We construct a geometric realization of representations for $\text{PSL}(2, \mathbb{F}_p)$ by the defining ideals of rational models $\mathcal{L}(X(p))$ of modular curves $X(p)$ over $\mathbb{Q}$, which gives rise to a Rosetta stone for geometric representations of $\text{PSL}(2, \mathbb{F}_p)$. The defining ideal of a modular curve, i.e., an anabelian counterpart of the Eisenstein ideal, is the anabelianization of the Jacobian of this modular curve and is a reification of the fundamental group $π_1$. We show that there exists a correspondence among the defining ideals of modular curves over $\mathbb{Q}$, reducible $\mathbb{Q}(ζ_p)$-rational representations $π_p: \text{PSL}(2, \mathbb{F}_p) \rightarrow \text{Aut}(\mathcal{L}(X(p)))$ of $\text{PSL}(2, \mathbb{F}_p)$, and $\mathbb{Q}(ζ_p)$-rational Galois representations $ρ_p: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{Aut}(\mathcal{L}(X(p)))$ as well as their modular and surjective realization. It is an anabelian counterpart of the global Langlands correspondence for $\text{GL}(2, \mathbb{Q})$ by the étale cohomology of modular curves as well as an anabelian counterpart of Artin's conjecture, Serre's modularity conjecture, the Fontaine-Mazur conjecture and the $p$-adic local Langlands correspondence for $\text{GL}(2, \mathbb{Q}_p)$. It is an ideal theoretic (i.e. nonlinear) counterpart of Grothendieck's section conjecture and an ideal theoretic (i.e. nonlinear) reification of ``arithmetic theory of $π_1$'' expected by Weil for modular curves. It is also an anabelian counterpart of the theory of Kubert-Lang and Mazur-Wiles on the cuspidal divisor class groups and the Eisenstein ideals of modular curves.
Torsion of Drinfeld modules and equicharacteristic unimodular Galois covers
本文证明了群PSL（r，q^d）可以F_q（T）-正则地实现为纯超越域F_q（T）（t_1，.，t_{r-1}）如果r是偶数并且r/2与q^d-1互质。该方法是使用Drinfeld模的扭模方案（a la Shih）。如果人们对模形式有足够的控制，就可以把它变成一种算法来求显式方程。对于PSL（2，q^2），我们使用Drinfeld模形式来实现这一点。
There are several variants of the inverse Galois problem which involve restrictions on ramification. In this paper we give sufficient conditions that a given finite group $G$ occurs infinitely often as a Galois group over the rationals $\mathbb Q$ with all nontrivial inertia groups of order $2$. Notably any such realization of $G$ can be translated up to a quadratic field over which the corresponding realization of $G$ is unramified. The sufficient conditions are imposed on a parametric polynomial with Galois group $G$--if such a polynomial is available--and the infinitely many realizations co…
We study certain unitary CM fields whose Galois closure has Galois group $\operatorname{PSL}_2(\mathbb{F}_q) \times \mathbb{Z}/2\mathbb{Z}$. After investigating the CM types of these fields, we turn towards Colmez's conjectural formula on the Faltings heights of CM abelian varieties. We explicitly calculate the class functions appearing in the statement of the conjecture and then apply refinements of the recently proven average version to establish Colmez's conjecture in this case.
We compute the first explicit polynomials with Galois groups $G=PΓL_3(4)$, $PGL_3(4)$, $PSL_3(4)$ and $PSL_5(2)$ over $\mathbb{Q}(t)$. Furthermore we compute the first examples of totally real polynomials with Galois groups $PGL_2(11)$, $PSL_3(3)$, $M_{22}$ and $Aut(M_{22})$ over $\mathbb{Q}$. All these examples make use of families of covers of the projective line ramified over four or more points, and therefore use techniques of explicit computations of Hurwitz spaces. Similar techniques were used previously e.g. by Malle, Couveignes, Granboulan and Hallouin. Unlike previous examples, howeve…
Rationality problem for norm one tori for small non-Galois extensions: Symmetric, alternating and linear group cases including stably rational ${\rm PSL}_2(\mathbb{F}_{2^d})$ cases with $d=1,2,3$
Jambor--Liebeck--O'Brien showed that there exist non-proper-power word maps which are not surjective on $\mathrm{PSL}_{2}(\mathbb{F}_{q})$ for infinitely many $q$. This provided the first counterexamples to a conjecture of Shalev which stated that if a two-variable word is not a proper power of a non-trivial word, then the corresponding word map is surjective on $\mathrm{PSL}_2(\mathbb{F}_{q})$ for all sufficiently large $q$. Motivated by their work, we construct new examples of these types of non-surjective word maps. As an application, we obtain non-surjective word maps on the absolute Galoi…
We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the unramified inverse Galois problem. That is, we show that our methods can be used to determine that certain groups cannot be realized as the Galois groups of unramified extensions of certain number fields. To demonstrate the power of our methods, we give an infinite family of totally imaginary quadratic number fields such that $\text{Aut}(\text{PSL}(2,q^2))$ for $q$ a…
The constructive inverse Galois problem via Hilbert modular forms: realizing the transitive group 17T7
Massachusetts Institute of Technology（麻省理工学院）；University of Bristol（布里斯托大学）；Harvard University（哈佛大学）；Universität Würzburg（维尔茨堡大学）；Rijksuniveriteit Groningen（格罗宁根大学）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）；Dartmouth College（达特茅斯学院）
Ramification groups of Galois extensions over local fields of positive characteristic with Galois group isomorphic to the group of unitriangular matrices
In this paper we prove that for each integer of the form $n=4\varpi$ (where $\varpi$ is a prime between $17$ and $73$) at least one of the following groups: $PΩ^+_n(\mathbb{F}_{\ell^s})$, $PSO^+_n(\mathbb{F}_{\ell^s})$, $PO_n^+(\mathbb{F}_{\ell^s})$ or $PGO^+_n(\mathbb{F}_{\ell^s})$ is a Galois group of $\mathbb{Q}$ for almost all primes $\ell$ and infinitely many integers $s > 0$. This is achieved by making use of the classification of small degree representations of finite simple groups of Lie type in defining characteristic of F. Lübeck and a previous result of the author on the image of th…
We study origami $f: C \rightarrow E$ with $G$-Galois cover $Q_8$. For a point $P \in E(\mathbb{Q}) \backslash \left\{ \mathcal{O} \right\}$, we study the field obtained by adjoining to $\mathbb{Q}$ the coordinates of all of the preimages of $P$ under $f$. We find a defining polynomial, $f_{E, Q_8,P}$, for this field and study its Galois group. We give an isomorphism depending on $P$ between a certain subfield of this field and a certain subfield of the 4-division field of the elliptic curve.
There are 3 examples in these notes. The first one is the standard example of the cubic resolvent of a quartic. The second example is exactly from Adelmann \cite{Adelmann} and gives a defining polynomial corresponding to the unique $S_4$-quotient of $\mathrm{GL}_2(\mathbb{Z}/4\mathbb{Z})$. The splitting field of the Adelmann polynomial over $\mathbb{Q}$ is a subfield of the 4-division field of an elliptic curve, that contains the 2-division field of the elliptic curve. The third example is new and needed in the study of the field theory of quaternion origami. Associated to an elliptic curve de…
In this paper, we will define analogues of multiple zeta values by replacing the differential forms defining multiple zeta values with some $\mathbb{Q}$-rational differential forms on the Fermat curve $F_2$ of degree 2 and discuss their arithmetic properties. We also investigate a motivic structure of the motivic periods corresponding to our periods. However, in order to study them, the current theory for motivic zeta elements is insufficient, and it leads us to study the base extension of the space of the motivic periods $\mathcal{H}_4$ of level 4 and its Galois invariant part.
For an elliptic curve E over an abelian extension k/K with CM by K of Shimura type, the L-functions of its [k:K] Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a modular curve to E pulls back the 1-forms on E to give the Hecke theta functions. This article refines the study of our earlier work and shows that certain class of chiral correlation functions in Type II string theory with [E]_C (E as real analytic manifold) as a target space yield the same Hecke theta functions as objects on the modular curve. The Kahler paramet…
