Class Groups Of Number Fields Research Articles

Ideal Class Groups of Number Fields Associated to Modular Galois Representations

Ideal Class Groups of Number Fields Associated to Modular Galois Representations

Ideal Class Groups of Number Fields and Bloch-Kato’s Tate-Shafarevich Groups for Symmetric Powers of Elliptic Curves

For an elliptic curve E over ℚ, putting K=ℚ(E[p]) which is the p-th division field of E for an odd prime p, we study the ideal class group ClK of K as a Gal(K/ℚ)-module. More precisely, for any j with 1⩽j⩽p-2, we give a condition that ClK⊗Fp has the symmetric power SymjE[p] of E[p] as its quotient Gal(K/ℚ)-module, in terms of Bloch-Kato’s Tate-Shafarevich group of SymjVpE. Here VpE denotes the rational p-adic Tate module of E. This is a partial generalization of a result of Prasad and Shekhar for the case j=1.

ℓ-torsion bounds for the class group of number fields with an ℓ-group as Galois group

We describe the relations among the ℓ \ell -torsion conjecture, a conjecture of Malle giving an upper bound for the number of extensions, and the discriminant multiplicity conjecture. We prove that the latter two conjectures are equivalent in some sense. Altogether, the three conjectures are equivalent for the class of solvable groups. We then prove the ℓ \ell -torsion conjecture for ℓ \ell -groups and the other two conjectures for nilpotent groups.

Galois covers of ℙ1 and number fields with large class groups

For each finite subgroup [Formula: see text] of [Formula: see text], and for each integer [Formula: see text] coprime to [Formula: see text], we construct explicitly infinitely many Galois extensions of [Formula: see text] with group [Formula: see text] and whose ideal class group has [Formula: see text]-rank at least [Formula: see text]. This gives new [Formula: see text]-rank records for class groups of number fields.

Zeros of Rankin–Selberg $L$-functions at the edge of the critical strip (with an appendix by Colin J. Bushnell and Guy Henniart)

Let $\pi$ and $\pi_0$ be unitary cuspidal automorphic representations. We prove log-free zero density estimates for Rankin-Selberg $L$-functions of the form $L(s,\pi\times\pi_0)$, where $\pi$ varies in a given family and $\pi_0$ is fixed. These estimates are unconditional in many cases of interest; they hold in full generality assuming an average form of the generalized Ramanujan conjecture. We consider applications of these estimates related to mass equidistribution for Hecke-Maass forms, the rarity of Landau-Siegel zeros of Rankin-Selberg $L$-functions, the Chebotarev density theorem, and $\ell$-torsion in class groups of number fields.

Moments and interpretations of the Cohen–Lenstra–Martinet heuristics

The goal of this paper is to prove theorems that elucidate the Cohen-Lenstra-Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler statement of the conjectures. We show that the probabilities that arise are inversely proportional the to number of automorphisms of structures slightly larger than the class groups. We find the moments of the Cohen-Lenstra-Martinet distributions and prove that the distributions are determined by their moments. In order to apply these conjectures to class groups of non-Galois fields, we prove a new theorem on the capitulation kernel (of ideal classes that become trivial in a larger field) to relate the class groups of non-Galois fields to the class groups of Galois fields. We then construct an integral model of the Hecke algebra of a finite group, show that it acts naturally on class groups of non-Galois fields, and prove that the Cohen-Lenstra-Martinet conjectures predict a distribution for class groups of non-Galois fields that involves the inverse of the number of automorphisms of the class group as a Hecke-module.

Fitting Ideals in Number Theory and Arithmetic

We describe classical and recent results concerning the structure of class groups of number fields as modules over the Galois group. When presenting more modern developments, we can only hint at the much broader context and the very powerful general techniques that are involved, but we endeavour to give complete statements or at least examples where feasible. The timeline goes from a classical result proved in 1890 (Stickelberger’s Theorem) to a recent (2020) breakthrough: the proof of the Brumer-Stark conjecture by Dasgupta and Kakde.

On p-class groups of relative cyclic p-extensions

We prove a general stability theorem for $p$-class groups of number fields along relative cyclic extensions of degree $p^2$, which is a generalization of a finite-extension version of Fukuda's theorem by Li, Ouyang, Xu and Zhang. As an application, we give an example of pseudo-null Iwasawa module over a certain $2$-adic Lie extension.

Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields K K (the trivial bound being O ϵ , n ( | D i s c ( K ) | 1 / 2 + ϵ ) O_{\epsilon ,n}(|\mathrm {Disc}(K)|^{1/2+\epsilon }) coming from the bound on the entire class group). This yields corresponding improvements to: (1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves, (2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves, (3) bounds on the sizes of 2-Selmer groups and ranks of Jacobians of hyperelliptic curves, and (4) bounds of Baily and Wong on the number of A 4 A_4 -quartic fields of bounded discriminant.

The weak form of Malle’s conjecture and solvable groups

For a fixed finite solvable group G and number field K, we prove an upper bound for the number of G-extensions L / K with restricted local behavior (at infinitely many places) and $$\mathrm{inv}(L/K)<X$$ for a general invariant “$$\mathrm{inv}$$”. When the invariant is given by the discriminant for a transitive embedding of a nilpotent group $$G\subset S_n$$, this realizes the upper bound given in the weak form of Malle’s conjecture. For other solvable groups, the upper bound depends on the size of torsion of the class group of number fields with fixed degree. In particular, the bounds we prove realize the upper bound given in the weak form of Malle’s conjecture for the transitive embedding of a solvable group $$G\subset S_n$$ if we assume that for each finite abelian group A the average size of class group torsion $$|\mathrm{Hom}(\mathrm{Cl}(L),A)|$$ is smaller than $$X^{\epsilon }$$ as L / K varies over certain families of extensions with $$\mathrm{inv}(L/K)<X$$.

Elliptic surfaces over ℙ1 and large class groups of number fields

Given a non-isotrivial elliptic curve over [Formula: see text] with large Mordell–Weil rank, we explain how one can build, for suitable small primes [Formula: see text], infinitely many fields of degree [Formula: see text] whose ideal class group has a large [Formula: see text]-torsion subgroup. As an example, we show the existence of infinitely many cubic fields whose ideal class group contains a subgroup isomorphic to [Formula: see text].

Arithmetic of double torus quotients and the distribution of periodic torus orbits

We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus. Using the new invariants we significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank $S$-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adelic torus and are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer. The proof combines geometric invariant theory, Galois actions, local arithmetic estimates and homogeneous dynamics.

Classes of order 4 in the strict class group of number fields and remarks on unramified quadratic extensions of unit type

Let K be a number field of degree n over $$\mathbb {Q}$$ . Then the 4-rank of the strict class group of K is at least $$\text {rank}_2 \, ( E_{K}^{+} / E_K^2) - \lfloor n /2 \rfloor $$ where $$E_K$$ and $$ E_{K}^{+} $$ denote the units and the totally positive units of K, respectively, and $$\text {rank}_2$$ is the dimension as an elementary abelian 2-group. In particular, the strict class group of a totally real field K with a totally positive system of fundamental units contains at least $$(n-1)/2$$ (n odd) or $$n/2 -1$$ (n even) independent elements of order 4. We also investigate when units in K are sums of two squares in K or are squares mod 4 in K.

Generating subgroups of ray class groups with small prime ideals

Explicit bounds are given on the norms of prime ideals generating arbitrary subgroups of ray class groups of number fields, assuming the Extended Riemann Hypothesis. These are the first explicit bounds for this problem, and are significantly better than previously known asymptotic bounds. Applied to the integers, they express that any subgroup of index $i$ of the multiplicative group of integers modulo $m$ is generated by prime numbers smaller than $16(i\log m)^2$, subject to the Riemann Hypothesis. Two particular consequences relate to mathematical cryptology. Applied to cyclotomic fields, they provide explicit bounds on generators of the relative class group, needed in some previous work on the shortest vector problem on ideal lattices. Applied to Jacobians of hyperelliptic curves, they allow one to derive bounds on the degrees of isogenies required to make their horizontal isogeny graphs connected. Such isogeny graphs are used to study the discrete logarithm problem on said Jacobians.

On Torsion Subgroups in Class Groups of Number Fields

On Torsion Subgroups in Class Groups of Number Fields

On formal groups and Tate cohomology in local fields

Let $L/K$ be a Galois extension of local fields of characteristic $0$ with Galois group $G$. If $\mathcal{F}$ is a formal group over the ring of integers in $K$, one can associate to $\mathcal F$ and each positive integer $n$ a $G$-module $F_L^n$ which as a set is the $n$-th power of the maximal ideal of the ring of integers in $L$. We give explicit necessary and sufficient conditions under which $F_L^n$ is a cohomologically trivial $G$-module. This has applications to elliptic curves over local fields and to ray class groups of number fields.

Reflection principles for class groups

We present several new examples of reflection principles which apply to both class groups of number fields and picard groups of curves over P1/Fp. This proves a conjecture of Lemmermeyer [3] about equality of 2-rank in subfields of A4, up to a constant not depending on the discriminant in the number field case, and exactly in the function field case. More generally we prove similar relations for subfields of a Galois extension with group G for the cases when G is S3, S4, A4, D2l and Z/lZ⋊Z/rZ. The method of proof uses sheaf cohomology on 1-dimensional schemes, which reduces to Galois module computations.

A class group heuristic based on the distribution of 1-eigenspaces in matrix groups

We propose a modification to the Cohen–Lenstra prediction for the distribution of class groups of number fields, which should also apply when the base field contains non-trivial roots of unity. The underlying heuristic derives from the distribution of 1-eigenspaces in certain generalized symplectic groups over finite rings. The motivation for that heuristic comes from the function field case. We also give explicit formulas for the new predictions in several important cases. These are in close accordance with known data.

Buchmann-Williams Authenticated Key Agreement Protocol With Pre-shared Password

Based on Buchmann-Williams key exchange protocol, a Buchmann-Williams Authenticated Key Agreement (BWAKA) protocol with pre-shared password is proposed. Its security relies on the Discrete Logarithm Problem over class groups of number fields. It provides identity authentication, perfect forward secrecy and key validation.

Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points

Shyr derived an analogue of Dirichlet’s class number formula for arithmetic tori. We use this formula to derive a Brauer-Siegel formula for tori, relating the discriminant of a torus to the product of its regulator and class number. We apply this formula to derive asymptotics and lower bounds for Galois orbits of CM points in the Siegel modular variety A g , 1 A_{g,1} . Specifically, we ask that the sizes of these orbits grow like a power of the discriminant of the underlying endomorphism algebra. We prove this unconditionally in the case g ≤ 6 g\leq 6 , and for all g g under the Generalized Riemann Hypothesis for CM fields. Along the way we derive a general transfer principle for torsion in ideal class groups of number fields.