The Ellenberg–Jain–Venkatesh random-matrix model for λ-invariants of imaginary quadratic fields: proving the equidistribution conjecture and its refinements

Ellenberg, Jain and Venkatesh (2011) proposed a random-matrix model for the Iwasawa λ-invariants of imaginary quadratic fields: for a fixed odd prime $p$, among fields in which $p$ does not split, the proportion with λ-invariant $r$ is predicted to be $p^{-r}\prod_{t>r}(1-p^{-t})$ — the limiting law, as $n\to\infty$, of the number of eigenvalues in the open unit ball about 1 of a Haar-random element of $\mathrm{GL}_n(\mathbb{Z}_p)$, equivalently of the coset $\mathrm{GSp}_\alpha(2n,\mathbb{Z}_p)$ with $\alpha\not\equiv1\pmod p$. They proved the model-side limit, computed the degree-2 statistics of the model, and tested the conjecture by p-adic interpolation of classical L-values, observing a systematic deficit of large λ at finite discriminant. The task is to convert this heuristic into a rigorous, computationally accessible theory of the zero statistics of p-adic L-functions over families of quadratic characters: to prove or refute the arithmetic equidistribution conjecture, to explain the finite-range deficit, to justify the model's finer predictions (joint statistics of the Iwasawa polynomial, discriminant valuations of all degrees) in the number-field setting, to treat the excluded symmetry types, notably $\mathrm{GSp}_\alpha$ with $\alpha\equiv1\pmod p$, and to construct a genuinely p-adic algorithm matching the $O(d^{1/2+\varepsilon})$ cost of the classical-value route. No proof or disproof of the arithmetic conjecture is known to date.

Background

Iwasawa theory studies class groups in infinite towers of number fields. Fix a prime $p$ and a number field $K$. The cyclotomic $\mathbb{Z}_p$-extension $K_\infty/K$ is the infinite Galois extension with $\mathrm{Gal}(K_\infty/K)\cong\mathbb{Z}_p$ (the additive group of $p$-adic integers) contained in the field generated over $K$ by all $p$-power roots of unity. A foundational theorem of the subject states that the order of the $p$-primary part of the class group of the $n$-th finite layer $K_n$ grows like $p^{\lambda n+\mu p^n+\nu}$ for integers $\lambda,\mu,\nu\ge0$ independent of $n$; the integer $\lambda=\lambda_p(K)$, the Iwasawa λ-invariant, is the steady exponential rate at which $p$ divides class numbers up the tower and is the central object of the problem. For the fields studied here the $\mu$ term is known to vanish, so $\lambda$ carries the entire growth.

The analytic handle on $\lambda$ is the p-adic L-function. To an odd Dirichlet character $\chi$ — for the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-d})$, the quadratic character $\chi_d$ of conductor $d$ — one attaches the Kubota–Leopoldt p-adic L-function $L_p(s,\chi)$, a p-adic analytic function of $s\in\mathbb{Z}_p$ interpolating the classical special values $L(1-n,\chi)$ at nonpositive integers. There is a power series $f_\chi(T)\in\mathbb{Z}_p[[T]]$ such that $L_p(s,\chi)$ is obtained from $f_\chi$ by the change of variable $T=(1+p)^{-s}-1$ (sign conventions for $s$ vary between references). By the p-adic Weierstrass preparation theorem, $f_\chi$ factors uniquely as $p^{\mu}$ times a unit power series times a monic polynomial $P_\chi$ whose non-leading coefficients are all divisible by $p$ — a distinguished polynomial — and $\lambda(\chi)=\deg P_\chi$ equals the number of zeroes of $f_\chi$ in the open unit disc, equivalently the number of zeroes of the p-adic L-function, read in the multiplicative variable $w=1+T=(1+p)^{-s}$, in the open unit ball of $\overline{\mathbb{Q}}_p$ centered at $1$, counted with multiplicity. The Iwasawa main conjecture, a theorem in this setting, identifies $P_\chi$ with the characteristic polynomial of the associated Iwasawa module, so this $\lambda$ governs class-number growth in the cyclotomic tower and is related to, but typically exceeds, the $p$-rank of the class group of $K$ itself.

The Cohen–Lenstra heuristics are the standard probabilistic model for $p$-primary class groups of quadratic fields: they model the $p$-part as the cokernel of a random self-adjoint endomorphism, predicting for instance that the $p$-part of the class group of an imaginary quadratic field is trivial with probability $\prod_{i\ge1}(1-p^{-i})$ (about $0.560$ for $p=3$), and that a $p$-rank $r$ occurs with probability on the order of $p^{-r^2}$, so large ranks are rare.

Ellenberg–Jain–Venkatesh proposed modeling not just the class group but the full polynomial $P_\chi$, equivalently the zeroes of the p-adic L-function near $1$, by p-adic random matrix theory. Take $A$ Haar-random (uniform for the p-adic topology) in $\mathrm{GL}_n(\mathbb{Z}_p)$, or in the general symplectic group with multiplier $\alpha$, $\mathrm{GSp}_\alpha(2n,\mathbb{Z}_p)=\{g:g^tJg=\alpha J\}$ for a fixed symplectic form $J$ and fixed $\alpha\in\mathbb{Z}_p^{\times}$ with $\alpha\not\equiv1\pmod p$. The eigenvalues of $A$ congruent to $1$ modulo $p$ determine a distinguished polynomial $P_A$, and $\deg P_A$ counts the generalized eigenvalues of $A$ in the open unit ball about $1$. Their model theorem proves that as $n\to\infty$ the law of $\deg P_A$ converges: $\Pr[\deg P_A=r]\to p^{-r}\prod_{t>r}(1-p^{-t})$. This limit has a much heavier tail (of order $p^{-r}$) than the Cohen–Lenstra p-rank law (of order $p^{-r^2}$): λ-invariants are typically larger than class-group p-ranks, and the $r=0$ value $\prod_{i\ge1}(1-p^{-i})$ is exactly the Cohen–Lenstra constant for a trivial $p$-part. They conjectured the same law on the arithmetic side, and computed the distribution of the p-adic valuations of degree-2 discriminants of $P_A$ as a finer model prediction.

To test the conjecture they computed $\lambda(\chi_d)$ at scale: evaluate classical special values $L(1-n,\chi_d)$ (a transcendental computation of cost about $O(d^{1/2+\varepsilon})$ per value), p-adically interpolate at points of $p\mathbb{Z}_p$ with provable precision, recover the coefficients of $f_\chi$ by triangular inversion, and read off $P_\chi$ by Weierstrass preparation. For $p=3$, over the first $10^6$ admissible discriminants — those in which $p$ does not split, since a splitting $p$ produces an exceptional zero that shifts $\lambda$ — the observed frequency of $\lambda=0$ was about $0.590$ against a predicted $0.560$, with a corresponding deficit of large $\lambda$; the discrepancy decreases as the discriminant range grows but remains visible throughout.

The problem occupies, within Iwasawa theory, the position that random-matrix modeling of zero statistics occupies in the classical theory of L-functions, with two features making it distinct: the statistic is an ultrametric count of zeroes in a p-adic ball (not a spacing statistic in the complex plane), and the arithmetic side is built from generalized Bernoulli numbers obeying strong congruences, so the model's statistical independence is a genuine prediction about arithmetic rather than a formal consequence of the construction.

Problem Statement

The problem is the open core of the Ellenberg–Jain–Venkatesh program of modeling λ-invariants by p-adic random matrices, stated in the source as the principal questions the paper leaves open. All parts share one scientific objective — to convert the heuristic into a rigorous, computationally accessible theory of the zero statistics of p-adic L-functions over families of quadratic characters.

(1) Equidistribution conjecture. Prove or refute: for each fixed odd prime $p$ and each integer $r\ge0$, among imaginary quadratic fields $K$ in which $p$ does not split, ordered by the absolute value of the discriminant, the proportion with $\lambda_p(K)=r$ converges to $p^{-r}\prod_{t>r}(1-p^{-t})$, the limiting law of the number of generalized eigenvalues in the open unit ball about $1$ of a Haar-random element of $\mathrm{GL}_n(\mathbb{Z}_p)$ as $n\to\infty$.

(2) Finite-range deviations. Explain the observed systematic deficit of large λ at moderate discriminant: determine whether the convergence in (1) admits a secondary main term (expected, by analogy with class-group statistics where negative secondary main terms occur, to be negative and to bias toward small λ), and if so determine its order, sign, and shape; or otherwise account for the slow convergence.

(3) Refined statistics in the number-field setting. Justify or refute the model's predictions for finer invariants: the joint distribution of the coefficients of $P_\chi$ over the family of quadratic characters, and the distribution of the p-adic valuations of discriminants of $P_\chi$ in all degrees (the model side is known for degrees at most 2 from the original paper and in full generality from later work).

(4) Excluded symmetry types. Analyze the ensembles excluded from the model theorem — in particular $\mathrm{GSp}_\alpha(2n,\mathbb{Z}_p)$ with $\alpha\equiv1\pmod p$, where the eigenvalue-$1$ statistics degenerate — and determine the correct model and arithmetic prediction for these excluded ensembles.

(5) Genuinely p-adic computation. Construct an algorithm that computes the p-adic L-function (equivalently $P_\chi$ or $\lambda(\chi)$) intrinsically p-adically, with provable precision control, at asymptotic cost in the discriminant $d$ matching the $O(d^{1/2+\varepsilon})$ of the transcendental classical-value route that current implementations interpolate from.

References

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