Galois Orbits Research Articles

Galois orbits of torsion points near atoral sets

Galois orbits of torsion points near atoral sets

Linear system of hypersurfaces passing through a Galois orbit

Linear system of hypersurfaces passing through a Galois orbit

Galois equivariant functions on Galois orbits in large $p$-adic fields

Given a prime number p let \mathbb{C}_p be the topological completion of the algebraic closure of the field of p -adic numbers. Let O(T) be the Galois orbit of a transcendental element T of \mathbb{C}_p with respect to the absolute Galois group. Our aim is to study the class of Galois equivariant functions defined on O(T) with values in \mathbb{C}_p . We show that each function from this class is continuous and we characterize the class of Lipschitz functions, respectively the class of differentiable functions, with respect to a new orthonormal basis. Then we discuss some aspects related to analytic continuation for the functions of this class.

$p$-adic equidistribution of CM points

Let X be a modular curve and consider a sequence of Galois orbits of CM points in X , whose p -conductors tend to infinity. Its equidistribution properties in X(\mathbf{C}) and in the reductions of X modulo primes different from p are well understood. We study the equidistribution problem in the Berkovich analytification X_{p}^{\operatorname{an}} of X_{\operatorname{Q}_{p}} . We partition the set of CM points of sufficiently high conductor in X_{\operatorname{Q}_{p}} into finitely many explicit basins B_{V} , indexed by the irreducible components V of the \bmod\text{-}p reduction of the canonical model of X . We prove that a sequence z_{n} of local Galois orbits of CM points with p -conductor going to infinity has a limit in X_{p}^{\operatorname{an}} if and only if it is eventually supported in a single basin B_{V} . If so, the limit is the unique point of X_{p}^{\operatorname{an}} whose mod- p reduction is the generic point of V . The result is proved in the more general setting of Shimura curves over totally real fields. The proof combines Gross's theory of quasi-canonical liftings with a new formula for the intersection numbers of CM curves and vertical components in a Lubin–Tate space.

Lower bounds for Galois orbits of special points on Shimura varieties: a point-counting approach

Let S be a Shimura variety. We conjecture that the heights of special points in \(S(\overline{\mathbb {Q}})\) are discriminant negligible with respect to some Weil height function \(h:S(\overline{\mathbb {Q}})\rightarrow \mathbb {R}\). Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In particular, we give a new proof of a theorem of Tsimerman on lower bounds for Galois degrees of special points in Shimura varieties of abelian type. This gives a new proof of the André–Oort conjecture for such varieties that avoids the use of Masser–Wüstholz isogeny estimates, replacing them by a point-counting argument.

Galois orbits of TQFTs: symmetries and unitarity

We study Galois actions on 2+1D topological quantum field theories (TQFTs), characterizing their interplay with theory factorization, gauging, the structure of gapped boundaries and dualities, 0-form symmetries, 1-form symmetries, and 2-groups. In order to gain a better physical understanding of Galois actions, we prove sufficient conditions for the preservation of unitarity. We then map out the Galois orbits of various classes of unitary TQFTs. The simplest such orbits are trivial (e.g., as in various theories of physical interest like the Toric Code, Double Semion, and 3-Fermion Model), and we refer to such theories as unitary “Galois fixed point TQFTs”. Starting from these fixed point theories, we study conditions for preservation of Galois invariance under gauging 0-form and 1-form symmetries (as well as under more general anyon condensation). Assuming a conjecture in the literature, we prove that all unitary Galois fixed point TQFTs can be engineered by gauging 0-form symmetries of theories built from Deligne products of certain abelian TQFTs.

Power series expansions of modular forms and p-adic interpolation of the square roots of Rankin–Selberg special values

Let [Formula: see text] be a newform of even weight [Formula: see text] for [Formula: see text], where [Formula: see text] is a possibly split indefinite quaternion algebra over [Formula: see text]. Let [Formula: see text] be a quadratic imaginary field splitting [Formula: see text] and [Formula: see text] an odd prime split in [Formula: see text]. We extend our theory of [Formula: see text]-adic measures attached to the power series expansions of [Formula: see text] around the Galois orbit of the CM point corresponding to an embedding [Formula: see text] to forms with any nebentypus and to [Formula: see text] dividing the level of [Formula: see text]. For the latter we restrict our considerations to CM points corresponding to test objects endowed with an arithmetic [Formula: see text]-level structure. Also, we restrict these [Formula: see text]-adic measures to [Formula: see text] and compute the corresponding Euler factor in the formula for the [Formula: see text]-adic interpolation of the “square roots” of the Rankin–Selberg special values [Formula: see text], where [Formula: see text] is the base change to [Formula: see text] of the automorphic representation of [Formula: see text] associated, up to Jacquet-Langland correspondence, to [Formula: see text] and [Formula: see text] is a compatible family of grössencharacters of [Formula: see text] with infinite type [Formula: see text].

Equidistribution on Kuga–Sato varieties of torsion points on CM elliptic curves

A connected Kuga–Sato variety \mathbf{W}^r parametrizes tuples of r points on elliptic curves (with level structure). A special point of \mathbf{W}^r is a tuple of torsion points on a CM elliptic curve. A sequence of special points is strict if any CM elliptic curve appears at most finitely many times and no relation between the points in the tuple is satisfied infinitely often. The genus orbit of a special point is the \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}^{\mathrm{ab}}) -orbit. We show that genus orbits of special points in a strict sequence equidistribute in \mathbf{W}^r(\mathbb{C}) , assuming a congruence condition at two fixed primes. A genus orbit can be very sparse in the full Galois orbit. In particular, the number of torsion points on each elliptic curve in a genus orbit is not bounded below by the torsion order. A genus orbit corresponds to a toral packet in an extension of \mathbf{SL}_2 by a vector representation. These packets also arise in the study by Aka, Einsiedler and Shapira of grids orthogonal to lattice points on the 2-sphere. As an application we establish their joint equidistribution assuming two split primes.

On the number of Galois orbits of newforms

Counting the number of Galois orbits of newforms in S_k(\Gamma_0(N)) and giving some arithmetic sense to this number is an interesting open problem. The case N=1 corresponds to Maeda's conjecture (still an open problem) and the expected number of orbits in this case is 1, for any k \ge 16 . In this article we give local invariants of Galois orbits of newforms for general N and count their number. Using an existence result of newforms with prescribed local invariants we prove a lower bound for the number of non-CM Galois orbits of newforms for \Gamma_0(N) for large enough weight k (under some technical assumptions on N ). Numerical evidence suggests that in most cases this lower bound is indeed an equality, thus we leave as a question the possibility that a generalization of Maeda's conjecture could follow from our work. We finish the paper with some natural generalizations of the problem and show some of the implications that a generalization of Maeda's conjecture has.

An on-average Maeda-type conjecture in the level aspect

We present a conjecture on the average number of Galois orbits of newforms when fixing the weight and varying the level. This conjecture implies, for instance, that the central L L -values (resp. L L -derivatives) are nonzero for 100 100% of even weight prime level newforms with root number + 1 +1 (resp. − 1 -1 ).

Zeroes of quaternionic modular forms and central L-values

Values of quaternionic modular forms are related to twisted central L-values via periods and a theorem of Waldspurger. In particular, certain twisted L-values must be non-vanishing for forms with no zeroes. Here we study, theoretically and computationally, zeroes of definite quaternionic modular forms of trivial weight. Local sign conditions force certain forms to have trivial zeroes, but we conjecture that almost all forms have no nontrivial zeroes. In particular, almost all forms with appropriate local signs should have no zeroes. We show these conjectures follow from a conjecture on the average number of Galois orbits, and give applications to (non)vanishing of L-values.

Moment Maps and Galois Orbits in Quantum Information Theory

SIC-POVMs are configurations of points or rank-one projections arising from the action of a finite Heisenberg group on $\mathbb{C}^d$. The resulting equations are interpreted in terms of moment maps by focusing attention on the orbit of a cyclic subgroup and the maximal torus in $\mathrm{U}(d)$ that contains it. The image of a SIC-POVM under the associated moment map lies in an intersection of real quadrics, which we describe explicitly. We also elaborate the conjectural description of the related number fields and describe the structure of Galois orbits of overlap phases.

Finite simple groups with short Galois orbits on conjugacy classes

All finite simple groups are determined with the property that every Galois orbit on conjugacy classes has size at most 4. From this we list all finite simple groups G for which the normalized group of central units of the integral group ring ZG is an infinite cyclic group.

Counting rational points and lower bounds for Galois orbits

In this article we present a new method to obtain polynomial lower bounds for Galois orbits of torsion points of one dimensional group varieties.

The distribution of Galois orbits of points of small height in toric varieties

We study the distribution of Galois orbits of points of small height on proper toric varieties, and its application to the Bogomolov problem. We introduce the notion of monocritical toric metrized divisor. We prove that a toric metrized divisor D on a proper toric variety X over a global eld K is monocritical if and only if for every generic D-small sequence of algebraic points of X and every place v of K, the sequence of their Galois orbits on the analytic space X an converges to a measure. When this is the case, the limit measure is a translate of the natural measure on the compact torus sitting in the principal orbit of X. The key ingredient is the study of the v-adic modulus distribution of Ga- lois orbits of generic D-small sequences of algebraic points. In particular, we characterize all their cluster measures. We generalize the Bogomolov problem by asking when a subvariety of the principal orbit of a proper toric variety that has the same essential minimum than the ambient variety, must be a translate of a subtorus. We prove that the generalized Bogomolov problem has a positive answer for monocritical toric metrized divisors, and we give several examples of toric metrized divisors for which the Bogomolov problem has a negative answer.

Effective moments of Dirichlet L-functions in Galois orbits

Effective moments of Dirichlet L-functions in Galois orbits

Joint equidistribution of CM points

We prove the mixing conjecture of Michel and Venkatesh for toral packets with negative fundamental discriminants and split at two fixed primes; assuming all splitting fields have no exceptional Landau-Siegel zero. As a consequence we establish for arbitrary products of indefinite Shimura curves the equidistribution of Galois orbits of generic sequences of CM points all whose components have the same fundamental discriminant; assuming the CM fields are split at two fixed primes and have no exceptional zero. The joinings theorem of Einsiedler and Lindenstrauss applies to the toral orbits arising in these results. Yet it falls short of demonstrating equidistribution due to the possibility of intermediate algebraic measures supported on Hecke correspondences and their translates. The main novel contribution is a method to exclude intermediate measures for toral periods. The crux is a geometric expansion of the cross-correlation between the periodic measure on a torus orbit and a Hecke correspondence, expressing it as a short shifted convolution sum. The latter is bounded from above generalizing the method of Shiu and Nair to polynomials in two variables on smooth domains.

Refined dimensions of cusp forms, and equidistribution and bias of signs

We refine known dimension formulas for spaces of cusp forms of squarefree level, determining the dimension of subspaces generated by newforms both with prescribed global root numbers and with prescribed local signs of Atkin–Lehner operators. This yields precise results on the distribution of signs of global functional equations and sign patterns of Atkin–Lehner eigenvalues, refining and generalizing earlier results of Iwaniec, Luo and Sarnak. In particular, we exhibit a strict bias towards root number +1 and a phenomenon that sign patterns are biased in the weight but perfectly equidistributed in the level. Another consequence is lower bounds on the number of Galois orbits.

Arbitrarily large Galois orbits of non-homeomorphic surfaces

It has recently been proved that any element $$\sigma $$ of the absolute Galois group $$\mathrm{Gal}\,\overline{\mathbb {Q}}/{\mathbb Q}$$ different from the identity and complex conjugation conjugates some surface S into a surface $$S^{\sigma }$$ with non-isomorphic fundamental group. Here we use group theory to construct, for each integer $$n\geqslant 0$$ , an n-dimensional family of complex surfaces whose conjugates under $$\mathrm{Gal}\,\overline{\mathbb {Q}}/{\mathbb Q}$$ exhibit arbitrarily many non-isomorphic fundamental groups. These fundamental groups nevertheless have isomorphic profinite completions. The surfaces constructed are isogenous to higher products via actions of groups $$\mathrm{PGL}_2(p)$$ on algebraic curves.

Quantitative equidistribution of Galois orbits of small points in the N-dimensional torus

We present a quantitative version of Bilu's theorem on the limit distribution of Galois orbits of sequences of points of small height in the $N$-dimensional algebraic torus. Our result gives, for a given point, an explicit bound for the discrepancy between its Galois orbit and the uniform distribution on the compact subtorus, in terms of the height and the generalized degree of the point.