Indivisibility of ray class groups of real quadratic fields

Abstract: Let ${\ell}$, p $\ge$ 5 be primes such that p | (${\ell}$ -1). Let $Δ$ > 0 be the fundamental discriminant of a real quadratic field in which ${\ell}$ splits. We denote by h - ${\ell}$ ($Δ$) the order of the minus part (for the Galois action) of the ray class group of Q( $\sqrt$ $Δ$) of modulus ${\ell}$. In this paper, we study the indivisibility of h - ${\ell}$ ($Δ$) by p, and prove that under the assumption that this set is non-empty. This lower bound is made unconditional if ${\ell}$ = 2p + 1, i.e. if p is a Sophie Germain prime. Our result can be viewed as being in the continuity of the results of Kohnen-Ono, Ono, Byeon, Beckwith etc. regarding the class numbers of quadratic fields, in the sense that we rely on techniques from the theory of half-integral weight modular forms. Significant difficulties however arise in our study, as we have to study Eisenstein congruences for cuspforms of weight 3 2 , and use a generalized Shimura correspondence of Baruch-Mao. Combined with the results of Lecouturier-Wang, our result has implications eg. for the 5-part of BSD for even quadratic twists of X 0 (11).

• The paper proves that, assuming one suitable example exists, at least order √X/log X real quadratic fields with discriminant below X have ray class minus parts not divisible by p.
• The authors translate ray class indivisibility into the nonvanishing of class-number and fundamental-unit expressions, then connect these values to central L-values through Eisenstein congruences and weight-3/2 modular forms.
• For Sophie Germain primes, the paper constructs explicit examples unconditionally and applies the method to show that many quadratic twists of X₀(11) satisfy the 5-part of the Birch–Swinnerton-Dyer conjecture.

This paper by Lecouturier and Maire establishes lower bounds for the number of real quadratic fields whose ray class groups of a fixed prime modulus $\ell$ have minus part not divisible by a prime $p$ dividing $\ell-1$. The main result states that, under the sole assumption that at least one such field exists, the count of fundamental discriminants $0<\Delta<X$ with $\ell$ split and $p\nmid h_\ell^-(\Delta)$ grows at least on the order of $\sqrt{X}/\log X$. The result extends the indivisibility program initiated by Kohnen–Ono, Ono, and Byeon from ordinary class groups to ray class groups, and its proof requires new Eisenstein congruences for weight-$\frac{3}{2}$ modular forms built on the generalized Shimura correspondence of Baruch–Mao.

Arithmetic framework and reformulation

The object of study is the ray class group $\Cl_{K,\ell}$ of the real quadratic field $K=\Q(\sqrt{\Delta})$ of modulus $p$0, where $p$1 is a prime splitting in $p$2 and $p$3 divides $p$4. The Galois group $p$5 acts on $p$6, permuting the two primes above $p$7, and $p$8 denotes the order of the $p$9-eigenspace.

A key structural input is the Gras–Munnier theorem governing tame cyclic degree-$\ell-1$0 extensions: such an extension ramified exactly at a prescribed set of tame primes exists if and only if a corresponding linear relation among Frobenius elements in the governing field $\ell-1$1 vanishes. From this, the authors derive that there is no $\ell-1$2-extension of $\ell-1$3 unramified outside a single prime $\ell-1$4 precisely when $\ell-1$5 and the fundamental unit $\ell-1$6 is not a $\ell-1$7th power modulo $\ell-1$8. They then package these conditions into the quantity

$\ell-1$9

where $0<\Delta<X$0 is a fixed surjection, and prove that $0<\Delta<X$1 holds if and only if $0<\Delta<X$2. This equivalence is central: it converts the ray class group condition into a statement about class numbers and units amenable to modular-form techniques. It also implies the useful characterization that $0<\Delta<X$3 exactly when $0<\Delta<X$4 and no fundamental unit is a $0<\Delta<X$5th power modulo a prime above $0<\Delta<X$6.

Existence via explicit construction: the Sophie Germain case

The main theorem is conditional on non-emptiness of $0<\Delta<X$7, where $0<\Delta<X$8 is an auxiliary squarefree integer coprime to $0<\Delta<X$9 with $\ell$0. This hypothesis is removed when $\ell$1: for Sophie Germain primes $\ell$2, the authors construct explicitly a discriminant $\ell$3 with $\ell$4 inert in $\ell$5, $\ell$6 split, and $\ell$7.

The construction starts from integers $\ell$8 with $\ell$9 and considers $p\nmid h_\ell^-(\Delta)$0, so that $p\nmid h_\ell^-(\Delta)$1 is a unit of norm $p\nmid h_\ell^-(\Delta)$2. Choosing $p\nmid h_\ell^-(\Delta)$3 via the Chinese remainder theorem so that $p\nmid h_\ell^-(\Delta)$4 is inert and $p\nmid h_\ell^-(\Delta)$5 (with $p\nmid h_\ell^-(\Delta)$6) is not a $p\nmid h_\ell^-(\Delta)$7th power modulo $p\nmid h_\ell^-(\Delta)$8 — possible since for $p\nmid h_\ell^-(\Delta)$9 the $\sqrt{X}/\log X$0th powers are exactly $\sqrt{X}/\log X$1 — yields a field satisfying all local conditions. To force $\sqrt{X}/\log X$2, the authors combine the trivial bound $\sqrt{X}/\log X$3 with sharper explicit estimates valid when $\sqrt{X}/\log X$4 is inert: $\sqrt{X}/\log X$5 for $\sqrt{X}/\log X$6, improving to $\sqrt{X}/\log X$7 when additionally $\sqrt{X}/\log X$8 ramifies and $\sqrt{X}/\log X$9. For small $\frac{3}{2}$0 (all $\frac{3}{2}$1 Sophie Germain primes below $\frac{3}{2}$2), existence is verified numerically with discriminants $\frac{3}{2}$3. The contradiction argument shows that assuming $\frac{3}{2}$4 forces $\frac{3}{2}$5, which is impossible.

Eisenstein congruences in weight 2

The first analytic step relates $\frac{3}{2}$6 to central critical $\frac{3}{2}$7-values. By Mazur's theory of the Eisenstein ideal, since $\frac{3}{2}$8 there exists a newform $\frac{3}{2}$9 congruent to the Eisenstein series $\Cl_{K,\ell}$0 modulo a maximal ideal above $\Cl_{K,\ell}$1. Mazur's isomorphism $\Cl_{K,\ell}$2, combined with Merel's description via the $\Cl_{K,\ell}$3-part of homology and the winding element, produces a surjective homomorphism

$\Cl_{K,\ell}$4

satisfying $\Cl_{K,\ell}$5 for $\Cl_{K,\ell}$6 with lower-right entry $\Cl_{K,\ell}$7. Notably, this map depends only on the choice of $\Cl_{K,\ell}$8, not on the particular Eisenstein-congruent newform $\Cl_{K,\ell}$9 — a point requiring care because $K=\Q(\sqrt{\Delta})$0 may not be unique when $K=\Q(\sqrt{\Delta})$1; the authors handle general coefficient orders $K=\Q(\sqrt{\Delta})$2 that need not be maximal.

Combining Popa's explicit Waldspurger formula, expressed through Heegner cycles attached to oriented optimal embeddings of level $K=\Q(\sqrt{\Delta})$3, with the congruence properties of $K=\Q(\sqrt{\Delta})$4, they obtain:

$K=\Q(\sqrt{\Delta})$5

with $K=\Q(\sqrt{\Delta})$6 and reduction $K=\Q(\sqrt{\Delta})$7 in $K=\Q(\sqrt{\Delta})$8. Thus the indivisibility of $K=\Q(\sqrt{\Delta})$9 is equivalent to the non-vanishing modulo $p$00 of a normalized twisted central $p$01-value.

Transfer to weight 3/2 via Baruch–Mao

The most technically involved step lifts this congruence to a genuine congruence between modular forms of weight $p$02 and level $p$03. Applying the generalized Shimura correspondence of Baruch–Mao yields a form $p$04, lying in the Kohnen space, whose coefficients satisfy $p$05 for $p$06.

Two obstructions arise. First, $p$07 is not an eigenvector for $p$08, since the Kohnen space is not stable under this operator. Second, the Hecke operator at $p$09 must be controlled. The authors resolve both by working adelically: decomposing $p$10 as a linear combination $p$11 of two full Hecke eigenforms, where $p$12 is a root of $p$13 congruent to $p$14 modulo $p$15, and

$p$16

so that $p$17 and $p$18 modulo $p$19. Explicit computations of local Whittaker functionals at $p$20, following Waldspurger's metaplectic Hecke operators, give precise ratios $p$21 depending on $p$22 modulo $p$23. Gluing over Galois conjugates of $p$24 produces a form $p$25 with coefficients in $p$26, and the main weight-$p$27 theorem establishes:

• $p$28;
• $p$29, where $p$30 is the classical theta series;
• for nonsquare $p$31 with associated discriminant $p$32, $p$33;
• non-vanishing of $p$34 forces non-vanishing of $p$35.

The congruence $p$36 is interpreted as an Eisenstein (indeed higher Eisenstein) congruence compatible with the Shimura lifting, since the formal lift of $p$37 is exactly $p$38. A subsequent twisting argument isolating coefficients with $p$39 produces the working form $p$40 over $p$41 of level $p$42.

Proof of the counting bound

The final step follows Ono and Byeon. Using a Bruinier–Ono identity for $p$43 (generalized to arbitrary Nebentypus and coefficients in $p$44), the authors show the support of $p$45 cannot be contained in finitely many square-class orbits: otherwise a prime $p$46 with prescribed Legendre symbols would force $p$47. Hence one can fix $p$48 in the support whose squarefree part has a prime factor exceeding $p$49.

For each prime $p$50 in a positive-density set $p$51 with suitable splitting behavior, a Sturm-bound argument on the two forms $p$52 and $p$53 shows some $p$54 satisfies $p$55. Each such index yields a discriminant $p$56 with $p$57 and $p$58. A combinatorial argument (among any three primes $p$59, at least two discriminants differ) controls multiplicities, giving

$p$60

The dependence on the non-emptiness assumption enters only through the single nonzero coefficient $p$61; the method cannot produce unconditional results without an explicit witness except in the Sophie Germain situation.

Heuristics and numerical evidence

Guided by Cohen–Lenstra philosophy, the authors conjecture that among discriminants with $p$62, the density of those with $p$63 equals $p$64, independently of $p$65. Their heuristic explanation runs as follows: everything takes place in the minus part; passing from imaginary to real quadratic fields corresponds to quotienting by a random element (moving from $p$66 to $p$67); and passing from the ray class group of modulus $p$68 back to the class group quotients by the cyclic ramification subgroup at $p$69, returning to $p$70. Numerics at $p$71 support this: the observed proportions $p$72 ($p$73), $p$74 ($p$75), and $p$76 ($p$77) track $p$78, $p$79, and $p$80 closely, with near-independence of $p$81 (e.g., $p$82 for $p$83, $p$84 for $p$85). The authors note that Bartel–Pagano's recent work on Cohen–Lenstra heuristics for ray class groups of quadratic fields with fixed rational modulus provides a natural framework in which their conjecture could be revisited.

Application to BSD for even twists of $p$86

Taking $p$87, $p$88, the curve $p$89 is the $p$90-Eisenstein quotient of $p$91. By results of Lecouturier–Wang, for $p$92 with $p$93 and $p$94, the condition $p$95 is equivalent to $p$96, and in that case the $p$97-part of BSD holds for the twist. Since $p$98, the character can be taken quadratic and $p$99 has integral coefficients (expressible via generalized theta series), allowing a direct construction of the relevant weight-$\ell-1$00 form over $\ell-1$01 with the twist by $\ell-1$02 built in. The witness $\ell-1$03 exceeds the required constant $\ell-1$04, yielding unconditionally:

$\ell-1$05

Combined with the heuristic density $\ell-1$06, this suggests that roughly three-quarters of even quadratic twists of $\ell-1$07 with $\ell-1$08, $\ell-1$09 should satisfy $\ell-1$10.

Limitations and open questions

Several restrictions qualify the results. The exponent $\ell-1$11 falls far short of the conjectured positive proportion $\ell-1$12, and closing this gap appears out of reach with current methods. The main theorem remains conditional on $\ell-1$13 for general pairs $\ell-1$14; the unconditional Sophie Germain argument relies on delicate explicit upper bounds for $\ell-1$15 requiring $\ell-1$16 inert and, in the even-discriminant case, $\ell-1$17 ramified, together with numerical verification for all $\ell-1$18. The sign in the congruence $\ell-1$19 is a priori allowed to depend on $\ell-1$20, and the authors explicitly raise the question of whether it can be normalized to $\ell-1$21. Finally, the compatibility of the present framework with the Bartel–Pagano heuristics for ray class groups is left unexplored.

Conclusion

The paper extends indivisibility results for class numbers of quadratic fields to the setting of ray class groups of real quadratic fields with prime modulus, proving a $\ell-1$22 lower bound conditional on a single example, and unconditionally for Sophie Germain primes. Methodologically, its contribution lies in establishing an Eisenstein congruence at weight $\ell-1$23 — $\ell-1$24 modulo the Eisenstein ideal — via a careful adelic analysis of the Baruch–Mao correspondence at the places $\ell-1$25 and $\ell-1$26, thereby connecting the arithmetic of tame $\ell-1$27-extensions, Waldspurger-type $\ell-1$28-value formulas, and Sturm-bound sieving. The application to the $\ell-1$29-part of BSD for a positive-density family of even twists of $\ell-1$30 demonstrates the concrete Diophantine payoff of the technique.