Can Golod-Shafarevich Towers of Arithmetic Surfaces Over the Integers Be Constructed?

A tower whose section splits completely at every level

The paper's central object is not a single surface but an infinite tower. Theorem 1.1 produces a regular projective arithmetic surface S over Spec Z, a section s: Spec Z to S, and a chain of covers S_{n+1} to S_n down to S_0 = S. Each S_n to S is a connected finite etale Galois cover of degree 2^n, and each transition S_{n+1} to S_n has degree two. Every S_n is again a regular projective arithmetic surface with smooth geometrically connected generic fibre.

The section behaviour is the decisive feature. The fibre product of S_n with Spec Z over s is a disjoint union of 2^n copies of Spec Z, so s splits completely at every level. Because the generic fibre of S is geometrically connected, the image of the etale fundamental group of the generic fibre inside the etale fundamental group of S is infinite; the paper states that this image equals the whole etale fundamental group of S.

This is exactly the configuration Bost and Charles asked about in 2022, after proving finiteness results for fundamental groups of arithmetic surfaces. They allowed nonproper surfaces; the examples here are projective. The contrast with earlier finiteness results is sharp: for the connected Neron model of X(N) studied by Calegari, Dimitrov and Tang, the image of the generic-fibre fundamental group in the integral one is trivial, whereas here it is infinite.

The section, not the surface, carries the arithmetic content

A reader can easily conflate two different statements: that a surface has infinite etale fundamental group, and that a rational point splits completely in every level of a tower. The paper needs both, and they are logically distinct. The section s is what forces the tower to remain arithmetically nontrivial over Spec Z rather than collapsing after base change to Q.

The relevant group is defined as the image of the etale fundamental group of the generic fibre S_Q inside the etale fundamental group of S. For the surfaces constructed here this image is infinite. The paper notes that for the connected Neron model of X(N), the corresponding image is trivial, so the two constructions sit at opposite extremes of the same finiteness question.

On the generic fibres the paper works with the Q-rational geometric fundamental group of a curve C with a rational point P. This group is the quotient of the etale fundamental group of C by the closed normal subgroup generated by the image of the section induced by P. It classifies etale covers in which P splits completely, which is precisely the property the tower is built to preserve.

Realizable central extensions keep the tower going

The construction starts from a hyperelliptic curve C_{-1} over Q of genus g with an explicit etale cover C_0 to C_{-1} whose Galois group is (F_2)^g. A key property of C_{-1} is that it has bad reduction at no more than O(g / log g) primes. The authors build arithmetic models S_0 and S_{-1} with a finite map q: S_0 to S_{-1} whose generic fibre is C_0 to C_{-1}; S_0 is regular and carries a section s_0.

Each further step is controlled by group cohomology. Given S_n with G_n = Gal(S_n / S_{-1}), a class theta in H^2(G_n, F_2) corresponds to a central extension of G_n by F_2. Such a class is called realizable if there is a connected finite etale double cover S_{n+1} to S_n whose composite to S_{-1} is Galois with that extension group. The realizable classes, together with zero, form a subspace E_n of H^2(G_n, F_2).

The technical heart is a bound on the codimension of this subspace. The paper proves that the dimension over F_2 of H^2(G_n, F_2) / E_n is at most 5000 g^2 / log g, uniformly in n. This bound depends only on g, the genus of C_{-1}, and not on the level of the tower.

Golod-Shafarevich theory then supplies the other side of the comparison: dim_{F_2} H^2(G_n, F_2) exceeds g^2 / 4. For sufficiently large g, 5000 g^2 / log g is smaller than g^2 / 4, so a nonzero realizable class exists at every stage. That is what allows the tower to be continued indefinitely rather than terminating at some finite level.

Generic fibres give curves with infinite rational geometric fundamental groups

Passing to generic fibres converts the surface tower into a tower of curves over Q. Theorem 1.2 states that for every sufficiently large integer g there is a smooth projective geometrically connected curve C over Q of genus g, a point P in C(Q), and a tower of smooth projective geometrically connected curves D_{n+1} to D_n down to D_0 = C, where D_n to C is finite etale Galois of degree 2^n and P splits completely in every D_n.

The quantitative consequences are explicit. The genus satisfies g(D_n) - 1 = 2^n (g - 1), and the number of rational points satisfies #D_n(Q) at least 2^n. The Q-rational geometric fundamental group of the pair (C, P) is infinite. This answers questions posed by Ihara and by Frey, Kani and Volklein, and also the Diem-Frey question about whether geometrically connected finite etale Galois covers of unbounded degree can exist in characteristic zero with a point splitting completely.

The starting data are concrete. For m at least 32768 and b = 2^m - 1, the paper uses the smooth projective curve C_{-1} of genus b with affine equation y^2 equal to the product over a from 0 to 2b+1 of (x - a), together with the point P_+ at infinity where y / x^{b+1} = 1. The choice b + 1 = 2^m is what allows the initial etale cover C_0 to C_{-1} to extend to a finite map of arithmetic models S_0 to S_{-1} with S_0 regular, including over the prime 2.

Logarithmic conductor grows at most linearly in the genus

Venkatesh asked at PCMI in 2022 how small the logarithmic conductor of a curve over Q can be relative to its genus. For a smooth projective geometrically connected curve C over Q, the paper writes N(C) for the conductor of its Jacobian. Standard examples such as the modular curves X_0(p) give logarithmic conductor of order g log g: with g_p = g(X_0(p)), one has g_p asymptotic to p/12 and N(X_0(p)) = p^{g_p}.

Theorem 1.3 answers the question in the affirmative. There is a constant A > 0 and a sequence of smooth projective geometrically connected curves C_n over Q with g(C_n) tending to infinity and log N(C_n) at most A g(C_n) for all n at least 0. The curves can be arranged in a tower with finite etale Galois maps C_n to C_0 of degree 2^n in which a point of C_0(Q) splits completely at every level.

The mechanism behind the linear bound is local. At each bad prime p there is a finite extension E_p of Q_p, independent of n, over which every C_n has semistable reduction. For each such p the paper obtains a constant B_p depending only on E_p / Q_p such that the exponent a_p(C_n) of p in N(C_n) satisfies a_p(C_n) at most B_p g(C_n) for every n. Summing a_p(C_n) log p over the finitely many bad primes gives log N(C_n) = O(g(C_n)).

The paper also records the expected optimality context: assuming the completed L-function of the Jacobian of C is entire of order one and satisfies its expected functional equation, Mestre proves log N(C) > log(10) times g(C). The linear order of magnitude achieved here is therefore the expected optimal one, not merely an improvement on the g log g examples.

An independent proof reaches a related surface by a shorter route

The paper reports that Pablo Chenal independently and simultaneously proved the existence of a proper regular arithmetic surface over Spec Z with a section and an infinite pro-2 quotient of its etale fundamental group. His starting curve is closely related to the C_{-1} used here: both are hyperelliptic with rational Weierstrass points and O(g / log g) primes of bad reduction, where g is the genus. The main change in the respective Weierstrass equations is a factor 8 used in his analysis of the model at 2.

The two arguments differ in how the base of the tower is obtained. Chenal's cover, playing the role of C_0, is not determined explicitly and comes from an existence argument, which lets him avoid much of the detailed analysis of integral models and successive central extensions carried out here. His proof is described as substantially shorter and more elegant, working directly with the pro-2 geometric fundamental group of his starting curve and the Galois action induced by a rational point, combining specialization at 2 with Vinberg's refinement of the Golod-Shafarevich theorem.

The authors state that Chenal and they are working towards a unification of the two approaches and plan to merge the two papers into a single joint work. For a reader comparing the two, the trade-off is explicit: the present paper pays for its explicit integral models and codimension bounds with length and technical detail, while the independent argument pays for its brevity with a non-explicit base cover.

What the construction does and does not establish

The results are existence statements about specific towers. They do not claim that every arithmetic surface over Spec Z has infinite geometric etale fundamental group, nor that the finiteness results of Bost and Charles or of Calegari, Dimitrov and Tang are wrong; those results concern different models, such as models of elliptic curves, certain integral modular schemes, and the connected Neron model of X(N), where the image of the generic-fibre fundamental group is trivial.

The linear conductor bound is asymptotic in the tower index and depends on a constant A that the paper does not optimize. The bound log N(C_n) at most A g(C_n) is proved by summing local contributions over the finitely many bad primes, with each local constant B_p depending only on the extension E_p / Q_p over which semistable reduction is attained. The extensions E_p are chosen using C_0 and are independent of n, which is what makes the constant A independent of the tower level.

The starting genus is large: the construction takes m at least 32768 and b = 2^m - 1, and Theorem 1.2 applies for every sufficiently large genus g, with the proof noting g_0 = 3(2^{32768} - 1) as an example. The theorems are therefore existence results at large genus, not constructions at small genus, and the explicit numerical thresholds are part of the statement rather than incidental choices.

The priority is the section splitting completely, not the surface alone

Among the three questions answered, the one that most changes how a reader should evaluate related work is the Bost-Charles question, because it is the one where the paper's answer is a projective example with an infinite geometric etale fundamental group and a section that splits completely at every level. The infinite fundamental group alone would not distinguish this construction from a surface whose generic-fibre fundamental group dies in the integral one; the section is what rules that out.

The comparison to keep in hand is therefore between two configurations of the same finiteness question. In the connected Neron model of X(N), the image of the generic-fibre fundamental group in the integral fundamental group is trivial. In the surfaces built here, that image is infinite, and the tower S_n to S has degree 2^n at level n with the section splitting into 2^n copies of Spec Z. A reader assessing any future example should check which of these two behaviours it exhibits before treating it as progress on the same problem.

The conductor result should be read on its own terms rather than folded into the fundamental-group statement. It is a statement about the sequence of generic fibres C_n, with log N(C_n) at most A g(C_n) and g(C_n) tending to infinity, in contrast to the g log g order of the modular curves X_0(p). The two results share a construction but answer different questions, and the linear-in-genus bound is the one that speaks directly to Venkatesh's 2022 question.

Sources used in this answer