Weil Group Research Articles

The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves

Abstract We show that if F is $\mathbb{Q}$ or a multiquadratic number field, $p\in\left\{{2,3,5}\right\}$ , and $K/F$ is a Galois extension of degree a power of p, then for elliptic curves $E/\mathbb{Q}$ ordered by height, the average dimension of the p-Selmer groups of $E/K$ is bounded. In particular, this provides a bound for the average K-rank of elliptic curves $E/\mathbb{Q}$ for such K. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F. The central result is that: for each finite Galois extension $K/F$ of number fields and prime number p, as $E/\mathbb{Q}$ varies, the difference in dimension between the Galois fixed space in the p-Selmer group of $E/K$ and the p-Selmer group of $E/F$ has bounded average.

A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over [formula omitted]

We provide an algorithm for computing a basis of homology of elliptic surfaces over PC1 that is sufficiently explicit for integration of periods to be carried out. This allows the heuristic recovery of several algebraic invariants of the surface, notably the Néron–Severi lattice, the transcendental lattice, the Mordell–Weil group and the Mordell–Weil lattice. This algorithm comes with a SageMath implementation.

On local L-factors for Archimedean GL(n)

We show that the n-dimensional representations of the Weil group of C and R are uniquely determined by the local L-factors twisted by GL1 for the complex case and by GL1 and GL2 for the real case. This improves an old announced (though not published) result by Henniart.

Torsion points of elliptic curves over multi-quadratic number fields

We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves X1(M,MN) over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.

Local Galois representations associated to additive polynomials

For an additive polynomial and a positive integer, we define an irreducible smooth representation of a Weil group of a non-archimedean local field. We study several invariants of this representation. We obtain a necessary and sufficient condition for it to be primitive.

Ranks of elliptic curves in cyclic sextic extensions of [formula omitted]

For an elliptic curve E/Q we show that there are infinitely many cyclic sextic extensions K/Q such that the Mordell–Weil group E(K) has rank greater than the subgroup of E(K) generated by all the E(F) for the proper subfields F⊂K. For certain curves E/Q we show that the number of such fields K of conductor less than X is ≫X.

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Let K be a field, Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} a finite group of field automorphisms of K, F the Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-fixed field in K, and G≤GLv(K)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G\\le {{\\,\ extrm{GL}\\,}}_v(K)$$\\end{document} a finite matrix group. Then the action of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} defines a grading on the symmetric algebra of the F-space Kv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K^v$$\\end{document} which we use to introduce the notion of homogeneous Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-conjugate invariants of G. We apply this new grading in invariant theory to broaden the connection between codes and invariant theory by introducing Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-conjugate complete weight enumerators of codes. The main result of this paper applies the theory from Nebe, Rains, Sloane to show that under certain extra conditions these new weight enumerators generate the ring of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-conjugate invariants of the associated Clifford–Weil groups. As an immediate consequence, we obtain a result by Bannai et al. that the complex conjugate weight enumerators generate the ring of complex conjugate invariants of the complex Clifford group. Also the Schur–Weyl duality conjectured and partly shown by Gross et al. can be derived from our main result.

Mordell–Weil groups and automorphism groups of elliptic $K3$ surfaces

We present a method to calculate the action of the Mordell–Weil group of an elliptic K3 surface on the numerical Néron–Severi lattice of the K3 surface. As an application, we compute a finite generating set of the automorphism group of a K3 surface birational to the double plane branched along a 6-cuspidal sextic curve of torus type.

Extendible functions and local root numbers: Remarks on a paper of R. P. Langlands

AbstractThis paper refers to Langlands' big set of notes devoted to the question if the (normalized) local Hecke–Tate root number , where E is a finite separable extension of a fixed nonarchimedean local field F, and χ a quasicharacter of , can be appropriately extended to a local ε‐factor for all virtual representations ρ of the corresponding Weil group . Whereas Deligne has given a relatively short proof by using the global Artin–Weil L‐functions, the proof of Langlands is purely local and splits into two parts: the algebraic part to find a minimal set of relations for the functions Δ, such that the existence (and unicity) of will follow from these relations; and the more extensive arithmetic part to give a direct proof that all these relations are actually fulfilled. Our aim is to cover the algebraic part of Langlands' notes, which can be done completely in the framework of representations of solvable profinite groups, where two modifications of Brauer's theorem play a prominent role.

Twists of Albanese varieties of abelian and dihedral covers of ℙℓ with large ranks over function fields

In this paper, we construct twists of Albanese varieties of abelian and dihedral covers of projective spaces with large Mordell–Weil rank over function fields of certainly defined high dimensional varieties. We achieve this by using the theory of twists, a generalization of the notion of Prym variety to higher dimensions, and a structure theorem for the Mordell–Weil group of abelian varieties over function fields.

Arithmetic statistics and diophantine stability for elliptic curves

We study the growth and stability of the Mordell–Weil group and Tate–Shafarevich group of an elliptic curve defined over the rationals, in various cyclic Galois extensions of prime power order. Mazur and Rubin introduced the notion of diophantine stability for the Mordell–Weil group an elliptic curve $$E_{/{{\mathbb {Q}}}}$$ at a given prime p. Inspired by their definition of stability for the Mordell–Weil group, we introduce an analogous notion of stability for the Tate–Shafarevich group, called -stability. From the perspective of Iwasawa theory, it benefits us to introduce a stronger notion of diophantine stability for the Mordell–Weil group. It is shown that any non-CM elliptic curve of rank 0 defined over the rationals is strongly diophantine stable and -stable at $$100\%$$ of primes p. Next, we show that standard conjectures on rank distribution give lower bounds for the proportion of rational elliptic curves E that are strongly diophantine stable at a fixed prime $$p\ge 11$$ . Related questions are studied for rank jumps and growth of ranks Tate–Shafarevich groups on average in prime power cyclic extensions.

Deformation of integral sections in the Mordell–Weil group

Deformation of integral sections in the Mordell–Weil group

Plectic Stark–Heegner points

We propose a conjectural construction of determinants of global points on modular elliptic curves over arbitrary number fields, generalizing both the p-adic construction of Heegner points via Čerednik–Drinfeld uniformization and the definition of classical Stark–Heegner points. In alignment with Nekovář and Scholl's plectic conjectures, we expect the non-triviality of these plectic Stark–Heegner points to control the Mordell–Weil group of higher rank elliptic curves. We provide some indirect evidence for our conjectures by showing that higher order derivatives of anticyclotomic p-adic L-functions compute plectic invariants.

A local Langlands parameterization for generic supercuspidal representations of $p$-adic $G_2$

We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will construct a bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K) \] from the set of generic supercuspidal representations of $G_2(K)$ to the set of irreducible continuous homomorphisms $\rho : W_K \to G_2(\CC)$ with $W_K$ the Weil group of $K$. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix. The proof that the map is a bijection is arithmetic in nature, and specifically uses automorphy lifting theorems. These can be applied thanks to a recent result of Hundley and Liu on automorphic descent from $GL(7)$ to $G_2$.

Algebraic structure and characteristic ideals of fine Mordell–Weil groups and plus/minus Mordell–Weil groups

Algebraic structure and characteristic ideals of fine Mordell–Weil groups and plus/minus Mordell–Weil groups

Heron triangles with two rational medians and Somos-5 sequences

Triangles with integer length sides and integer area are known as Heron triangles. Taking rescaling freedom into account, one can apply the same name when all sides and the area are rational numbers. A perfect triangle is a Heron triangle with all three medians being rational, and it is a longstanding conjecture that no such triangle exists. However, Buchholz and Rathbun showed that there are infinitely many Heron triangles with two rational medians, an infinite subset of which are associated with rational points on an elliptic curve E({mathbb {Q}}) with Mordell–Weil group {mathbb {Z}},{times }, {mathbb {Z}}/2{mathbb {Z}}, and they observed a connection with a pair of Somos-5 sequences. Here we make the latter connection more precise by providing explicit formulae for the integer side lengths, the two rational medians, and the area in this infinite family of Heron triangles. The proof uses a combined approach to Somos-5 sequences and associated Quispel–Roberts–Thompson (QRT) maps in the plane, from several different viewpoints: complex analysis, real dynamics, and reduction modulo a prime.

Plectic p-adic invariants

For modular elliptic curves over number fields of narrow class number one, and with multiplicative reduction at a collection of p-adic primes, we define new p-adic invariants. Inspired by Nekovář and Scholl's plectic conjectures, we believe these invariants control the Mordell–Weil group of higher rank elliptic curves and we support our expectations with numerical experiments.

On the Mordell–Weil lattice of y^2 = x^3 + b x + t^{3^n + 1} in characteristic 3

We study the elliptic curves given by y^2 = x^3 + b x +t^{3^n+1} over global function fields of characteristic 3 ; in particular we perform an explicit computation of the L-function by relating it to the zeta function of a certain superelliptic curve u^3 + b u = v^{3^n + 1}. In this way, using the Néron–Tate height on the Mordell–Weil group, we obtain lattices in dimension 2 cdot 3^n for every n ge 1, which improve on the currently best known sphere packing densities in dimensions 162 (case n=4) and 486 (case n=5). For n=3, the construction has the same packing density as the best currently known sphere packing in dimension 54, and for n=1 it has the same density as the lattice E_6 in dimension 6.

Geometric Realization of the Local Langlands Correspondence for Representations of Conductor Three

We prove a realization of the local Langlands correspondence for two-dimensional representations of a Weil group of conductor three in the cohomology of Lubin–Tate curves by a purely local geometric method.

Cubic and Quartic Points on Modular Curves Using Generalised Symmetric Chabauty

Abstract Answering a question of Zureick-Brown, we determine the cubic points on the modular curves $X_0(N)$ for $N \in \{53,57,61,65,67,73\}$ as well as the quartic points on $X_0(65)$. To do so, we develop a “partially relative” symmetric Chabauty method. Our results generalise current symmetric Chabauty theorems and improve upon them by lowering the involved prime bound. For our curves a number of novelties occur. We prove a “higher-order” Chabauty theorem to deal with these cases. Finally, to study the isolated quartic points on $X_0(65)$, we rigorously compute the full rational Mordell–Weil group of its Jacobian.