An Open Image Theorem based on the Small Heisenberg Group

Geometry Talk 2: An Open Image Theorem based on the Small Heisenberg Group
Mike Fried, Emeritus Professor UC Irvine

Talk 1 was about alternating groups, Hurwitz spaces from 3-cycles, and lift invariant values describing the Hurwitz space components. Maybe that didn't strike you as modular curve-like topic. Even if you studied modular curves, you didn't hear about lift invariants.

When we have r = 4 conjugacy classes, C, defining a reduced Hurwitz space, then each component is an upper half-plane quotient. Further, by changing variables we find it ramifies only over the points 0, 1 and ∞ of the classical j-line, P¹_j.

Part A: Main Modular Tower Conjecture: I simplify by using the case r = 4. K denotes a number field.

I will show each of the 1st Talk spaces are level 0 of a sequence that resembles

(*) ^… → X₀(p^k⁺²) → X₀(p^k+¹) → ^… → X₀(p) → P¹_j (k ≥ 0 the level), a modular curve sequence.

For p a prime, a collection in G is p' if each element has order prime to p. The construction gives a sequence like (*) if G is p perfect, and C are p' classes whose elements generate G. With G = G₀,

(**) M(G, p, C): ^… → H(G_k₊₂, p, C) → H(G_k₊₁, p, C) → ^… → H(G, p, C) → P¹_j (if r > 4 over a configuration space J_r).

Our Main M(G, p, C) Conjecture: There are no K points on H(G_k₊₁, p, C) for k large.

A Modular Tower on M(G, p, C) is any (nonempty) projective sequence of components on (**). The lift-invariant of the 1st Talk is an aid to identify Modular Towers on M(G, p, C).

The M(G, p, C) conjecture shows us why the inverse Galois problem is so hard. It generalizes this fact:

Modular curves, outside their cusps, have no K points at high levels.

When r = 4, the main conjecture holds. We can see that by locating a level on each Modular Tower whose genus exceeds 1. Then, that case follows from Weil's theorem that the eigenvalues of the Frobenius acting on the 1st cohomology of a curve have a certain explicit absolute value. We use p-Frattini properties of Modular Tower cusps to see that genus grow.

Part B: Using Part A to Generalize Serre's Open Image Theorem:

I constructed all modular curves from two ''easy'' groups: D = Zx^sZ/2 (for dihedral), semidirect product of Z and Z/2 (order 2 group), and ²D = (Z)²x^sZ/2. What happens when we replace D and ²D with G = (Z)²x^sZ/3 and ²G = (Z)⁴x^sZ/3, p≠ 3?

We find here a lift invariant value appears in a Heisenberg group, instead of in a Spin group. Each nonzero value of the lift invariant gives one component.

We have a new phenomenon: There are several HM components (definition from Talk 1). Those always have trivial lift invariant. So, separating covers in these components requires a new idea.

An OIT starts by generalizing to Modular Towers a 1967 Serre result for modular curves. I explained Serre's result in the colloquium talk. I expand here to the conjectured property.

Our M(G, p, C) monodromy statement: Any Modular Tower has geometric monodromy (over P¹_j when r = 4) that is eventually p-Frattini, and for almost all p it is actually p-Frattini.

We base this talk on The Small Heisenberg Group, and l-adic representations from Hurwitz Spaces.

Generalizing beyond Modular curves explicit properties of an Open Image Theorem

Remainder of the Theta Function Footnotes

t13. Riemann developed ϑ functions to generalize Abel's Theorem – constructing analytic functions on an elliptic curve – to an arbitrary compact Riemann surface X. Attempts to make the ϑ s canonical, lead to two types: even and odd, referring to what happens when you change ϑ(x) to ϑ(-x). Further, we get several of each type if X has genus > 1.

t14. HM components of Hurwitz spaces are defined by braid orbits containing HM representatives. That [Fr95, Thm. 3.28] shows respect that action of G_Q respects HM components (permutes them). This implies that each H_±^*,rd → J_r has definition field Q. This is a special corollary of a general result that plays a much bigger role in this talk. Name comes from the natural combination of [Mu72] and [H84].

t15. The Jacobian Jac(Wp) is linear equivalence classes of divisors on Wp of degree 0.

t15a. An important lemma notes that reduced equivalence respects the linear equivalence class of dφ/2. That is: The divisors (dφ)/2 and d((aφ + b)/(cφ +d))/2 differ by the divisor of a function [Fr10, Lem. 6.1].

t15b. Sometimes we can assure even ϑ s produce non-zero ϑ-nulls (evaluate ϑ(x) at 0, all along the Hurwitz space locus; ϑ-nulls formed from odd theta are always zero). These Hurwitz-Torelli functions are like automorphic functions on the parameter space of the covers.

Start of footnotes for the Small Heisenberg group test for generalizing the OIT

1. How, can I show in one talk there is a true generalization of modular curve considerations? I stay as close as possible to modular curves – for comparison – by taking r = 4 while demonstrating this is a generalization and a necessary one in the evolution of the OIT. Recall: When r = 4, reduced Hurwitz space components are upper half-plane quotients; ramifying only over 0, 1, ∞ on P¹_j.

2. D_p^k_⁺¹ is p-perfect for p odd. Any simple (non-cyclic) group G is p-perfect for p | |G|.

3. The name for the projective limit of the sequence G₀ ← G₁← G₂ ← … of group coves on the slide is the universal abelianized p-Frattini cover of G. Notions:

p-Frattini: Frattini cover, with kernel a p-group.
Eventually p-Frattini (p-Frattini from some point on).

4. Commonly, for simple groups, there is only one (nonzero) value v(G,p) ( = v(G,p)_max) that works. For G = A₅, this is the case for p = 2 and 3, where v(G,p)_max is (resp. 5 and 4). For p = 5, the possible values of v(G,p) are 3 and 6.^{[Fr95,
Part B]}. For G = A₄, and p = 2, v(G,p)_max = 5, but there is also v(G,p) = 2.

5. Schur-Zassenhaus implies p' conjugacy classes lift uniquely to same order classes in all the G_k s. Ditto for H ≤ G that is p'. So, if H (is p') defines the permutation representation T_H at level 0, then it canonically defines a coset rep. at every level.

6. The cases abs (resp. inn) where G = D_p and C = C_2⁴ are X₀(p^k⁺¹) (resp. X₁(p^k⁺¹)).

The preliminary step is group theoretic: Assure the levels are non-empty. Then, the spaces have (complex) dimension r - 3. Example: For n > 5, A_n, C = C_3^r. Exact condition to conclude nonempty is that r ≥ n - 1.

7. Each of H₊(A_n, C_3^r)ⁱⁿⁿ and H_–(A_n, C_3^r)ⁱⁿⁿ has definition field Q. A dense subset of Q points on H_±(A_n, C_3^r)^abs give (A_n , S_n, C_3^r) geometric/arithmetic monodromy realizations.

8. Once a MT level has genus > 1, Faltings' Theorem implies it has but finitely many points. That gives a projective system of points on the levels. Then, that case follows from Weil's Theorem: Action of the Frobenius on the 1st Cohomology of a curve. The genus conclusion results from p-Frattini properties of Modular Tower cusps.[Fr06, Fratt. Princ. 3.1]

9. Denote by H(x_p^•) the projective sequence of the groups H(x_p,k+1). The largest H(x_p^•) can be is the arithmetic monodromy of the cover over K. Serre's result was stated without the <± I> quotient. He didn't use moduli properties of spaces. Choosing from ±x with x a p-division point on an elliptic curve translates to Serre's result.

10. Rank t: The rank of (Z)^t with H acting on it.

Explain Universal Hilbert Subset S: Dense set of j' with the decomposition fiber over j' the whole arithmetic monodromy for all but finitely many covers over K. Since H(x_p^•) is always a subgroup of the arithmetic monodromy of the projective sequence of moduli space covers, once it maps onto the k₀ level, the p-Frattini property implies it is the whole arithmetic monodromy.

11. It is all about the geometric monodromy, with the conclusions on this page and the previous applying once the M(G,p,C) monodromy statement holds. A basic lemma that says you cannot even get a weak OIT without the monodromy statement holding. I suppress the 2nd coordinate p if the notation for G is G_p.

12. Remind of the Basic Hurwitz space theorem: A Hurwitz space (with its extra structure) is defined over Q if and only if C is Q-rational (generalizes to any number field).^{[FrV91, Thm. 1]}

13. Two things to prove:

That α extends to ℋ(Z/p^k⁺¹).
That the universal central extension of (Z/p^k⁺¹)² x^s Z/3 is ℋ(Z/p^k⁺¹).

14. _w,3g=(α₀ ,_w_₂α₀, _w_₃α₀): replace α₀ by (α₀^-1, α₀^-1), and braid to the form _vg. For the actual value of the lift invariant we get f(x, y)=(x²-xy+y²)/3 = f(x, y)^α. where w₃=(x, y). This allows computing the lift invariant in related situations.

15. Further clarifying points show the value of concentrating on elements in T_p,±±,1-deg.

Those Nielsen class elements with trivial lift invariant consist precisely of HM and shift of HM reps.

There are K_p of each type – _v,1,3g and _v,2,4g – of double identity elements.

Works because if g has HM-depth 0, and p ≣ 2 mod 3, there is a unique u₃ mod p^k⁺¹ with (g)q₃^2u₃ 1-degenerate. If p ≣ 1 mod 3. Adjust for eigenvectors at level k = 0. The (g)q₃^2u₃ (resp. (g)q₂^2u₂) orbit contains a (unique) 1-degenerate element if and only if v₂ (resp. v₄) is not an α eigenvector.