be a supersingular genus-2 curve over an algebraically closed field of characteristic 3. We show that if C is not isomorphic to the curve y^2 = x^5 + 1 then up to isomorphism there are exactly 20 degree-3 maps phi from C to the…

All the results in this paper are conditional on the Riemann Hypothesis for the L-functions of elliptic curves. Under this assumption, we show that the average analytic rank of all elliptic curves over Q is at most 2, thereby improving a…