Gemini Deep Think Reveals Eigenweights for Arithmetic Hirzebruch Proportionality in -Functions

Scientists have long sought to fully understand the arithmetic volumes of moduli stacks of shtukas, a problem central to arithmetic geometry. Tony Feng from Google DeepMind, alongside Tony Feng from Google, and collaborators, now provide a crucial step towards resolving this, determining the elusive “eigenweights” that underpin the (Higher) Arithmetic Hirzebruch Proportionality Principle established by Feng, Yun, Zhang. This research, documented using a custom system built upon Gemini Deep Think, significantly advances the field by connecting these eigenweights to the representation theory of symmetric groups and calculating them for all classical groups, offering a general solution previously unavailable and opening new avenues for exploring arithmetic volumes.

The team achieved this by establishing a connection between the eigenweights and the representation theory of symmetric groups, allowing for a general formula applicable to all classical groups.

This innovative approach bypasses the need for case-by-case calculations, offering a unified framework for understanding these crucial constants. The research establishes a clear definition of eigenweights within the context of the augmentation filtration on the invariant ring of the Weyl group, providing a rigorous mathematical foundation for their computation.
Experiments show that for the group GLn, the eigenweights associated with power sums can be determined using a closed-form expression, extending previous results. Specifically, for n ≥ 2 and a minuscule coweight of (1, 0n−1), the eigenweight is consistently −1 for all powers from 1 to n. Furthermore, for n ≥ 3 and μ = (12, 0n−2), a more complex formula yields the eigenweights, demonstrating the method’s ability to handle more intricate cases.

This breakthrough not only advances theoretical understanding but also has implications for computing arithmetic volumes of Shimura varieties and extending the Higher Siegel, Weil formula to singular terms. The work opens avenues for further research in Arakelov geometry and the study of automorphic vector bundles, offering a uniform explanation for constants appearing in related formulas.

By automating the complex calculations with an AI agent, the researchers demonstrate a powerful new approach to mathematical discovery, paving the way for future explorations in arithmetic geometry and related fields. The study meticulously defines the mathematical framework, beginning with a smooth, projective curve X of genus g over a finite field Fq, and a semisimple reductive group G of rank n over X.

Researchers constructed a multivariable L-function LX,G(s1, . . . , sn) associated with G, which recovers the L-function of the motive of G when s1 = . . . = sn = s. They then defined Shtμr G as the moduli stack of G-shtukas on X with r legs of type μ, and introduced a cohomology class η ∈H2N(BPμ), where N = dim G/Pμ + 1.

Experiments involved evaluating η on tautological “Hodge” bundles to obtain characteristic classes, and summing these classes to form a top cohomology class on Shtμr G. The team demonstrated that vol(Shtμr G, r Y i=1 ev∗ i η) = #π0(BunG) · q(g−1) dim G · dr s1=.=sn=0LX,G(s1, . . . , sn), where d is the differential operator d = (−log q)−1 n X i=1 εi(η, μ) ∂ ∂si.

The weighting constants εi(η, μ) represent the eigenweights, calculated as eigenvalues of a “local” operator ∇ η μ. To define these eigenweights, scientists established a graded ring R:= H∗(BT) = SymQ(X∗(T)Q), where X∗(T) is the character group of a split maximal torus T ⊂G. They then defined V as the first associated graded for the augmentation filtration on the invariant ring RW, relating it to Gross’s “motive of G”.

The operator ∇ η μ on V was defined as ∇ η μ(f) = Z G/Pμ η · ∂μ(f), with eigenweights corresponding to its eigenvalues. For a Casimir element Ω∈Sym2(X∗(T)Q)W, the team set η:= t1+dim G/Pμ μ, ensuring the preservation of grading during eigenweight calculation. Calculations for groups of rank n utilized the Casimir element Ω:= 1/2 Σi x2i, and the team derived closed-form descriptions of eigenweights for type A groups, specifically for minuscule coweights like (1m, 0n−m).

Eigenweight calculations for minuscule coweights in classical groups are notoriously difficult

Researchers have determined eigenweights for minuscule coweights, resolving a longstanding problem in the arithmetic of moduli stacks of shtukas. For $n\ge 2$ and the coweight μ=(1,0n−1)\mu = (1,0^{n-1})μ=(1,0n−1), it is shown that the eigenweights satisfy εk(Ω,μ)=(−1)n−1\varepsilon_k(\Omega,\mu) = (-1)^{n-1}εk​(Ω,μ)=(−1)n−1 for all $k=1,\dots ,n$. For $n\ge 3$ and μ=(12,0n−2)\mu = (1^2,0^{n-2})μ=(12,0n−2), explicit formulas for the eigenweights are derived for all $k$, extending earlier results. More generally, for coweights of the form μ=(1m,0n−m)\mu = (1^m,0^{n-m})μ=(1m,0n−m) with $m\ge 3$, the eigenweights are computed in closed form using characters of irreducible representations of the symmetric group, yielding a uniform formula that depends on $m$, $n$, and the index $k$.

The study further establishes explicit eigenweight formulas for all classical groups. In Type B, corresponding to $\mathrm{SO}(2n+1)$, the eigenweights associated with the fundamental coweights are constant and equal to $-4$. In Type C, for $\mathrm{PSp}(2n)$, the eigenweights for the spin coweight are expressed through alternating sums of symmetric-group characters and depend explicitly on the rank $n$. In Type D, for $\mathrm{PSO}(2n)$, the standard coweight yields eigenweights equal to $4$ for $k=1,\dots ,n-1$, with the Pfaffian weight equal to $2$, while the spinor coweights admit distinct formulas depending on whether $n$ is even or odd.

These results are obtained using methods from algebraic combinatorics and the representation theory of symmetric groups, linking eigenweights to partitions arising from skew Young diagrams. The Hook Length Formula plays a central role in evaluating the relevant character expressions and explains how differences between specific skew partitions determine the final values of the eigenweights. While the analysis focuses on classical groups, the framework provides a unified and systematic approach to eigenweight calculations, extending and strengthening previous results in the field.