On p-Adic Artin L-Functions II

We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

In this article, we study the growth of (fine) Selmer groups of elliptic curves in certain infinite Galois extensions where the Galois group G is a uniform, pro-p, p-adic Lie group. By comparing the growth of (fine) Selmer groups with that of class groups, we show that it is possible for the $$\mu $$ -invariant of the (fine) Selmer group to become arbitrarily large in a certain class of nilpotent, uniform, pro-p Lie extension. We also study the growth of fine Selmer groups in false Tate curve extensions.

Let f and g be two cuspidal modular forms and let ${\mathcal {F}}$ be a Coleman family passing through f, defined over an open affinoid subdomain V of weight space $\mathcal {W}$ . Using ideas of Pottharst, under certain hypotheses on f and $g,$ we construct a coherent sheaf over $V \times \mathcal {W}$ that interpolates the Bloch–Kato Selmer group of the Rankin–Selberg convolution of two modular forms in the critical range (i.e, the range where the p-adic L-function $L_p$ interpolates critical values of the global L-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$ .

Plectic points were introduced by Fornea and Gehrmann [Adv. Math. 414 (2023)] as certain tensor products of local points on elliptic curves over arbitrary number fields F F . They are constructed in rank r ≤ [ F : Q ] r\leq [F:\mathbb {Q}] -situations, and they conjecturally come from p p -adic regulators of basis of the Mordell-Weil group defined over dihedral extensions of F F . In this article we define anticyclotomic p p -adic L-functions with variables of type weight and level attached to a family of overconvergent modular symbols defined over totally real fields F F and a quadratic extension K / F K/F . Their restriction to the weight space provides Hida-Rankin p p -adic L-functions. If such a family passes through an overconvergent modular symbol attached to a modular elliptic curve E E , we obtain a p p -adic Gross-Zagier formula that computes higher derivatives of such Hida-Rankin p p -adic L-functions in terms of plectic points. This result generalizes that of Bertolini and Darmon [Ann. of Math. (2) 170 (2009), pp. 343–370], which has been key to the recent approach towards the algebraicity of Darmon points.

A characterization of number fields by p-closed extensions

Let X be a subgroup of a group Y. The \\emphinterval (X, Y) is the set of subgroups of Y that contain X including X and Y. By local class field theory the interval (N K/k K*, k*) contains a finite number of norm groups for any finite extension K of a 𝔭-adic number field k. In the present work we investigate the number of norm groups in the interval (N K/k K*, k*) for a given finite extension K/k of algebraic number fields. We prove that if K/k is an extension of prime degree, or of degree n such that the normal closure of K over k has the Galois group isomorphic to A n or S n , then the interval (N K/k K*, k*) contains only the obvious two norm groups. Also, the interval (N K/k K*, k*) contains a finite number of norm groups for any Galois extension of degree 4, and there are extensions with Galois groups order 8 for which the corresponding intervals contain a finite number of norm groups. The main theorem in our earlier work states that the interval (N K/k K*, k*) contains infinitely many norm groups for any Galois extension of even degree that is not a 2-extension, the so-called 2𝔫-\\emphextensions. In the present work we generalize the main theorem to non-Galois 2𝔫-extensions K/k, and determine some subintervals of (N K/k K*, k*) that contain infinitely many norm groups. We then use this theorem to prove that the interval (N K/k K*, k*) contains infinitely many norm groups for any Galois 2-extension K/k with the Galois group that either contains an element of order 8 or contains the quaternion group Q8 of order 8, or Q8 is a homomorphic image of the Galois group.

We prove that the number of unimodular integral n × n matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group Sn is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases. More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field F . Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes developed by Jouve, Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice point counting result, and a non-concentration phenomenon for lattice points in algebraic subvarieties of the group variety, both established previously by the authors. The results answer a question posed by Rivin and by Jouve, Kowalski and Zywina, who have considered Galois groups of random products of elements in algebraic groups.

Let N ( d , G , X ) N(d, G, X) be the number of degree d d number fields K K with Galois group G G and whose discriminant D K D_K satisfies | D K | ≤ X |D_K| \le X . Under standard conjectures in diophantine geometry, we show that N ( 4 , A 4 , X ) ≪ ϵ X 2 / 3 + ϵ N(4, A_4, X) \ll _\epsilon X^{2/3+\epsilon } , and that there are ≪ ϵ N 3 + ϵ \ll _\epsilon N^{3+\epsilon } monic, quartic polynomials with integral coefficients of height ≤ N \le N whose Galois groups are smaller than S 4 S_4 , confirming a question of Gallagher. Unconditionally we have N ( 4 , A 4 , X ) ≪ ϵ X 5 / 6 + ϵ N(4, A_4, X) \ll _\epsilon X^{5/6 + \epsilon } , and that the 2 2 -class groups of almost all Abelian cubic fields k k have size ≪ ϵ D k 1 / 3 + ϵ \ll _\epsilon D_k^{1/3+\epsilon } . The proofs depend on counting integral points on elliptic fibrations.

(English) The goal of this thesis has been to represent the p-adic valuation of the special values of p-adic L-functions via Euler Characteristics using different cohomology theories in various settings. More specifically, homotopy theoretic invariants such as the K(1)-local K-theory spectrum have been used to describe these values in the setting of commutative and noncommutative totally real fields and also in some cases for imaginary quadratic fields. The most successful case has been in the commutative and totally real case, where the padic valuation of the Deligne-Ribet p-adic L-function can now be understood using étale cohomology, K(1)-local K-theory, and also as the fiber of the (K(1)-local) cyclotomic trace map, therefore extending the work of Hesselholt, which only covered F = Q. In addition to this, the foundation has been layed to extend these results in other directions, such as in the case of function fields, or more precisely, to the case of Abelian varieties with semistable reduction over function fields. (Català) L'objectiu d'aquesta tesi ha estat representar la valoració p-àdica dels valors especials de funcions-L p-àdiques mitjançant característiques d'Euler utilitzant diferents teories de cohomologia en diversos contextos. Més específicament, s'han utilitzat invariants homotòpics com l'espectre K(1)-local de K-teoria per descriure aquests valors en el context de cossos reals totalment commutatius i no-commutatius, i també en alguns casos per a cossos quadràtics imaginaris. El cas més exitós ha estat el commutatiu i totalment real, on ara es pot entendre la valoració p-àdica de la funció-L p-àdica de Deligne-Ribet utilitzant cohomologia ètale, K-teoria K(1)-local, i la fibra del mapa de traça ciclotòmica (K(1)-local), estenent així el treball de Hesselholt, que només cobreix el cas F = Q. Finalment, s'ha establert la base per estendre aquests resultats en altres direccions, com ara en el cas de cossos de funcions, o més concretament, en el cas de varietats abelianes amb reducció semiestable sobre cossos de funcions.

Let K be a number field of degree n over ℚ and let d, h, and R be the absolute values of the discriminant, class number, and regulator of K, respectively. It is known that if K contains no quadratic subfield, then $$ h\;R\gg \frac{d^{1/2}}{ \log d}, $$ where the implied constant depends only on n. In Theorem 1, this lower estimate is improved for pure cubic fields. Consider the family $$ {\mathcal{K}}_n $$ , where K ∈ $$ {\mathcal{K}}_n $$ if K is a totally real number field of degree n whose normal closure has the symmetric group S n as its Galois group. In Theorem 2, it is proved that for a fixed n ≥ 2, there are infinitely many K ∈ $$ {\mathcal{K}}_n $$ with $$ h\gg {d}^{1/2}{\left( \log \log d\right)}^{n-1}/{\left( \log d\right)}^n, $$ where the implied constant depends only on n. This somewhat improves the analogous result h ≫ d 1/2/(log d) n of W. Duke [MR 1966783 (2004g:11103)].

Are number fields determined by Artin L-functions?

Given an abelian algebraic group A over a global field F, α ∈ A(F), and a prime ℓ, the set of all preimages of α under some iterate of [ℓ] generates an extension of F that contains all ℓ-power torsion points as well as a Kummer-type extension. We analyze the Galois group of this extension, and for several classes of A we give a simple characterization of when the Galois group is as large as possible up to constraints imposed by the endomorphism ring or the Weil pairing. This Galois group encodes information about the density of primes p in the ring of integers of F such that the order of (α mod p) is prime to ℓ. We compute this density in the general case for several classes of A, including elliptic curves and one-dimensional tori. For example, if F is a number field, A/F is an elliptic curve with surjective 2-adic representation and α ∈ A(F) with α ∉ 2A(F(A[4])), then the density of p with (α mod p) having odd order is 11/21.

A finite transitive permutation group G is called a Frobenius group if every element of G other than 1 leaves at most one letter fixed, and some element of G other than 1 leaves some letter fixed. It is proved in [20] (and sketched below) that if k is a number field such that SL(2, 5) and one other nonsolvable group Ŝ5 of order 240 are realizable as Galois groups over k, then every Frobenius group is realizable over k. It was also proved in [20] that there exists a quadratic (imaginary) field over which these two groups are realizable. In this paper we prove that they are realizable over the rationals Q, hence we ObtainTHEOREM 1. Every Frobenius group is realizable as the Galois group of an extension of the rational numbersQ.

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = 1, 2), S4 × Cm (m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers of an imaginary quadratic field contained in K. Suppose that the -rank over a Galois extension F is 1 or 2. If ≠ and and h (class number of ) is odd, we show that E acquires positive -rank over a cyclic extension of K or over a field whose Galois group is one of SL2(/3), an extension of SL2(/3) by /2, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L-functions.

The equivariant main conjecture of Iwasawa theory is shown to hold for a Galois extension K/k of totally real number fields with Galois group an l -adic pro- l Lie group of dimension 1 containing an abelian subgroup of index l , provided that Iwasawa's \mu -invariant \mu(K/k) vanishes.