The Generalized Sato-Tate Conjecture for Higher-Dimensional Abelian Varieties

The Generalized Sato-Tate Conjecture for Higher-Dimensional Abelian Varieties

Definitions and mathematical statement

For an abelian variety $A/K$ of dimension $g$ and a good finite place $v$, Frobenius on its first cohomology has eigenvalues of absolute value $(Nv)^{1/2}$. Divide them by this factor and use the resulting conjugacy class $c_v$ in the source's compact Sato--Tate group $\operatorname{ST}(A)$. The conjecture is \[ \frac1{\pi_K(x)}\sum_{Nv\le x,\ v\text{ good}}F(c_v) \longrightarrow\int_{\operatorname{ST}(A)}F(g)\,dg \] for every continuous class function $F$, with normalized Haar measure and $\pi_K(x)$ the number of prime ideals of norm at most $x$. The full group, including its components, and its Frobenius-class construction use the source convention.

Formulation and concept references: [S1].

Short English statement

Are normalized Frobenius symmetries of every abelian variety distributed according to Haar measure on its Sato--Tate group?

Research attempt

Editorial note on the formulation reference.

The DOI in [S1] resolves to Gan--Takeda, The local Langlands conjecture for $\mathrm{GSp}(4)$, not the displayed authors or a definition of the Sato--Tate group for arbitrary abelian varieties. The original reference and statement are retained. In this attempt fix $\ell$ and an embedding $\overline{\mathbb{Q}}_\ell\hookrightarrow{\mathbb{C}}$ and use Sutherland's compact-group and normalized-Frobenius construction, Sections 3.2--3.3, Conjecture 3.6. This is a declared convention where [S1] supplies none; no independence of all choices is assumed. [E1][E2]

Current frontier.

Sato--Tate for non-CM Hilbert modular forms is an established specialized result, not the arbitrary higher-dimensional statement. [E3] Classification of Sato--Tate groups of abelian threefolds, examples and moment computations likewise do not establish equidistribution for every variety with one of those groups. [E4] The analytic route requires control of the nontrivial representation $L$-functions, not just knowledge of the algebraic monodromy group. [E2]

Attack route.

Combine a tensor description of the compact group with prime averages. Possible bridges are automorphy and boundary nonvanishing for all irreducible $L$-functions, direct cancellation of projected tensor traces, or finite moment reconstruction. We prove the exact sufficiency of the second bridge and test the third. The deductions do not claim a new arithmetic cancellation theorem.

Attempt: projected tensor moments suffice.

Let $G\subset U(V)$ be compact with faithful $V$, and put \[ T_{a,b}=V^{\otimes a}\otimes\overline V^{\otimes b},\qquad \Pi_{a,b}=\int_G T_{a,b}(g)\,dg. \] This finite-dimensional integral is the orthogonal projection onto invariant vectors. For every $G$-equivariant orthogonal projection $P$ on $T_{a,b}$, \[ F_P(g)=\operatorname{tr}(P T_{a,b}(g)) \] is a continuous class function with Haar mean $\operatorname{tr}(P\Pi_{a,b})$. We claim that the prime-average condition, for every fixed $a,b,P$, \begin{equation}\label{wsref-0-33-30-30-30-36-37-34-35-3a-6d-6f-6d-65-6e-74-73} \frac1{\pi_K(x)}\sum_{Nv\le x}F_P(c_v) \longrightarrow\operatorname{tr}(P\Pi_{a,b}) \end{equation} implies the target equidistribution for this group. Finitely many omitted bad primes have no effect on the normalized averages.

Matrix coefficients of $V$ and $\overline V$ generate a self-adjoint algebra separating elements of $G$, hence are uniformly dense in $C(G)$ by Stone--Weierstrass. Products of coefficients are coefficients of the tensors $T_{a,b}$. If an irreducible representation $W$ occurred in none of these, Schur orthogonality would make its nonzero coefficients orthogonal to a dense subspace of $L^2(G)$, a contradiction. Thus some tensor contains $W$. Projection onto one invariant copy of $W$ commutes with $G$ and its $F_P$ is $\chi_W$. Condition (30006745:moments) gives average zero for each nontrivial irreducible character and one for the trivial character.

To complete the limiting step, conjugation-average a uniformly approximating coefficient polynomial. Averaging contracts the sup norm; Schur's lemma makes the conjugation average of an irreducible matrix coefficient a scalar multiple of its character. Finite character combinations are consequently dense in the continuous class functions. Approximate a test within $\epsilon$, use the finitely many character limits, then send $\epsilon\downarrow0$. No infinite series and prime limit have been interchanged. This proves the claim.

Why the projections carry essential information.

Taking only $P=I$ measures powers of $\operatorname{tr}V$ and its conjugate. Trace need not separate conjugacy classes. For $G=SU(2)\times SU(2)$ and $V=V_1\oplus V_2$, swapping two different component traces preserves the total trace but changes the $G$-conjugacy class. Even all characteristic-polynomial coefficients distinguish only ambient unitary conjugacy, which can be coarser than conjugacy in a subgroup. The individual equivariant projections in (30006745:moments) avoid that loss. They are not claimed to be easy arithmetic observables.

The group also matters for numerical moments. For diagonal $SU(2)$ on $V_1\oplus V_1$, the trace is $2t$ and its second Haar moment is $4$. For the product group on $V_1\oplus V_2$ it is $t_1+t_2$ and the second moment is $2$. Indeed $\int t=0$ and $\int t^2=1$, by the absence of invariants in $V_1$ and the one-dimensional invariant subspace of $V_1^{\otimes2}$. Replacing the trace law of $E^2$ by two independent elliptic trace laws would thus give the wrong moment even if each marginal were known perfectly.

Stress test: every fixed truncation can be fooled.

On $U(1)$, for $D\ge0$ and $0<\epsilon<1$, the probability measure \[ \frac{1+\epsilon\cos((D+1)\theta)}{2\pi}\,d\theta \] is non-Haar and positive, but agrees with Haar on $z^a\overline z^b$ for all $a+b\le D$. Orthogonality of integer exponentials proves the identity exactly. Thus a finite moment test is insufficient even without numerical error. This constructs probability measures, not a counterexample Frobenius sequence.

Passing to a field where the group becomes connected does not by itself prove all averages on the original disconnected group. The component-field construction is described in Sutherland, Theorem 3.12. [E2] The criterion above retains representations of the whole group, including its finite quotient. Finally, tensor constructions can have repeated Hodge--Tate weights. Regularity hypotheses in potential automorphy theorems must be checked, not assumed for every tensor factor. No required continuation or nonvanishing theorem for all those $L$-functions is proved here.

Outcome.

Outcome: Conditional reduction. All projected tensor prime averages in (30006745:moments) imply the full class equidistribution target, with the declared group convention. The proof retains component information and handles approximation uniformly. Finite trace moments, independent-product heuristics and connected-component arguments do not supply the hypothesis. The missing assertion remains arithmetic cancellation for all the required projected traces.

Sources

• S1 T. Barnet-Lamb, D. Geraghty, M. Harris, R. Taylor, Ann. of Math. 173$3$:1841-1869, 2011. https://doi.org/10.4007/annals.2011.173.3.12. Catalogue formulation source.
• E1 W. T. Gan and S. Takeda, The local Langlands conjecture for $\mathrm{GSp}(4)$, Annals of Mathematics 173 $2011$, 1841--1882; metadata for the DOI in [S1]. https://annals.math.princeton.edu/2011/173-3/p12.
• E2 A. V. Sutherland, Sato--Tate distributions, Contemporary Mathematics 740 $2019$, 197--248, Sections 2, 3.2--3.3, Conjecture 3.6, Theorem 3.12; arXiv:1604.01256. https://math.mit.edu/~drew/conm14904.pdf.
• E3 T. Barnet-Lamb, T. Gee and D. Geraghty, The Sato--Tate conjecture for Hilbert modular forms, JAMS 24 $2011$, 411--469; arXiv:0912.1054, main theorem. https://arxiv.org/abs/0912.1054.
• E4 F. Fité, K. S. Kedlaya and A. V. Sutherland, Sato--Tate groups of abelian threefolds, arXiv:2106.13759, abstract and classification statement. https://arxiv.org/abs/2106.13759.