ANALOGUE OF THE DEGREE CONJECTURE OVER FUNCTION FIELDS

MIHRAN PAPIKIAN

Abstract. We prove an upper bound on the degree of modular parametrizations of elliptic curves by Drinfeld modular curves which is the function field analogue of the conjectured bound over the rational numbers.

1. Introduction

Let E be a semi-stable elliptic curve over ℚ of conductor N_{E}, H be the complex upper half-plane, and Γ_{0}(N_{E}) be the Hecke congruence subgroup of level N_{E}. Denote the compactification of the quotient Γ_{0}(N_{E}) \ H by X_{0}(N_{E}). One can view X_{0}(N_{E}) as an algebraic curve over ℂ (and in fact over ℚ). From the work of Wiles and Taylor it is known that there is a non-constant map

℘ : X_{0}(N_{E}) → E

called modular parametrization. We always choose a modular parametrization of minimal degree. Assume further that E is optimal, that is, the map induced by ℘ between the Albanese varieties has connected (and smooth) kernel. It is known that every ℚ-isogeny class of elliptic curves contains a unique such curve. The optimal curve in a given ℚ-isogeny class can also be characterized as the curve for which deg ℘ is minimal. G. Frey made the following remarkable conjecture

Conjecture 1.1 (Degree conjecture). For any ε > 0 there is a constant c(ε) depending only on ε such that

deg ℘ ≤ c(ε)N_{E}^{2+ε}.

This conjecture is known to imply the celebrated ABC conjecture, and also implies a bound on congruence primes of the newform corresponding to E; see [14], [15]. In this paper, under a certain assumption, we will prove the function field analogue of this conjecture.

Let 𝔽_{q} be the finite field of q elements, and let C/𝔽_{q} be a smooth, projective, geometrically irreducible curve over 𝔽_{q}. Denote by F the function field of C, and by A the subring of F consisting of functions which are regular outside a fixed place ∞ on C. (F and A are the analogues of ℚ and ℤ respectively.) In this setting the role of the complex upper half-plane is played by the Drinfeld upper half-plane

Ω = ˆ \overline{F}_{∞} − F_{∞},

where F_{∞} is the completion of F at ∞, and ˆ \overline{F}_{∞} is the completion of the algebraic closure of F_{∞}. The upper half-plane Ω can be endowed with a structure of a one-dimensional rigid-analytic space. Fix an ideal 𝔫 of A and denote by Γ_{0}(𝔫) the subgroup of GL_{2}(A) whose image in GL_{2}(A/𝔫) consists of upper-triangular matrices. The subgroup Γ_{0}(𝔫) acts on Ω through linear fractional transformations. Drinfeld showed that the quotient Γ_{0}(𝔫)\Ω is the underlying rigid-analytic space of an affine algebraic curve defined over F_{∞}. The compactification X_{0}(𝔫) of this curve is called the Drinfeld modular curve for Γ_{0}(𝔫). The analogue of the modularity conjecture over function fields is a consequence of some deep results of Deligne, Drinfeld and Zarhin, cf. [10]: Let E be an elliptic curve over F with conductor 𝔫_{E} = 𝔫 · ∞ and split multiplicative reduction at ∞. Then there is a non-constant morphism

Supported by a fellowship from the European Postdoctoral Institute.

℘ : X_{0}(𝔫) → E_{/F_{∞}}.

In fact this map can be defined over certain finite Galois extension of F. As over ℚ, we choose ℘ of minimal degree. Let T denote the Bruhat-Tits tree of PGL_{2}(F_{∞}). The group Γ := Γ_{0}(𝔫) naturally acts on T , and the homology group H_{1}(Γ \ T , ℤ) of the quotient graph is free of ℤ-rank equal to the genus of X_{0}(𝔫). There is an injective homomorphism ι : H_{1}(Γ \ T , ℤ) → \underline{H}_{!}(T , ℤ)^{Γ}, where \underline{H}_{!}(T , ℤ)^{Γ} is the group of ℤ-valued Drinfeld cusp forms for Γ (also called harmonic cochains). It is known that this latter group is free and the cokernel of ι is finite; see [10, §3]. One can associate to our elliptic curve E a primitive element v_{E} ∈ H_{1}(Γ \ T , ℤ), well-defined up to a sign. (Primitive means that v_{E} is not an integer multiple of any other element except −v_{E}.) We make the assumption (∗) that the element ι(v_{E}) is primitive. Also assume E is semi-stable and optimal. The main result of this paper is the following bound.

Theorem 1.2. With previous notation and assumptions we have

deg ℘ < q^{(14g+5+deg ∞)} · N (𝔫_{E}) · (deg 𝔫_{E})^{3},

where N (𝔫_{E}) := q^{deg 𝔫_{E}}, g is the genus of the curve C and deg denotes the degree of Weil divisors on C.

Taking g and deg(∞) as the fixed data attached to F and A, we see that the bound in the theorem is somewhat better than the conjectured bound over ℚ as it is analogous to the upper bound N_{E}(log N_{E})^{3}. The degree of optimal modular parametrization over function fields can be shown to be divisible by the congruence primes of E; see [7]. Hence our theorem proves the analogue of Murty’s conjecture [15, Conj. D] (modulo the assumptions we made). Using the argument in the proof of [17, Thm.6.4], one can show that deg ℘ is bounded from below by α · N (𝔫_{E})/(deg 𝔫)^{2}, where α is a constant depending only on q, g and deg ∞. Thus, log(deg ℘) ≍ deg 𝔫_{E} and the bound in the theorem is essentially the best possible.

We make few comments on our assumptions. The assumption that E is semistable is not essential. The only reason we made this assumption is to avoid some technical nuances in the analytic calculations in §4; see Remark 4.3. On the other hand, the assumption that E is optimal cannot be discarded. Over ℚ the degree conjecture can be stated without assuming the elliptic curve is optimal. This is equivalent to the original conjecture since by Mazur’s fundamental result any two elliptic curves in the same ℚ-isogeny class can be related by an isogeny of degree at most 163. Due to the existence of the Frobenius morphism, this is not the case over function fields. In fact, any isogeny class of non-isotrivial elliptic curves over F contains infinitely many curves and the degrees of minimal possible isogenies between them can be arbitrarily large. Hence to ask a meaningful question on how large deg ℘ can be one has to restrict to optimal curves. Finally about (∗). In [10, (6.4.5)] Gekeler and Reversat conjecture that the homomorphism ι is always bijective. It is easy to see that this implies (∗). The conjecture itself is known to be true if A = 𝔽_{q}[t], see [9], so in this case our theorem is unconditional. Note that in the statement of Conjecture 1.1 one is implicitly using the results of Mazur and Raynaud which say that for semi-stable elliptic curves Manin’s constant c_{E} is 1 or 2 (and Manin conjectures that c_{E} is always equal to 1). Otherwise the bound in the conjecture has to be multiplied by c_{E}^{2}; see [14]. Our assumption (∗) is of similar nature. If ι(v_{E}) is a c̃_{E} multiple of some primitive element, c̃_{E} ∈ ℤ, then we can drop (∗) by multiplying the right hand-side of the inequality in Theorem 1.2 by (c̃_{E})^{2}. In both cases the reason for the appearance of the constant c^{2} is that the degree of optimal modular parametrization is proportional to the Petersson norm of c · f_{E}, with f_{E} being the normalized newform associated to E. In fact, the proof of Theorem 1.2 proceeds by first establishing the following formula for deg ℘, which might be of some independent interest:

Proposition 1.3. Assume E is optimal. Then

deg ℘ = q^{2g−2}N (𝔫_{E}) · \frac{(c̃_{E})^{2} · (#coker ℘_{∗}^{Φ})^{2}}{#Φ_{E,∞}} · L(1, Sym^{2}π_{E}).

Here Φ_{E,∞} is the group of connected components of the Néron model of E over ∞, ℘_{∗}^{Φ} is the functorial homomorphism Φ_{J,∞} → Φ_{E,∞} between the component groups of Néron models of the Jacobian of X_{0}(𝔫) and E induced by ℘, and π_{E} is the automorphic cuspidal representation attached to E.

Following [10, (6.4.4)], it is possible to give an explicit but rather large bound on c̃_{E}. We note that Proposition 1.3 generalizes the formula [18, (27)] for the case of 𝔽_{q}(t) to general function fields.

The main obstacle in proving the degree conjecture over ℚ lies in giving lower bounds on the area of the fundamental parallelogram of the lattice of periods of E, ∫_{E(ℂ)} |ω_{E}|, in terms of the conductor of E, cf. [15]. (Here ω_{E} is a Néron differential of E.) Over function fields this quantity gets replaced by #Φ_{E,∞}, and hence obviously is larger than or equal to 1. This trivial lower bound on #Φ_{E,∞} turns out to be sufficient to deduce the bound in Theorem 1.2. Note that by a theorem of Tate #Φ_{E,∞} is essentially ∫_{E(F_{∞})} |ω_{E}|, hence one could say that the degree conjecture is possible to prove over function fields because a certain complex period gets replaced by a non-archimedean period.

The challenge in proving Theorem 1.2 is primarily in relating deg ℘ to the Petersson norm of f_{E}. Over ℂ one derives the analogous formula by pulling back via ℘ a holomorphic differential form on E to X_{0}(N_{E}). This approach does not work over F since it has positive characteristic and we essentially see deg ℘ only modulo the characteristic. The remedy consists of looking at the induced maps between integral homology groups (or rather their analogues in rigid-analytic setting). The idea of doing this is due to Gekeler [7]. We used his results to prove in [18] a slightly weaker version of Theorem 1.2 for the case of A = 𝔽_{q}[t]. In this paper we deal with general function fields by using Grothendieck’s monodromy pairing instead, which seems better adapted to this situation.

The organization of the paper is as follows: Section 2 is algebraic in nature and gives a formula for deg ℘ in terms of Grothendieck’s monodromy pairing. Section 3 contains the proof of the main theorem 1.2. We left all analytic calculations required in the proof of the main theorem for the last Section 4. This consists of computing certain Rankin-Selberg integrals and bounding the special values of symmetric power L-functions for GL(2).

Acknowledgements. This paper was completed while I was visiting Max-Planck-Institut für Mathematik in Bonn. I thank the staff and the members of the institute for their hospitality. I also thank C. Skinner, K. Soundararajan, and especially B. Conrad for helpful discussions related to the subject of the paper.

2. Monodromy pairing and a formula for degree

In this section we let R be a complete discrete valuation ring, K be its field of fractions, and k be the residue field.

2.1. Néron models of abelian varieties with toric reduction. Let A be an abelian variety over K. Denote by A its Néron model over R and denote by A_{k}^{0} the connected component of the identity of the closed fibre A_{k} of A. We have an exact sequence

(2.1)  0 → A_{k}^{0} → A_{k} → Φ_{A} → 0,

where Φ_{A} is a finite étale group-scheme over k called the component group of A. In other words, Φ_{A} is a commutative finite group equipped with a continuous action of Gal(k̄/k). We say that A has (split) toric reduction if A_{k}^{0} is a (split) algebraic torus over k.

Proposition 2.1. Let f_{K} : A → B be a closed immersion of abelian varieties over K, i.e., A is an abelian subvariety of B. Assume B has toric reduction. If R is equicharacteristic then the canonical morphism f : A → B extending f_{K} is a closed immersion.

Proof. First we need a simple lemma.

Lemma 2.2. Assume R is equicharacteristic. Let G_{K} be a commutative finite flat group-scheme over K of prime order p. If G_{K} extends to a finite flat group-scheme G over R such that the closed fibre G_{k} is either étale or multiplicative then such extension is unique.

Proof. If p is not the characteristic of k then G is clearly unique and must be étale. Now assume p is the characteristic of k. Using Cartier duality, we can assume that G_{K} extends to a finite flat group-scheme G over R such that G_{k} is étale. This implies that G is étale, and in particular, G_{K} is étale. By considering normalizations, one easily checks that étale extensions are unique. Thus, we are reduced to proving the following statement: If G_{K} is étale then it cannot be extended to a finite flat group-scheme over R with multiplicative closed fibre. Indeed, assume G_{k} is multiplicative. The Cartier dual of G will have an étale closed fibre, so has to be étale. This implies that the Cartier dual of G_{K} is étale, so G_{K} ≅ μ_{p}. This is a contradiction as μ_{p} is not étale in characteristic p. (Note that the statement of the lemma is false without assuming R is equicharacteristic as the example of μ_{2} over ℚ_{2} shows.) □

Now we return to the proof of the proposition. By slightly modifying the argument in [1, Lem.7.4/2], one can show that A also has toric reduction. Since the formation of Néron models commutes with passing to the strict henselization or completion of R, we can assume that R is both strictly henselian and complete.

First we show that f is a monomorphism; i.e., that H := ker f is trivial. For this it is enough to show that the special fibre H_{k} of H is trivial. The kernel of f_{k} : A_{k} → B_{k} is finite, as A is isogenous (in fact isomorphic) to a subvariety of B. Hence H_{k} is finite, and if we show that it has trivial ℓ-torsion for any prime ℓ, we could conclude that H_{k} is trivial. As H_{k}[ℓ] ⊆ A[ℓ]_{k}, it is enough to show that f restricted to quasi-finite and flat group-scheme A[ℓ] is a monomorphism. Since we are assuming R is henselian, there is a unique decomposition A[ℓ] = G ∐ G_{0} into disjoint subgroup-schemes which are both open and closed in A[ℓ], and such that G is finite and flat over R and G_{0} has empty closed fibre; cf. [1, §7.3]. Hence it is enough to prove that f is injective when restricted to G. Further, by restricting to a finite flat subgroup of G of prime order, we can assume that G has prime order. Let G′ be the schematic closure of G_{K} in B (here we are using the assumtion that f_{K} is a closed immersion). As both A and B have toric reduction, the groupschemes G_{k} and G′_{k} are either étale or multiplicative, cf. (2.1). Hence G and G′ are two extensions of G_{K} over R satisfying the assumptions of Lemma 2.2. The lemma implies that f restricted to G gives an isomorphism f : G ⟶^{∼} G′, i.e., f is a monomorphism. The rest of the proof is the same as in [1, Thm.7.5/4] except that the use of Cor.7.5/6 in [1] must be replaced by Lemma 2.2. □

Let A and B be as in Proposition 2.1. Consider the commutative diagram induced by f_{k}:

(2.2)  \xymatrix{0 \ar[r] & A_{k}^{0} \ar[r] \ar[d]^{f_{k}^{0}} & A_{k} \ar[r] \ar[d]^{f_{k}} & Φ_{A} \ar[r] \ar[d]^{f_{Φ}} & 0 \\ 0 \ar[r] & B_{k}^{0} \ar[r] & B_{k} \ar[r] & Φ_{B} \ar[r] & 0.}

Corollary 2.3. The morphism f_{k}^{0} is a closed immersion and ker(f_{Φ}) can be identified with a Gal(k̄/k)-subgroup of the quotient torus B_{k}^{0}/A_{k}^{0}.

2.2. Monodromy pairing and component groups. Let A be an abelian variety over K with toric reduction. Let A^{∨} be the dual abelian variety of A. The character group

M_{A} = Homk̄(A0 ; k̄, 𝔾_{m,k̄})

is a free abelian group contravariantly associated to A; its ℤ-rank is equal to the dimension of A. Let M_{A^{∨}} be the analogous group associated to A^{∨}. In [12, §§9-10] Grothendieck defined a canonical bilinear Gal(k̄/k)-equivariant pairing,

u_{A} : M_{A} × M_{A^{∨}} → ℤ,

which he called the monodromy pairing. This pairing is uniquely characterized by the property that its extension of scalars u_{A} ⊗ ℤ_{ℓ}, for a prime ℓ ≠ char(k), can be expressed in terms of the ℓ-adic Weil pairing on T_{ℓ}(A) × T_{ℓ}(A^{∨}) via a formula given in [12, (9.1.2)]. The monodromy pairing has the following properties:

Theorem 2.4 (Grothendieck).

(i) u_{A} is non-degenerate. (ii) u_{A} is bifunctorial.

(iii) If ξ : A^{∨} → A is a polarization, then

u_{A,ξ} : M_{A} × M_{A} ⟶^{1×ξ} M_{A} × M_{A^{∨}} ⟶^{u_{A}} ℤ

is symmetric and positive-definite. (iv) There is a Gal(k̄/k)-equivariant exact sequence

0 ⟶ M_{A^{∨}} ⟶^{u_{A}} Hom(M_{A}, ℤ) ⟶ Φ_{A} ⟶ 0.

Proof. Once u_{A} is constructed compatibly with the Weil pairing (this is quite nontrivial), properties (i)–(iii) follow from the well-known facts about the latter pairing; see [12, Thm. 10.4]. For the proof of (iv) see [12, Thm. 11.5]. □

Definition 2.5. The abelian variety A is an optimal quotient of the abelian variety B if there is a faithfully flat morphism π : B → A whose functorial kernel is represented by an abelian subvariety of B (that is, ker π is connected and smooth).

From now on we will assume that R is equicharacteristic, so as to have at our disposal Corollary 2.3. Let π : B → A be as in Definition 2.5, and assume B, hence also A, have toric reduction. Consider the commutative diagram induced by π on the sequence in Theorem 2.4(iv)

(2.3)  \xymatrix{0 \ar[r] & M_{B^{∨}} \ar[r]^{u_{B}} \ar[d]^{(π^{∨})^{∗}} & Hom(M_{B}, ℤ) \ar[r] \ar[d]^{Hom(π^{∗},ℤ)} & Φ_{B} \ar[r] \ar[d]^{π_{Φ}} & 0 \\ 0 \ar[r] & M_{A^{∨}} \ar[r]^{u_{A}} & Hom(M_{A}, ℤ) \ar[r] & Φ_{A} \ar[r] & 0,}

where π^{∨} : A^{∨} → B^{∨} is the dual morphism of π, and π^{∗}, (π^{∨})^{∗} are the functorially induced homomorphisms between the character groups. It is a general fact that the duals of optimal quotients are closed immersion, so π^{∨} is a closed immersion. By Corollary 2.3, (A^{∨})_{k}^{0} → (B^{∨})_{k}^{0} is a closed immersion, so (π^{∨})^{∗} is surjective. From the snake lemma we get

(2.4)  #coker(π_{Φ}) = #Ext_{ℤ}^{1}(M_{B}/π^{∗}(M_{A}), ℤ) = #(M_{B}/π^{∗}(M_{A}))_{tor}.

There is a bifunctorial pairing Φ_{A} × Φ_{A^{∨}} → ℚ/ℤ, defined by Grothendieck in [12, §1.2] for any abelian variety A over K, which turns out to be perfect when A is semi-stable (in particular, when A has toric reduction); see [12, §11.3]. Applied to the situation at hand, this pairing induces canonical isomorphisms between Φ_{A} and the Pontrjagin dual (Φ_{A^{∨}})^{∗} of Φ_{A^{∨}}, and between ker(π_{Φ}^{∨}) and the Pontrjagin dual coker(π_{Φ})^{∗}. In particular, #coker(π_{Φ}) = # ker(π_{Φ}^{∨}).

Proposition 2.6. With notation as above, assume further that B has split toric reduction. Let q := #k, and let s be the minimum of dim(A) and dim(B)−dim(A). Then #coker(π_{Φ}) divides (q − 1)^{s}.

Proof. If B has split toric reduction, then so do A and A^{∨}. Hence the character groups M_{A} and M_{A^{∨}} have trivial Gal(k̄/k)-action. From Theorem 2.4(iv) we conclude that Φ_{A} and Φ_{A^{∨}} are constant étale group-schemes over k with at most dim(A) generators. We know that #coker(π_{Φ}) = # ker(π_{Φ}^{∨}). By Corollary 2.3 there is an injection of ker(π_{Φ}^{∨}) into the split torus T := (B^{∨})_{k}^{0}/(A^{∨})_{k}^{0}, which has dimension dim(B) − dim(A). In fact, since ker(π_{Φ}^{∨}) is constant, it must inject into T (k) ≅ (k^{×})^{dim(B)−dim(A)}. The proposition follows by combining all these facts. □

2.3. Formula for the degree. Let X be a proper, smooth, and geometrically connected curve over K. Assume X(K) ≠ ∅. Also assume that X has a semistable model over R with a k-split degenerate closed fibre. This last condition means that there is a proper flat curve X over R such that X_{K} ≅ X, and all the irreducible components of X_{k} are isomorphic to ℙ_{k}^{1} intersecting transversally at krational points. Let J = Pic_{X/K}^{0} be the Jacobian variety of X, and let J be the Néron model of J over R. From general theory of Néron models one knows that J has split toric reduction; see [1, Ch.9]. Fix a point ∞ ∈ X(K). Let E be an elliptic curve over K. Assume there is a non-trivial morphism ℘ : X → E which sends ∞ to the origin of E. Assume further that the homomorphism π : J → E induced from ℘ by the Albanese functoriality realizes E as an optimal quotient of J. The morphism ℘ can be recovered as the composite of the canonical closed immersion X ↪ J, which sends ∞ to the origin, with the quotient map π. Using the canonical principal polarizations on J and E, the homomorphism E → J induced from ℘ by the Picard functoriality can be identified with π^{∨}.

Lemma 2.7. The endomorphism π ∘ π^{∨} ∈ End_{K}(E) is multiplication by deg(℘).

Proof. Let more generally f : X → Y be a finite morphism of smooth projective geometrically irreducible curves over a field K, with deg f = n ≥ 1. The map induced on Picard varieties f^{∗} : Pic_{Y/K} → Pic_{X/K} by Picard functoriality is given by L ↦ f^{∗}L. The map f_{∗} : Pic_{X/K} → Pic_{Y/K} induced by Albanese functoriality is given by W ↦ det(f_{∗}W) ⊗_{O_{Y}} det(f_{∗}O_{X})^{−1}. We need to show that f_{∗} ∘ f^{∗} : Pic_{Y/K} → Pic_{Y/K} is just multiplication by n. By the Projection Formula, for L ∈ Pic_{Y/K} we have f_{∗}(f^{∗}L) ≅ L ⊗_{O_{Y}} f_{∗}O_{X}. Hence

det(f_{∗}(f^{∗}L)) ⊗_{O_{Y}} det(f_{∗}O_{X})^{−1} ≅ det(L ⊗_{O_{Y}} f_{∗}O_{X}) ⊗_{O_{Y}} det(f_{∗}O_{X})^{−1}

≅ L^{⊗n} ⊗_{O_{Y}} det(f_{∗}O_{X}) ⊗_{O_{Y}} det(f_{∗}O_{X})^{−1} ≅ L^{⊗n}.
□

Since E is isomorphic to a subvariety of J which has split toric reduction, E has split multiplicative reduction. Its character group M_{E} is isomorphic to ℤ. Denote by ρ a generator of M_{E} (this is well-defined up to a sign). Let

(2.5)  v_{E} := (π^{∗}(ρ) ⊗_{ℤ} ℚ) ∩ M_{J} and c := #(M_{J}/π^{∗}(M_{E}))_{tor}.

In particular, π^{∗}(ρ) = c · v_{E}. Denote the canonical principal polarization on Jacobian varieties by θ.

Proposition 2.8. We have the formula

#Φ_{E} · deg ℘ = #coker(π_{Φ})^{2} · u_{J,θ}(v_{E}, v_{E}).

Proof. Indeed, according to Theorem 2.4(iv) #Φ_{E} = u_{E,θ}(ρ, ρ). Using bifunctoriality and bilinearity of the monodromy pairing, along with Lemma 2.7, we have

deg ℘ · u_{E,θ}(ρ, ρ) = u_{E,θ}(π_{∗} ∘ π^{∗}ρ, ρ) = u_{J,θ}(π^{∗}ρ, π^{∗}ρ) = c^{2} · u_{J,θ}(v_{E}, v_{E}).

On the other hand, we know from (2.4) that c = #coker(π_{Φ} : Φ_{J} → Φ_{E}). This finishes the proof. □

Corollary 2.9. Let q = #k. Then deg ℘ ≤ (q − 1) · u_{J,θ}(v_{E}, v_{E}).

Proof. Obviously #Φ_{E} ≥ #coker(π_{Φ}). From Proposition 2.6 we have #coker(π_{Φ}) ≤ (q − 1). The claim follows from Proposition 2.8. □

3. Proof of the main theorem

Notation: Let C be a smooth, projective, geometrically connected curve over the finite field of q elements 𝔽_{q}. Let F be the field of rational functions on C. Fix a closed point ∞ on C of degree d_{∞}. Denote by A the ring of functions on C which are regular away from ∞. Let K be the completion of F at ∞, R be the ring of integers in K, and k be the residue field. Let q_{∞} := q^{d_{∞}}(= #k). Finally, let G := GL(2).

3.1. Drinfeld modular curves. The function field analogue of the complex upper half-plane is the Drinfeld upper half-plane

Ω = ℙ_{K}^{1} − ℙ_{K}^{1}(K).

It has a natural structure of a one-dimensional rigid-analytic space over K. Let Λ be a projective A-submodule of rank two in F^{2} and

G(Λ) = {γ ∈ G(F) | Λγ = Λ}.

Fix an ideal 𝔫 of A, and let G(𝔫) = ker(G(Λ) → G(Λ/𝔫Λ)) be the 𝔫-th congruence subgroup of G(Λ). An arithmetic subgroup Γ of G(F) is an intermediate group G(𝔫) ⊂ Γ ⊂ G(Λ) for suitable Λ and 𝔫. The group Γ acts on Ω through linear fractional transformations with finite stabilizers, hence the quotient Γ \ Ω exists as a rigid-analytic space over K. In fact, it is a theorem of Drinfeld [6] that there is a unique smooth, geometrically irreducible affine algebraic curve Y_{Γ}/K such that its analytification Y_{Γ}^{an} is isomorphic to Γ \ Ω. The curve Y_{Γ} is called Drinfeld modular curve for Γ. The compactification of Y_{Γ} will be denoted by X_{Γ}. From some general facts about algebraic curves with analytic uniformization it follows that X_{Γ} is a Mumford curve, that is, X_{Γ} has a flat model over R with k-split degenerate closed fibre; see [22]. Certain properties of X_{Γ} are closely related to the combinatorics of the Bruhat-Tits tree T of PGL_{2}(K) and its quotients. There is a pure affinoid covering of Ω whose associated analytic reduction has intersection graph isomorphic to T ; see [10, §§1-2]. We already mentioned that an arithmetic group Γ acts on Ω via fractional linear transformations. This action respects the pure covering of Ω, and hence acts on T . (The group Γ also naturally acts on T as the space of equivalence classes of rank two R-submodules of K^{2}. One can show that the resulting two actions are canonically isomorphic; see loc.cit.) The quotient graph

(3.1)  Γ \ T ≅ G ∪ ⋃ h_{i},

is a union of a finite graph G and a finite number of infinite half-lines h_{i}. In §3.2 we will see how G is related to X_{Γ}.

3.2. Monodromy pairing on J_{Γ}. Let J_{Γ} be the Jacobian variety of X_{Γ}. As we indicated in §2, J_{Γ} will have a split toric reduction over R. We would like to describe the θ-polarized monodromy pairing u_{J,θ} on J_{Γ} explicitly in terms of the arithmetic group Γ and its action on Ω. We state a result from [16, §4] which gives such description. The details being somewhat lengthy, we simply indicate the main ingredients of the proof.

First, one needs to construct a semi-stable model of X_{Γ} over Spec(R). It possible to construct such a model using the rigid-analytic uniformization of X_{Γ}. More precisely, one needs an explicit affinoid covering of X_{Γ}^{an}. This is the subject of a paper by Reversat [20]. Associating a formal affine scheme to each of these affinoids in a standard manner, and then gluing those formal affines, one obtains a proper flat one-dimensional formal scheme over Spf(R). Then Grothendieck’s algebraization theorem yields a proper flat curve X_{Γ} over Spec(R) which is a semistable model of X_{Γ}. We understand X_{Γ} fairly well due to the explicit nature of its construction. For example, the dual graph of the closed fibre (X_{Γ})_{k} is isomorphic to G appearing in (3.1); see [16, Prop.4.3].

Once we have X_{Γ} it is possible to make the monodromy pairing u_{J,θ} explicit using the Picard-Lefschetz formula [12, (12.10)]. Consider G as an oriented graph, that is, keep track of the ordering of the irreducible components through each singular point of (X_{Γ})_{k}. Let α be an oriented cycle in G. One can describe α by a sequence of oriented edges {e_{1}, …, e_{n}} satisfying t(e_{i}) = o(e_{i+1}) for i = 1, …, n − 1 and t(e_{n}) = o(e_{1}), where t(e) (resp. o(e)) denotes the terminus (resp. origin) of the edge e. Define a pairing ϕ(e_{1}, e_{2}) between the oriented edges of G by

ϕ(e_{1}, e_{2}) = \begin{cases}1 & if e_{1} = e_{2}, \\ −1 & if e_{1} = ē_{2}, \\ 0 & otherwise. \end{cases}

(Here the bar denotes the opposite orientation.) For the cycle α put

ϕ_{α}(e) = ∑_{i=1}^{n} ϕ(e, e_{i}).

One easily checks that ϕ_{α} depends only on the class of α in H_{1}(G, ℤ), and ϕ_{α+β} = ϕ_{α} + ϕ_{β}. The action of Γ on Ω and on T factors through its image Γ̃ in PGL_{2}(F). Also, since Γ acts discontinuously, the stabilizer in Γ of each edge of T is finite. Given an edge e ∈ Ed(G), denote by e′ some preimage of e in T , and let n(e) = #Stab_{e′} Γ̃. It is easy to check that this is well-defined.

Theorem 3.1. Let M be the character group of J_{Γ}^{0} × k, where J_{Γ} is the Néron model of J_{Γ} over R. Then M ≅ H_{1}(G, ℤ) and the θ-polarized monodromy pairing M × M → ℤ is given by

u_{J,θ}(α, β) = ∑_{e∈Ed(G)} n(e)ϕ_{α}(e)ϕ_{β}(e).

Proof. [16, Cor. 4.4.]. □

3.3. Harmonic cochains for Γ. Let \underline{H}_{!}(T , ℤ)^{Γ} be the group of ℤ-valued cuspidal harmonic cochains for Γ. This is a group of ℤ-valued currents on the edges of T which are invariant under the action of Γ and have compact support modulo Γ. For the definition and the properties of this group we refer to [10, §3]. It is known that \underline{H}_{!}(T , ℤ)^{Γ} is a free abelian group of ℤ-rank equal to the genus of X_{Γ}. There is a natural injective homomorphism loc.cit.

(3.2)  ι : H_{1}(G, ℤ) → \underline{H}_{!}(T , ℤ)^{Γ}.

As a consequence of a theorem of Drinfeld [6, Thm.10.3], see also [10, §4], the group \underline{H}_{!}(T , ℤ)^{Γ} is a lattice in the space of cusp forms on G(𝔸_{F}) whose corresponding cuspidal representations are special at ∞. (Here 𝔸_{F} is the ring of adeles of F.) In particular, there is a positive definite (a priori) ℂ-valued pairing on \underline{H}_{!}(T , ℤ)^{Γ} which comes from the Petersson inner product on the cusp forms.

Using the strong approximation theorem, one obtains a push-forward Haar measure on the edges of Γ \ T from a Haar measure on G(𝔸_{F}), and the Petersson product on \underline{H}_{!}(T , ℤ)^{Γ} with respect to such measure μ takes the form, cf. [10, (4.8)],

(3.3)  (φ_{1}, φ_{2})_{μ} = ∑_{e∈Ed(G)} φ_{1}(e)φ_{2}(e)μ(e).

With notation as in Theorem 3.1, assign a weight to an edge e of the quotient graph

(3.4)  μ_{gr}(e) = \frac{1}{2}n(e)^{−1}.

It is known that μ_{gr} is a push-forward Haar measure on Γ \ T ; see [10, (4.8)]. We have the following crucial fact.

Theorem 3.2. Under the isomorphism of Theorem 3.1, we have

u_{J_{Γ},θ}(α, β) = (ι(α), ι(β))_{μ_{gr}}.

Proof. The homomorphism in (3.2) has a simple description [10, (3.2.6)]. This allows to give an explicit expression for (ι(α), ι(β))_{μ_{gr}}; see the proof of [10, (5.7.1)]. This latter formula matches the formula for u_{J_{Γ},θ}(α, β) in Theorem 3.1. □

3.4. Main theorem. Let E be an elliptic curve over F of conductor 𝔫_{E} = 𝔫 · ∞ and split multiplicative reduction at ∞, where 𝔫 is an ideal of A. It is known [10] that E_{/K} is a quotient of J_{Γ}, where Γ := Γ_{0}(𝔫) is the Hecke congruence subgroup of G(A) of level 𝔫

Γ_{0}(𝔫) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ G(A) | c ∈ 𝔫\} .

Assume E_{/K} is an optimal quotient as in Definition 2.5. Let

℘ : X_{Γ} → E_{/K}

be the non-trivial morphism obtained by composing the canonical embedding X_{Γ} ↪ J_{Γ}, which sends the cusp ∞ to the origin, with the optimal quotient map.

Let v_{E} ∈ H_{1}(G, ℤ) be the primitive element (2.5) corresponding to E under the isomorphism M ≅ H_{1}(G, ℤ) in Theorem 3.1. There is a natural action of the Hecke algebra on H_{1}(G, ℤ), and this action preserves ℤv_{E}. It is known that the map in (3.2) is equivariant under the action of the Hecke algebra; see [10, (9.3.2)]. Hence ι(v_{E}) is an eigenform corresponding to E. Assume ι(v_{E}) is primitive. (As we mentioned in the introduction, according to a conjecture of Gekeler and Reversat this always should be the case; the conjecture is known to be true for A = 𝔽_{q}[t].)

Theorem 3.3. Assume E is semi-stable. Then

deg ℘ ≤ (q_{∞} − 1)q^{14g+5}q^{deg 𝔫_{E}}(deg 𝔫_{E})^{3}.

Proof. By Corollary 2.9 and Theorem 3.2 we have the bound

deg ℘ ≤ (q_{∞} − 1)(ι(v_{E}), ι(v_{E}))_{μ_{gr}}.

Let π_{E} be the cuspidal automorphic representation corresponding E, cf. [10, (8.2)]. Since we are assuming ι(v_{E}) is primitive, Corollary 4.5 gives

deg ℘ ≤ (q_{∞} − 1)q^{2g−2}q^{deg 𝔫_{E}}L(1, Sym^{2}π_{E}).

The Langlands correspondence preserves the conductors, so the conductor of π_{E} is 𝔫_{E}. Thus, the claim follows from Theorem 4.6. □

4. Rankin-Selberg method

The goal of this section is to bound from above the Petersson norm of a newform form in terms of the norm of its conductor. This bound is the required analytic input in the proof of Theorem 3.3. We first apply the Rankin-Selberg method to relate the Petersson norm to the special value of a certain L-function, and then use convexity estimates to bound this latter special value. The final result should be well-known to the specialists, but in absence of a convenient reference we will sketch some of the details.

4.1. Conventions. Unless specified otherwise, the following notation is fixed till the end of the paper:

C is a smooth, projective, geometrically irreducible curve over the finite field 𝔽_{q}; F is the field of rational functions on C; ∞ is a fixed closed point on C; A = H^{0}(C − ∞, O) is the ring of regular functions on C − {∞}; F_{v} is the completion of F at the place v; ϖ_{v} is a uniformizer of F_{v}; ord_{v} is the canonical valuation on F_{v} normalized by ord_{v}(ϖ_{v}) = 1; O_{v} = {x ∈ F_{v} | ord_{v}(x) ≥ 0} is the ring of integers in F_{v}; 𝔭_{v} = ϖ_{v}O_{v} is the maximal ideal of O_{v}; k_{v} is the residue field O_{v}/𝔭_{v}; deg(v) = [k_{v} : 𝔽_{q}]; q_{v} = q^{deg(v)} is the order of the finite field k_{v}; ζ_{v}(s) = (1 − q_{v}^{−s})^{−1}, s ∈ ℂ, is the zeta function of F_{v}; ζ_{F} (s) = ∏_{v} ζ_{v}(s) is the zeta function of F; 𝔸 is the ring of adeles of F; 𝔸^{×} is the group of ideles of F; G = GL(2); B is the Borel subgroup of upper-triangular matrices of G; Z is the center of G; K_{v} = G(O_{v}) is the standard maximal compact subgroup of G(F_{v});

K_{0}(𝔫_{v}) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ K_{v} | c ∈ 𝔫_{v}\}, where 𝔫_{v} is an ideal of O_{v}, is the Hecke con-

gruence subgroup of K_{v} of level 𝔫_{v}; K = ∏_{v} G(O_{v}) is the standard maximal compact subgroup of G(𝔸); 𝔫 = ∏_{v} 𝔫_{v} = ∏_{v} 𝔭_{v}^{n_{v}}, n_{v} ∈ ℤ and almost all n_{v} are equal to 0, is a Weil divisor on C written multiplicatively; deg(𝔫) = ∑_{v} n_{v} deg(v); N (𝔫) = q^{deg 𝔫}; K_{0}(𝔫) = ∏_{v} K_{0}(𝔫_{v}), 𝔫 is a non-negative Weil divisor, is the Hecke congruence subgroup of K of level 𝔫;

By a quasi-character χ of a group H we mean a homomorphism χ : H → ℂ^{×}. A character is a unitary quasi-character. If H is a topological group we will understand that (quasi-)characters are required to be continuous (with ℂ^{×} given its usual topology).

We specify Haar measures on several important groups. The local Haar measure dx_{v} of F_{v} will be normalized so that ∫_{O_{v}} dx_{v} = 1. We will likewise normalize the Haar measure d^{×}x_{v} on F_{v}^{×} by the condition ∫_{O_{v}^{×}} d^{×}x_{v} = 1; explicitly d^{×}x_{v} =

(1 − \frac{1}{q_{v}} )^{−1} \frac{dx_{v}}{|x_{v}|_{v}}. The measures which we will use on the adeles 𝔸 and ideles 𝔸^{×} will be the restricted product measures dx = ∏_{v} dx_{v} and d^{×}x = ∏_{v} d^{×}x_{v}.

Now we can normalize a Haar measure on G(F_{v}). Indeed, G is reductive and hence unimodular, and by the Iwasawa decomposition G(F_{v}) = B(F_{v})K_{v}. Since K_{v} is compact, and hence unimodular, we normalize the Haar measure dk_{v} to give total volume 1. On the other hand, B = (Z × T_{1}) ⋉ N as algebraic groups. Here T_{1} denotes the subgroup of diagonal matrices whose lower right entry is 1 and N the unipotent upper-triangular group. Since Z ≅ T_{1} ≅ 𝔾_{m}, on F_{v}-points we take dz_{v} and dt_{v} to be the transfer of the measure d^{×}x_{v} from F_{v}^{×} with previous normalization. On F_{v}-points of N ≅ 𝔾_{a} we take the transfer measure dn_{v} from dx_{v}, again with previous normalization. Now as one easily checks, cf. [2, p.426], d_{L}b_{v} = dz_{v}dt_{v}dn_{v} is a left Haar measure on B(F_{v}). Finally on G(F_{v}) we take the measure dg_{v} = dk_{v}d_{L}b_{v}, and on G(𝔸) we take the restricted product measure ∏_{v} dg_{v}.

Let H be a locally compact Hausdorff topological group with right Haar measure μ, and let Γ be a discrete subgroup of H. The canonical mapping φ : H → H/Γ is a local homeomorphism. It is easily seen that there is a unique measure μ′ on H/Γ such that, whenever X is a measurable subset of H which is mapped by φ bijectively onto its image X′ = φ(X) in H/Γ, μ′(X′) = μ(X). We will call μ′ the push-forward measure of μ.

4.2. Eisenstein series. For s ∈ ℂ, let δ_{v}(b, s) be the representation of B(F_{v}) given by its modular quasi-character

δ_{v} ((\begin{matrix}y_{1} & x \\ 0 & y_{2} \end{matrix}) , s) = |\frac{y_{1}}{y_{2}} |_{v}^{s} .

We consider the induced representation I_{v}(s) = Ind_{B(F_{v})}^{G(F_{v})}δ_{v}(s) consisting of locally constant functions Φ_{v}(g, s) on G(F_{v}) such that

(4.1)  Φ_{v}(bg, s) = δ_{v}(b, s)Φ_{v}(g, s)

for b ∈ B(F_{v}) and g ∈ G(F_{v}). Note that such functions automatically have compact support modulo B, and that Z acts trivially in this representation. Moreover, by the Iwasawa decomposition G(F_{v}) = B(F_{v})K_{v} any Φ_{v}(g, s) is uniquely determined by its restriction to K_{v}. For an ideal 𝔫_{v} of O_{v} we will denote Φ_{𝔫_{v}} the unique vector whose restriction to K_{v} is the characteristic function of K_{0}(𝔫_{v}) ⊆ K_{v}. This is well-defined since δ_{v} = 1 on B(F_{v}) ∩ K_{v} = B(F_{v}) ∩ K_{0}(𝔫_{v}).

Now we consider the global induced representation I(s) = ⊗_{v}I_{v}(s) of G(𝔸), consisting of smooth, right K-finite functions on G(𝔸) which satisfy the global analogue of (4.1). For an entire section Φ(g, s) ∈ I(s), we have the Eisenstein series

(4.2)  E(g, s, Φ) = ∑_{γ∈B(F)\G(F)} Φ(γg, s),

which is absolutely convergent for Re(s) > 1. As is well-known, this series has a meromorphic continuation to the whole s-plane with a simple pole at s = 1. Given a non-negative divisor 𝔫 on C, we will denote by Φ_{𝔫}(g, s) the factorizable section ⊗_{v}Φ_{𝔫_{v}}(g_{v}, s). We allow 𝔫 to be equal to 1 in which case Φ_{𝔫} restricted to K is the constant function 1.

For a finite set S of places of F define

D_{S}(s) = ∏_{v∈S} ζ_{v}(2s)/ζ_{v}(s).

If S is empty we put D_{S}(s) = 1.

Lemma 4.1. Given a non-negative divisor 𝔫 on C, let S be the set of places in the support of 𝔫. Then

Res_{s=1} E(g, s, Φ_{𝔫}) = \frac{D_{S}(1)}{N (𝔫)} \frac{1}{ζ_{F} (2)} Res_{s=1} ζ_{F} (s).

Proof. According to Prop. 3.7.5 in [2] any E(g, s, Φ) has a simple pole at s = 1 and the residue is independent of g. Moreover, as follows from the proof of that proposition the residue in question is equal to

Res_{s=1} ∫_{𝔸} Φ ((\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) , s) dx.

We compute this latter quantity for Φ_{𝔫}. Since Φ_{𝔫} is factorizable, we have

∫_{𝔸} Φ_{𝔫} ((\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}1 & x \\ 0 & 1 \end{matrix}) , s) dx = ∏_{v} ∫_{F_{v}} Φ_{𝔫_{v}} ((\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}1 & x_{v} \\ 0 & 1 \end{matrix}) , s) dx_{v}.

So it is enough to compute the local integrals. First consider a place where 𝔫_{v} = 1 (which is the case for almost all v). When ord_{v}(x_{v}) ≥ 0 the expression in the local integral is equal to 1 by definition. On the other hand, if ord_{v}(x_{v}) < 0 then the integrand is equal to |x_{v}|_{v}^{−2s} since

(\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}1 & x_{v} \\ 0 & 1 \end{matrix}) = (\begin{matrix}x_{v}^{−1} & −1 \\ 0 & x_{v} \end{matrix}) (\begin{matrix}1 & 0 \\ x_{v}^{−1} & 1 \end{matrix}) .

The volume of 𝔭_{v}^{−m} − 𝔭_{v}^{−(m−1)}, m > 1, is equal to q_{v}^{m}(1 − q_{v}^{−1}), as we normalized the local Haar measure by ∫_{O_{v}} dx_{v} = 1. Finally,

∫_{F_{v}} Φ_{𝔫_{v}} ((\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}1 & x_{v} \\ 0 & 1 \end{matrix}) , s) dx_{v} = 1 + ∑_{m=1}^{∞} \frac{1}{q_{v}^{2ms}} q_{v}^{m}(1 − q_{v}^{−1})

= \frac{ζ_{v}(2s − 1)}{ζ_{v}(2s)}.

A similar calculation for the case ord_{v}(𝔫_{v}) > 0 gives

∫_{F_{v}} Φ_{𝔫_{v}} ((\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) (\begin{matrix}1 & x_{v} \\ 0 & 1 \end{matrix}) , s) dx_{v} = (1 − q_{v}^{−1})\frac{ζ_{v}(2s − 1)}{q_{v}^{ord_{v}(𝔫_{v})(2s−1)}} .

From these two calculations the claim of the lemma easily follows. □

It is perhaps instructive to describe the Eisenstein series E(g, s, Φ) more explicitly as functions on Bruhat-Tits trees and to compare them with the series considered in [8] and [18]. We give one example.

Example 4.2. Let A = 𝔽_{q}[t] be the ring of polynomials with coefficients in 𝔽_{q}, F = 𝔽_{q}(t) be the fraction field of A, and ∞ = 1/t be the place of F associated to the degree of polynomials as a valuation on A. It is clear that F is the function field of ℙ_{𝔽_{q}}^{1}. Let 𝔫 be a non-negative divisor with support in Spec(A). Denote by K_{f} the restricted product ∏_{v≠∞} K_{v}; similarly denote K_{0,f}(𝔫) = ∏_{v≠∞} K_{0}(𝔫_{v}). Let Γ =

G(A) = G(F) ∩ K_{f}, and Γ_{0}(𝔫) = \{(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ | c ≡ 0 mod 𝔫\} = G(F) ∩ K_{0,f}(𝔫).

As det : K_{0,f}(𝔫) → O_{f}^{×} is surjective, by the strong approximation theorem,

G(𝔸) = G(F)G(F_{∞})K_{0,f}(𝔫).

Since E(g, s, Φ_{𝔫}) is left G(F) and right K_{0}(𝔫) invariant, it suffices to evaluate it for g ∈ G(F_{∞}), which we view as an element of G(𝔸) with trivial components at all

places except for ∞. Observe that B(F)\G(F) ≅ Δ\Γ, where Δ = \{(\begin{matrix}a & b \\ 0 & d \end{matrix}) ∈ Γ\}.

Let γ ∈ Γ. Then Φ_{𝔫}(γg, s) = (∏_{v≠∞} Φ_{𝔫_{v}}(γ, s)) · Φ_{∞}(γg, s) is equal to Φ_{∞}(γg, s) or 0 depending on whether γ ∈ Γ_{0}(𝔫) or not. So

E(g, s, Φ_{𝔫}) = ∑_{γ∈Δ\Γ_{0}(𝔫)} Φ_{∞}(γg, s).

It remains to make Φ_{∞} explicit. Since Φ_{∞} in invariant under the right action of Z(F_{∞})K_{∞}, we may regard it as a function on

G(F_{∞})/Z(F_{∞})K_{∞} ≅ B(F_{∞})/(B(F_{∞}) ∩ K_{∞})Z(F_{∞}).

We easily see that each matrix in this set is represented by a unique matrix of the

form (\begin{matrix}ϖ_{∞}^{k} & u \\ 0 & 1 \end{matrix}), where k ∈ ℤ, and u ∈ F_{∞} mod 𝔭_{∞}^{k}. By definition

Φ_{∞} ((\begin{matrix}ϖ_{∞}^{k} & u \\ 0 & 1 \end{matrix}) , s) = q^{−ks}.

From this point of view E(g, s, Φ_{𝔫}) is a function on the vertices of the Bruhat-Tits tree of PGL_{2}(F_{∞}), an infinite tree in which every vertex is adjacent to exactly q +1 other vertices.

4.3. Rankin-Selberg integral. Let (π, V) be an infinite-dimensional irreducible cuspidal representation of G(𝔸) with central character ω. Define the conductor 𝔫_{π} of π as in [3]; see also [11, Thm. 4.24]. Since π is almost everywhere unramified, 𝔫_{π} is a non-negative Weil devisor on C. The local L-functions of π are well-understood, cf. [2, §4.7]. If (π_{v}, V_{v}) = π(χ_{1}, χ_{2}) is an irreducible principal series representation then

L(s, π_{v}) = (1 − α_{1}q_{v}^{−s})^{−1}(1 − α_{2}q_{v}^{−s})^{−1},

where α_{i} = χ_{i}(ϖ_{v}) if χ_{i} is unramified and α_{i} = 0 otherwise. If (π_{v}, V_{v}) = σ(χ_{1}, χ_{2}), where (χ_{1}χ_{2}^{−1})(y) = |y|_{v}^{−1}, is a special representation then

L(s, π_{v}) = (1 − α_{2}q_{v}^{−s})^{−1},

with α_{2} as above. Finally, if π_{v} is supercuspidal then L(s, π_{v}) = 1.

If π_{v} ≅ π(χ_{1}, χ_{2}) is a principal series representation of G(F_{v}) with unramified quasi-characters χ_{i} on F_{v}^{×} then by the Ramanujan conjecture, proved for GL(2) over function fields by Drinfeld, α_{1,v} and α_{2,v} are complex conjugates of each other with absolute value 1.

Write L(s, π_{v}) = (1 − α_{1,v}q_{v}^{−s})^{−1}(1 − α_{2,v}q_{v}^{−s})^{−1} uniformly for all types of local representations π_{v} (so α_{1,v} = 0 if π_{v} is special and α_{1,v} = α_{2,v} = 0 if it is supercuspidal). Define

L(s, Sym^{2}π_{v}) := (1 − α_{1,v}^{2}q_{v}^{−s})^{−1}(1 − α_{1,v}α_{2,v}q_{v}^{−s})^{−1}(1 − α_{2,v}^{2}q_{v}^{−s})^{−1}

and

L(s, Sym^{2}π) := ∏_{v} L(s, Sym^{2}π_{v}).

Let ϕ ∈ V be a newform attached to π. The automorphic form ϕ is unique up to a scalar [3], and we will always assume that it is normalized by requiring that the local Whittaker functions W_{ϕ,v}(1) = 1 at all places v (we implicitly fix a non-trivial additive character ψ of the compact group 𝔸/F). Such ϕ is well-defined. Let ϕ̄ be the complex conjugate of the newform ϕ. The product ϕ(g)ϕ̄(g)E(g, s, Φ_{𝔫_{π}}) is invariant under Z(𝔸)G(F) acting on the left and under K_{0}(𝔫_{π}) acting on the right.

Assume that the conductor 𝔫_{π} is square-free, and let S be the set of places in the support of 𝔫_{π} (that is, the set of places where π is ramified). The basic formula, which follows from the Rankin-Selberg unfolding method, is

∫_{Y(𝔫_{π})}ϕ(g)ϕ̄(g)E_{𝔫_{π}}(g, s)dg = q^{(g−1)s}D_{S}(s)\frac{ζ_{F} (s)}{ζ_{F} (2s)}L(s, Sym^{2}π), (4.3)

where Y(𝔫_{π}) := Z(𝔸)G(F) \ G(𝔸)/K_{0}(𝔫_{π}) and g on the right hand-side is the genus of C. This relation can be proved by essentially repeating the calculation in [2, §3.8] and taking special care of the ramified places. We omit the details of these calculations. Denote

(ϕ, ϕ) := ∫_{Y(𝔫_{π})} ϕ(g)ϕ̄(g)dg.

Taking the residues at s = 1 on both sides of (4.3) and using Lemma 4.1, we get

(4.4)  (ϕ, ϕ) = q^{g−1}N (𝔫_{π})L(1, Sym^{2}π).

Remark 4.3. The assumption on 𝔫_{π} being square-free is not crucial for computing the Rankin-Selberg integral (although it does simplify the calculation). We make this assumption mostly for L(s, Sym^{2}π) to have a functional equation; this is used in §4.4. If we let ρ be the 2-dimensional ℓ-adic representation of Gal(F^{sep}/F) associated to π via Langlands correspondence, then, in notations of §4.4, L(s, Sym^{2}π_{v}) = L_{v}(Sym^{2}ρ, s) if and only if (Sym^{2}V_{ℓ})^{I_{v}} = Sym^{2}(V_{ℓ}^{I_{v}}) for all v, where I_{v} is the inertial group at v. This happens, for example, if the conductor is square-free. Even without assuming 𝔫_{π} is square-free, it is possible to deduce a bound on |L(1, Sym^{2}π)| (which is what we are interested in) from a bound on |L(Sym^{2}ρ, 1)| by studying the local factors L_{v}(Sym^{2}ρ, s) at the places where ord_{v}(𝔫_{π}) ≥ 2. Hence the assumption on square-freeness can be completely avoided at the expense of making the argument somewhat longer.

Let W_{sp}(𝔫) be the space of automorphic cusp forms whose corresponding representations have conductor 𝔫 · ∞, for some ideal 𝔫 in A, and are special at ∞. This space is finite dimensional. Let K_{0,f}(𝔫) = ∏_{v≠∞} K_{0}(𝔫_{v}), and let 𝔸_{f} be the finite adeles, i.e., 𝔸 = 𝔸_{f} · F_{∞}. The strong approximation theorem implies that the determinant map induces a bijection

(4.5)  G(F) \ G(𝔸_{f})/K_{f,0}(𝔫) ⟶^{∼} Pic(A).

Choose a system of representatives ℵ for this finite set. Let

Γ_{x} := G(F) ∩ xK_{0,f}(𝔫)x^{−1}

be the intersection in G(𝔸_{f}). It is not hard to show, cf. [10, (4.5.4)], that

Z(F_{∞})G(F) \ G(𝔸)/K_{0}(𝔫 · ∞) ≅ ∐_{x∈ℵ} Z(F_{∞})Γ_{x} \ G(F_{∞})/K_{0}(𝔭_{∞})

= ∐_{x∈ℵ} Ed(Γ_{x} \ T ).

One can think of W_{sp}(𝔫) as the space of ℂ-valued functions on the above double coset set which satisfy a cuspidality condition, cf. [10, (4.4)]. The following key fact is due to Drinfeld [10, (4.7.6)]

(4.6)  W_{sp}(𝔫) ≅ ⨁_{x∈ℵ} \underline{H}_{!}(T , ℂ)^{Γ_{x}}.

Now let E be an elliptic curve over F of conductor 𝔫_{E} := 𝔫 · ∞. Assume the reduction of E at ∞ is split multiplicative. Let π_{E} be the cuspidal representation corresponding to E. The conductor of π_{E} is 𝔫_{E}, and (π_{E})_{∞} is the special representation. Let ϕ_{E} be the (normalized) newform attached to π_{E}. Then ϕ_{E} ∈ W_{sp}(𝔫) and is an eigenform for the Hecke algebra. It is conventional to normalize the Langlands correspondence in such a way that π_{E} has trivial central character, so ϕ_{E} is invariant under the action of Z(𝔸); see [10, (8.2.4)]. Using (4.6), we can write ϕ_{E} = ∑_{x∈ℵ} ϕ_{x} with ϕ_{x} ∈ \underline{H}_{!}(T , ℂ)^{Γ_{x}}. In fact, since ϕ_{E} is a newform with rational Hecke eigenvalues, it can be shown that ϕ_{x} ∈ \underline{H}_{!}(T , ℤ)^{Γ_{x}} and all these harmonic cochains are primitive; cf. [10, (9.1)].

We have fixed a Haar measure on G(𝔸) in §4.1 which we denote by μ_{can}. This measure induces a certain push-forward measure on each Γ_{x} \ T which we again denote by μ_{can}. On the other hand, in §3.3 we explicitly described a measure on the edges of the quotient graph, which we denoted by μ_{gr}. Since the Haar measure on G(𝔸) is unique up to a constant multiple, μ_{can} and μ_{gr} differ by a fixed constant. It is easy to see that the Petersson norms (ϕ_{x}, ϕ_{x})_{μ}, x ∈ ℵ, are all equal to each other (with respect to either measure). Moreover, we have the following equality.

Proposition 4.4. Fix some x ∈ ℵ. Then

(ϕ_{E}, ϕ_{E}) = q^{1−g}(ϕ_{x}, ϕ_{x})_{μ_{gr}},

where g is the genus of the base curve C.

Proof. From what was said, we deduce that (ϕ_{E}, ϕ_{E}) and (ϕ_{x}, ϕ_{x})_{μ_{gr}} differ by a non-zero constant which depends possibly on C and ∞ (but is independent of ϕ). To get this constant it is enough to compare the volume of Y(𝔫 · ∞) with respect to μ_{can} to the volume of Ed(Γ_{x} \ T ) with respect to μ_{gr} for some fixed x ∈ ℵ. For simplicity we fix x = 1 and omit it from notation. First we assume Γ = G(A), i.e., 𝔫 = 1. From [21, §II.2.3] we have

(4.7)  Vol_{gr}(Γ \ T ) = (q_{∞} + 1)ζ_{F} (−1).

Let M > 1 be a fixed integer. Define a function on B(𝔸) by

δ_{M} ((\begin{matrix}y_{1} & x \\ 0 & y_{2} \end{matrix}) , s) = \begin{cases}|\frac{y_{1}}{y_{2}} |^{s} & if |\frac{y_{1}}{y_{2}} | ≤ M, \\ 0 & otherwise. \end{cases}

Let Φ_{∞}^{M}(g, s) be the factorizable locally constant function on G(𝔸) whose restriction to K is the characteristic function of K_{0}(∞), and which satisfies

Φ_{∞}^{M}(bg, s) = δ_{M}(b, s)Φ_{∞}^{M}(g, s)

for g ∈ G(𝔸) and b ∈ B(𝔸). One easily checks that this is well-defined. Define a truncated Eisenstein series

E_{∞}^{M}(g, s) = ∑_{γ∈B(F)\G(F)} Φ_{∞}^{M}(γg, s).

We modify the Eisenstein series in this manner to make the integrals below convergent. The Rankin-Selberg method [2, §3.8] yields

∫_{Y(∞)}E_{∞}^{M}(g, s)dg = Vol_{can}(𝔸/F)Vol_{can}(𝔸^{×}/F^{×})\frac{M^{s−1}}{1 − q^{1−s}} .

Since Res_{s=1} E_{∞}^{M}(g, s) = Res_{s=1} E(g, s, Φ_{∞}), taking the residues on both sides of this expression and using Lemma 4.1, we get

\frac{Res_{s=1} ζ_{F} (s)}{(1 + q_{∞})ζ_{F} (2)} · Vol_{can}(Y(∞)) = Vol_{can}(𝔸/F)Vol_{can}(𝔸^{×}/F^{×})\frac{1}{log(q)}.

On the other hand, as is well-known [23]

Vol_{can}(𝔸/F) = q^{g−1} and Vol_{can}(𝔸^{×}/F^{×}) = \frac{#Pic_{C/𝔽_{q}}^{0}(𝔽_{q})}{q − 1}.

Since #Pic_{C/𝔽_{q}}^{0}(𝔽_{q})/(q−1) = log(q)·Res_{s=0} ζ_{F} (s) and the functional equation of ζ_{F} (s)

is ζ_{F} (1 − s) = (q^{1−2s})^{1−g}ζ_{F} (s), we get Vol_{can}(Y(∞)) = q^{1−g}(q_{∞} + 1)ζ_{F} (−1). This proves the proposition for 𝔫 = 1.

For an arbitrary 𝔫, it is not hard to check that

Vol_{can}(Y(𝔫 · ∞)) = [K_{0}(∞) : K_{0}(𝔫 · ∞)]Vol_{can}(Y(∞))

and

Vol_{gr}(Γ_{0}(𝔫) \ T ) = [Γ : Γ_{0}(𝔫)]Vol_{gr}(Γ \ T ).

Since [K_{0}(∞) : K_{0}(𝔫 · ∞)] = [Γ : Γ_{0}(𝔫)], the proposition follows in general. □

Let ι(v_{E}) be as in §3.4. Since we are assuming ι(v_{E}) is primitive, it is equal (up to a sign) to ϕ_{x} with x = 1.

Corollary 4.5. There is the equality

(ι(v_{E}), ι(v_{E}))_{μ_{gr}} = q^{2g−2}N (𝔫_{E})L(1, Sym^{2}π_{E}).

Proof. Follows from (4.4) and Proposition 4.4. □

4.4. Bounds on special values of L-functions. Let π be a cuspidal representation of G(𝔸). Assume π̂ ≅ π, where π̂ is the contragredient of π. (This is the case, for example, when π has trivial central character.) In this subsection we give an upper bound on |L(1, Sym^{2}π)| in terms of the degree of the conductor, which finishes the analytic calculations required in the proof of the main theorem 3.3.

Theorem 4.6. Let g be the genus of the base curve C, let 𝔫 be the conductor of π, and let n ≥ 1. If 𝔫 is square-free and deg 𝔫 > 0 then

|L(s, Sym^{n}π)| ≤ q^{(4g+2)(n+1)+ \frac{n}{2}} N (𝔫)^{\frac{n}{2} (1−Re(s))}(deg 𝔫)^{n+1}

for 0 ≤ Re(s) ≤ 1. In particular,

|L(1, Sym^{n}π)| ≤ q^{(4g+2)(n+1)+ \frac{n}{2}} (deg 𝔫)^{n+1}.

Proof. We only indicate the main steps which go into the proof of this theorem and refer to [17, Thm. A.4] for the details.

It seems to be more convenient here to work with Galois representations rather than with cuspidal representations. Since Lafforgue proved the Langlands correspondence for GL(n) over function fields, this will be an equivalent problem; see also Remark 4.3. Choose a separable closure F^{sep} of F and let G = Gal(F^{sep}/F) be the absolute Galois group of F. Fix a prime ℓ not equal to the characteristic of F. Let ρ : G → GL(V_{ℓ}) be an irreducible ℓ-adic representation of G, where V_{ℓ} is a 2-dimensional vector space over some extension of ℚ_{ℓ}. Assume ρ is self-dual and the determinant quasi-character det(ρ) has finite order. Denote by Sym^{n}ρ the irreducible (n + 1)-dimensional ℓ-adic representation of G obtained from the action of G on the symmetric tensors of V_{ℓ}^{⊗n} via ρ. Let L(Sym^{n}ρ, s) be Grothendieck’s L-function for Sym^{n}ρ.

Drinfeld’s proof of the Langlands conjecture for GL(2) over function fields implies that ρ is pure of weight 0, i.e., for any place v where ρ is unramified, the images of the geometric Frobenius eigenvalues α_{1,v} and α_{2,v} under any embedding of \overline{ℚ}_{ℓ} into ℂ are of absolute value 1. Thus, for Re(s) = 1 + ϵ, with ϵ > 0,

|L_{v}(Sym^{n}ρ, s)| ≤ ζ_{F_{v}}(1 + ϵ)^{n+1}.

The same estimate is also valid at the ramified places by [5, Lem. 1.8.1] (in fact the estimates at the ramified places are even somewhat better). Putting the local factors together, we get |L(Sym^{n}ρ, s)| ≤ ζ_{F} (1 + ϵ)^{n+1} for Re(s) = 1 + ϵ. From Grothendieck’s theory of L-functions of ℓ-adic representations, L(Sym^{n}ρ, s) is an entire function [4, §10]; in fact it is a polynomial in q^{−s}, since we assumed ρ is irreducible. Moreover, as we also assumed ρ to be self-dual, Grothendieck’s functional equation for L(Sym^{n}ρ, s) takes the form

L(Sym^{n}ρ, s) = ε(Sym^{n}ρ, s)L(Sym^{n}ρ, 1 − s),

where ε(Sym^{n}ρ, s) is a monomial in ℂ[q^{−s}]. By restricting ρ to each local Weil group W_{F_{v}} ↪ Gal(F^{sep}/F), we obtain an ℓ-adic representation ρ_{v} of W_{F_{v}}. Next, one carefully studies the ε-factor of Sym^{n}ρ_{v} and obtains a lower bound on its absolute value in terms of the degree of the conductor 𝔫_{v} of ρ_{v}. Here the formulae in [4, §§4-5] are indispensable. These estimates imply a bound on the absolute value of the global ε-factor in terms of deg 𝔫_{ρ}, as by a theorem of Laumon [13, Thm. 0.4] the global ε-factor can be decomposed as a product of local factors. From this bound and the functional equation we get an explicit upper bound on |L(Sym^{n}ρ, 1 − s)| for Re(s) = 1 + ϵ. Since L(Sym^{n}ρ, s) is entire and bounded in vertical strips one can apply Rademacher’s version of Phragmén-Lindelöf theorem, [19, Thm.2], to conclude that in the vertical strip −ϵ ≤ Re(s) ≤ 1 + ϵ

|L(Sym^{n}ρ, s)| ≤ (q^{(g−1)(n+1)}N (𝔫_{ρ})^{\frac{n}{2}} )^{(1+ϵ−Re(s))} ζ_{F} (1 + ϵ)^{n+1}.

Recall that by Weil’s theorem

ζ_{F} (s) = \frac{P (s)}{(1 − q^{−s})(1 − q · q^{−s})},

where P (s) = ∏_{i=1}^{2g}(1 − a_{i}q^{−s}), with |a_{i}| = \sqrt{q}. Now it is easy to deduce the required bounds for L(Sym^{n}ρ, s) by choosing ϵ = (deg 𝔫_{ρ})^{−1}. If we let ρ be the ℓ-adic representation attached to π via Langlands correspondence, then the claim follows for π too as 𝔫_{ρ} = 𝔫_{π}. □

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Department of Mathematics, Stanford University, Stanford, CA 94305

   E-mail address: papikian@math.stanford.edu