$\overline{M}_{1,n}$ is usually not uniruled in characteristic $p$

Using étale cohomology, we define a birational invariant for varieties in characteristic p that serves as an obstruction to uniruledness a variant on an obstruction to unirationality due to Ekedahl. We apply this to M1,n and show that M1,n is not uniruled in characteristic p as long as n ≥ p ≥ 11. To do this, we use Deligne’s description of the étale cohomology of M1,n and apply the theory of congruences between modular forms. The aim of this paper is to show that M 1,n is not uniruled in characteristic p whenever n ≥ p − 1 ≥ 11 or p = 11 and n ≥ 11 (Theorem 3.2). We will also discuss the related concepts of rational chain-connectedness and unirationality. For smooth projective varieties X , H (X,OX) is known to be a birational invariant for all i, in characteristic 0 by [12], and in characteristic p by [5]. If f : Y → X is a separable morphism, then dimH (Y,OY ) ≥ dimH (X,OX), so if X is separably unirational then H (X,OX) vanishes for i > 0 and if X is separably uniruled than H (X,OX) vanishes for i = dimX . However X can be inseparably unirational in characteristic p even if H (X,OX) 6= 0, for instance when X is a supersingular Kummer surface and i = 2 [20, Theorem 1.1]. Using étale cohomology, Ekedahl [9] defined a birational invariant that fixes this problem. This is unsurprising as étale cohomology is invariant under inseparable morphisms. Roughly speaking, his invariant measures the multiplicity of all eigenvalues of Frobenius on the étale cohomology with compact supports, except for those eigenvalues that “look like” the eigenvalues of Frobenius with a lower-dimensional variety because we quotient by the contributions of lower-dimensional varieties, we obtain a birational invariant. We will apply this invariant to M 1,n. Due to the close relationship between the cohomology of M 1,n and modular forms, we are able to show that M 1,n,Fp is not unirational, or even uniruled, whenever there is a p-ordinary cusp form of level 1 and weight k for some k ≤ n+1 (Theorem 2.3). Applying the classical theory of modular forms modulo p, we show that M 1,n,Fp is not uniruled for n ≥ p− 1 ≥ 11 or p = 11 and n ≥ 11 (Theorem 3.2). However, for a given value of p, it is likely possible to get a much better value of n by explicitly computing coefficients of modular forms until an ordinary one is found, except for p = 2, 3, 5, 7 where no such modular forms exist. The author was supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zurich Foundation. The author was also in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Spring 2017 semester and supported by the National Science Foundation under Grant No. DMS-1440140.

This invariant is defined for varieties that are not necessarily smooth or proper. This enables us to avoid the use of stacks when working with M 1,n , though we do not expect there would be any great difficulty in extending these results to Deligne-Mumford stacks. In the smooth and projective case, this obstruction to unirationality is the same as that given by Esnault [10,Theorem 1.1], who also showed it was an obstruction even to the weaker property of having a trivial Chow group of zero-cycles.
The unirationality of M 1,n was completely understood in characteristic zero. It is unirational for n ≤ 10, because nine general points in P 2 determine a genus one curve with nine marked points, and the hyperplane class minus twice the first point gives a tenth marked point [2]. It is not even uniruled for n > 10, because its Kodaira dimension is zero for n = 11 and one for n > 11 [3,Theorem 3]. In characteristic p, the Kodaira dimension is not an obstruction to unirationality, so only the n ≤ 10 case remains valid, and to my knowledge this is all that was known.
I would like to thank Daniel Litt, Bhargav Bhatt, and John Lesieutre for helpful conversations.
We will always take a variety to be a geometrically integral separated scheme of finite type over a field, and the field will always be a finite field F q .

A Birational Invariant
We will actually define an invariant that is a slight variant of Ekedahl's. To makeétale cohomology into a birational invariant, we simply quotient by the maximum subspace which could come from a variety of lower dimension. We define this using the (geometric) Frobenius action: Definition 1.1. Let X be a variety over F q of dimension d. Let H i tdF (X) be the quotient of H i c (X F q , Q ℓ ) by the maximal Frob q -stable subspace on which the eigenvalues of Frob q all divide q d−1 in the ring of algebraic integers. (The eigenvalues lie in the ring of algebraic integers by [8,XXI Corollary 5.5.3].) The notation tdF is short for top-dimensional Frobenius -i.e. the part of cohomology where Frobenius does not act by eigenvalues that could come from a lower-dimensional variety via the excision exact sequence (see Proposition 1.2 for a precise version of this).
Ekedahl's invariant [9, Equation (1)] can be viewed as the formal sum of the eigenvalues of Frobenius acting on H i tdF (X) in the free group on the elements of Q, with multiplicity the multiplicity of the eigenvalue. This gives the same information as the characteristic polynomial of Frobenius acting on this vector space. The vector space carries slightly more information, which might prove useful, but analyzing it is not any more difficult. Indeed, many of our proofs are essentially the same as Ekedahl's, though some are new. Because the proofs are so short, we felt it was worth repeating them in this different context. Proposition 1.2. Let X and Y be two varieties that are birationally equivalent. Then . Z has dimension at most d − 1, so by [8,XXI Corollary 5.5.3(iii)], all the eigenvalues of Frob q acting on its compactly supported cohomology are algebraic integers dividing q d−1 . Thus modulo the maximal subspace on which the eigenvalues are algebraic integers dividing q d−1 , the excision map Proof. First assume that f is a finite morphism. The composition of the adjunction and trace maps by the nth power of Frob q , so its eigenvalues are the nth powers of the eigenvalues of Frob q . The eigenvalue λ n divides (q n ) d−1 in the ring of algebraic integers if and only if λ divides q d in the ring of algebraic integers, because the nth root of an algebraic integer is an algebraic integer. So these two vector spaces are manifestly isomorphic.
Using these, we can prove that H i tdF (X) is an obstruction to unirationality: Corollary 1.5. Let X be a variety over F q of dimension d that is unirational over F q . Then H i tdF (X) = 0 for i < 2d. This is not exactly stated in [9] but is very similar to [9, Corollary 3(iii)].
Proof. We have a dominant rational map P N Fq → X for some N. A general n-dimensional subspace of P N Fq will intersect the generic fiber and thus will still map dominantly to X, so we may assume N = n. Because X is unirational over F q , it is unirational over F q n for some n. When if H 2 tdF (X) = 0 then λ divides q so q/λ is an algebraic integer. Because all Galois conjugates σ(λ) of λ are also eigenvalues, they also have absolute value |q|, so |σα| = |q/σ(λ)| = q/q = 1 for all σ. Because α is an algebraic integer all whose conjugates have norm 1, it is a root of unity. Hence every eigenvalue of Frob q is q times a root of unity. Let n be the lcm of the orders of this root of unity. Then the eigenvalues of Frob q n on H 2 (X Fq , Q ℓ ) are all equal to q n . Under the Tate conjecture, that implies the cohomology group H 2 (X Fq , Q ℓ ) is generated by classes of cycles defined over F q n , so X is supersingular in Shioda's sense. Under Shioda's conjecture, because X is supersingular and simply-connected, it is unirational [20, Conjecture on pp. 167].
We can even show H d tdF (X) is an obstruction to uniruledness: Proof. Because X is uniruled over F q , it is uniruled over F q n for some n. Because X F q n is uniruled, it is dominated by Y × P 1 for some Y . By repeatedly taking a general hyperplane slice of Y , we may assume that Y has dimension d − , and all other cohomology groups of P 1 vanish (see e.g. [16,Example 16  For smooth varieties, we can express H i tdF (X) in a different way using Poincaré duality. This connects it to an argument of [10] and implies that H i tdF (X) is an obstruction, not just to unirationality, but to some weaker conditions, the weakest of which is that X admits a decomposition of the diagonal.
by the maximal subspace on which Frob q acts by eigenvalues not divisible by q.
Proof. By Poincaré duality [1, XVIII, Theorem 3.2.5], H i c (X F q , Q ℓ ) and H 2d−i (X F q , Q ℓ ) are dual, with Frob q acting on the pairing by multiplication by q d .
The notion of generalized eigenspaces is helpful here. The space H i c (X F q , Q ℓ ) splits into the direct sum of generalized eigenspaces of Frob q , and hence can be written as a sum of the sum of all generalized eigenspaces associated to eigenvalues that divide q d−1 and the set of all generalized eigenvalues that do not divide q d−1 . The second is H i tdF (X), and thus H i tdF (X) is dual to the sum of the generalized eigenspaces of H 2d−i (X F q , Q ℓ ) associated to eigenvalues λ where q d /λ does not divide q d−1 , i.e. where λ is not a multiple of q. This is exactly the quotient by the generalized eigenspaces of eigenvalues divisible by q. Remark 1.9. Katz's "Newton above Hodge" conjecture, proved by Mazur [15] and Ogus [17], and the comparison betweenétale and crystalline cohomology together imply that, if X is a smooth projective variety over F q and H i (X, O X ) = 0, then all eigenvalues of Frob q on H i (X, Q ℓ ) are divisible by q. Thus the invariant H 2d−i tdF (X) is closely connected to H i (X, O X ). Proposition 1.10. Let X be a smooth projective variety over F q . Assume one of the following conditions holds: • Ch 0 (X k(X) ) = Z.
• X admits a decomposition of the diagonal. Then H i tdF (X) = 0 for i < 2d .
Proof. The first condition implies the second condition by [4, [10, p. 190], the main argument of [10] works in the ℓ-adic setting (or one could deduce the ℓ-adic result from the p-adic one by the comparison of the characteristic polynomials of Frobenius in the two settings). This implies the vanishing of H i by Poincaré duality as in Proposition 1.8.

Application to the Moduli Space of Elliptic Curves
We will apply H i tdF to the Deligne-Mumford moduli space M 1,n of genus 1 curves with n marked points, a singular projective variety. (We will avoid the use of Deligne-Mumford stacks as they are unnecessary for this problem).
It turns out that H i tdF (M 1,n ) is controlled by modular forms: We say an algebraic integer is prime to p if some coefficient of its characteristic polynomial other than the first is nonzero modulo p. We say a Hecke eigenform is ordinary at p if its pth Hecke eigenvalue is prime to p. First we sketch the proof. Deligne, in [7], constructed two-dimensional Galois representations associated to modular forms. These representations are defined as subspaces of a certain sheaf cohomology group on a modular curve. The Frobenius eigenvalues of these representations can be related to the Hecke eigenvalues of the modular forms. Using the Leray spectral sequence, we may write theétale cohomology of M 1,n in terms of sheaf cohomology on a modular curve, and by comparing the relevant sheaves we can show that the two-dimensional Galois representations defined by Deligne occur as subquotients of the cohomology of M 1,n . To show that the Frobenius eigenvalues on these Galois representations are prime to p, and thus to show via Proposition 1.8 that H i tdF (M 1,n,Fp ) = 0, it suffices to show that the Hecke eigenvalues are prime to p.
We need the following lemma, which is surely well-known, though we did not find a suitable reference in the literature: Lemma 2.2. Let X be a variety and G a finite group acting on X. Let X/G be the quotient space. Then for all i, H i c (X/G, Q ℓ ) = H i c (X, Q ℓ ) G . Proof. Let π be the projection X → X/G. The morphism π is finite, so Rπ * Q ℓ ∼ = Rπ ! Q ℓ , and the fiber of π over any point consists of a single orbit. To check that the adjunction map Q ℓ → (Rπ * Q ℓ ) G = (Rπ ! Q ℓ ) G of sheaves on X/G is an isomorphism, it suffices to check it on stalks, so by proper base change [1, XIII, Corollary 5.2(iii)] it suffices to check when X is a single orbit and X/G is a single point, where it is obvious. Therefore (M 1,n,Fp ) = 0. We will show that all eigenforms in S k (Γ(1)) have Hecke eigenvalue divisible by p. If they are divisible by p, they cannot be prime to p, giving the desired contradiction.
Let m ≥ 3 be prime to p. Following Deligne [7], let M m,Fp be the fine moduli space of elliptic curves with full level m structure over

This contains as a summand
which has as a quotient the parabolic cohomologỹ .

So all eigenvalues of Frobenius onH
are divisible by p. Note that we need the assumption n ≥ k + 1 so that (n − 1) − (k − 2) ≥ 0.
Deligne [7, pp. 10] defines an action of the Hecke operator T p onH 1 (M m,Fp , Sym k−2 (R 1 f m * Q ℓ )), and shows that T p acts as F + I * p V where F is the geometric Frobenius, V is its transpose, and I * p is the action of the diagonal element p 0 0 p ∈ GL 2 (Z/m) [7,Proposition 4.8].
The action of I * p factors through the action of GL 2 (Z/m), so on the GL 2 (Z/m)-invariant subspace, T p = F + V . Note that all eigenvalues of F are divisible by p, and V is the transpose of F so all eigenvalues of V are divisible by p. Thus because F and V commute [7,Proposition 4.8(3)], all eigenvalues of T p onH 1 (M m,Fp , Sym k−2 (R 1 f m * Q ℓ )) GL 2 (Z/m) are divisible by p.
By [7,Corollary 4.2], the sheaf R 1ã (M m , Sym k−2 (R 1 f m * Q ℓ )) on Spec Z[1/mℓ] is lisse, and its stalk at the geometric point F p isH 1 (M m,Fp , Sym k−2 (R 1 f m * Q ℓ )). The Hecke operators and GL 2 (Z/m) also act on this sheaf. Because the sheaf is lisse, the stalks at F p and Q are isomorphic as vector spaces with a GL 2 (Z/m) and Hecke operator action. Because all Hecke eigenvalues on the GL 2 (Z/m)-invariant part the stalk of H 1 (M m,Fp , Sym k−2 (R 1 f m * Q ℓ )) at F p are divisible by p, all Hecke eigenvalues on the GL 2 (Z/m)invariant part of the stalk ofH 1 (M m,Fp , Sym k−2 (R 1 f m * Q ℓ )) at Q are divisible by p.
Deligne [7] uses several different spaces W of modular forms, indexed by subscripts and superscripts. The upper-left index is the weight, which is equal to the usual weight of modular forms minus two. The lower-left index is the level (where we always take modular forms for the full congruence subgroup Γ(n) of that level). The lower-right index defines the coefficient field, where ℓ represents Q ℓ , ∞ represents C, and an omitted subscript represents Q. The upper-right superscript may be used to denote the invariants under a group action (a standard notation, which we have used previously in this proof).

Calculations with Modular Forms
For any particular p and k we can compute the eigenvalues of the Hecke operator T p acting on S k (Γ(1)) and determine whether any eigenforms are ordinary. For instance it appears that for almost all p, the Ramanujan ∆ function, a cusp form of weight 12, is ordinary. This is true for all p < 10 10 except p = 2, 3, 5, 7, 2411, and 7758337633 [14]. For all such p, M 1,11,Fp is not uniruled. This is best possible in those characteristics as M 1,10 is always unirational. However a method based only on modular forms of weight 12 is unlikely to work in general, as it is not even known that there are infinitely many p at which ∆ is ordinary. Yet we can still prove a general existence result for an ordinary form, albeit of considerably higher weight than 12: Proposition 3.1. Let p > 11 be a prime. Then the space S p−1 (Γ(1)) of cusp forms of weight p − 1 and level 1 contains a p-ordinary eigenform.
Proof. In [18, 1.2], Serre definesM k as the space of power series mod p that are the reductions mod p of modular forms of weight k and level 1 with integer coefficients. So the lattice inside S p−1 (Γ(1)) consisting of modular forms of weight p − 1 and level 1 with integer coefficients maps toM p−1 , and this map commutes with the Hecke operator T p .
OnM p−1 , the action of T p is the same as the action of U [18, 2.2], which acts bijectively [18, Theorem 6(ii)], so all its eigenvalues are nonzero (mod p). Hence to show that one of the eigenvalues of T p on S p−1 (Γ(1)) is nonzero mod p, it suffices to show that the image of this map is nonzero. This follows from the existence of any cusp form of weight p − 1 with integral coefficients, since we can always divide it by p until at least one coefficient is nonzero mod p.
The dimension of the space of cusp forms of weight k and level 1 is ⌊k/12⌋, unless k ≡ 2 mod 12, in which case it is ⌊k/12⌋ − 1. This is easily seen to be greater than 1 if k > 14 or k = 12, and so is greater than 1 when k = p − 1 for any primes p > 11. Since there is an integral basis for the space of cusp forms (for an elementary proof, observe that the basis for all modular forms constructed in [11, Chapter 5, Proposition 1] restricts to a basis of cusp forms), there exists a cusp form of weight p − 1 with integer coefficients for all such k.
Theorem 3.2. Let p be a prime and n a natural number. If p > 11 and n ≥ p − 2 or p = 11 and n ≥ 11, then M 1,n,Fp is not uniruled.
We suggest two open questions that may be interesting: Question 3.3. For any p, is M 1,n,Fp uniruled (or even unirational) for infinitely many n?
We have given a negative answer for p > 7, so the remaining possibilities are p = 2, 3, 5, 7. A tantalizing fact is that our proof does not just fail by random chance -using the theory of congruences between modular forms, one can show without calculation that for p < 11 there are no p-ordinary cusp forms of level 1 and any weight k and thus Theorem 2.3 will never prove that M 1,n,Fp is not uniruled.
Indeed, Hida showed that the dimension of the space of p-ordinary cusp forms of weight k on Γ 0 (p) depends only on p mod k −1 for k ≥ 2 [11, Chapter 7, Theorem 1], and is equal to the dimension of the space of p-ordinary cusp forms on Γ(1) for k ≥ 3 [11, Chapter 7, Proposition 2]. Since for each residue class mod p − 1, we may find a representative in the interval [3, p + 1], it follows that if there is any p-ordinary cusp form at all, then there is one of weight ≤ p + 1. But the cusp form of smallest weight is 12, so if there are any p-ordinary cusp forms then p ≥ 11.
If this reason can be turned into a "geometric" explanation of the p-divisibility of the eigenvalues somehow, then that geometric explanation might prove a stronger statement as well, possibly even unirationality. Alternatively, if a plausible higher-dimensional analogue of Shioda's conjecture [20, Conjecture on pp. 167] is made, this argument could be used to show that M 1,g is unirational in characteristic p < 11 for all g conditionally on that conjecture.