Isogenies of elliptic curves over function fields
Richard Griffon

Fabien Pazuki

Abstract – We prove two theorems concerning isogenies of elliptic curves over function fields. The first
one describes the variation of the height of the j-invariant in an isogeny class. The second one is an“isogeny estimate”, providing an explicit bound on the degree of a minimal isogeny between two isogenous
elliptic curves. We also give several corollaries of these two results.
Keywords: Elliptic curves over function fields, Heights, Isogenies, Isogeny estimates.
2020 Math. Subj. Classification: 11G05, 11G35, 11G50, 11R58, 14G17, 14G25, 14G40, 14H52, 14K02.

Introduction
Let k be a perfect field, and C be a smooth projective and geometrically irreducible curve over k. We
write K := k(C) for the function field of C, and fix an algebraic closure K of K.
It is well-known that two elliptic curves defined over K are isomorphic if and only if they have thesame j-invariant. One may wonder, more generally, about arithmetic relations between the j-invariants of
two elliptic curves which are linked by an isogeny of arbitrary degree.
Denoting the Weil height on K by h(.), we prove the following.
Theorem A. Let E1 , E2 be two non-isotrivial elliptic curves defined over K with respective j-invariantsj(E1 ), j(E2 ). Assume that there exists an isogeny ϕ : E1 → E2 , and denote its dual by ϕb : E2 → E1 .
We havedegins (ϕ)
h(j(E2 )) =h(j(E1 )).
(1)degins (ϕ)
bHere degins (ϕ) and degins (ϕ)
b denote the inseparability degrees of ϕ and ϕb (see §4.1 for the definition); if K
has characteristic 0, these should be interpreted as 1.

In the particular case where the field K has characteristic 0, the above result states that isogenies
preserve the height of the j-invariant.
Theorem A should be compared to its analogue for elliptic curves over number fields: inspired byclassical work of Faltings and Silverman, Theorem 1.1 in [Paz19] asserts that the j-invariants of two
elliptic curves E1 , E2 defined over Q which are linked by an isogeny ϕ satisfy:|ht(j(E1 )) − ht(j(E2 ))| ≤ 9.204 + 12 log deg ϕ,(2)where ht(.) is the logarithmic Weil height on Q. Results of Szpiro–Ullmo [SU99] imply that the displayed
upper bound is almost optimal – in the sense that the Weil heights of j-invariants of elliptic curves relatedby an isogeny can actually differ by a quantity of the same order of magnitude as a multiple of log deg ϕ.
This latter statement in turn implies that, given an elliptic curve E/Q without CM, the set

J (E, B) := j(E 0 ) ∈ Q : E 0 /Q is isogenous to E, and ht(j(E 0 )) ≤ B
is finite for all B ≥ 1. Theorem A provides a much tighter control on the variation of the Weil heightof the j-invariant in isogeny classes of elliptic curves over function fields than (2) does for elliptic curves
over number fields. Which leads to the surprising consequence that the sets which are natural analogues
for J (E, B) are infinite in the function field setting (see Proposition 5.10)!
After introducing the relevant objects in sections 1–3 and proving some preliminary results about
isogenies in section 4, we show Theorem A in section 5. Our proof relies on a comparison of modular
and differential heights of elliptic curves, and on a result of Parshin stating that the differential height isinvariant under isogeny of degree coprime to the characteristic. As far as we know (see Remark 5.5), thisis the most direct approach.
We refer to the recent [BPR21] for an analogue of Theorem A for isogenous Drinfeld modules.
1Let us now turn to the second main theorem of this article, which is an isogeny estimate for elliptic
curves over function fields. The result (Theorem 6.1) may be stated as follows:
Theorem B. Let E1 and E2 be two non-isotrivial elliptic curves defined over a function field K ofgenus g, with respective j-invariants j(E1 ), j(E2 ). Assume that E1 and E2 are isogenous. Then there
exists a K-isogeny ϕ0 : E1 → E2 with
degins j(E1 ) degins j(E2 )deg ϕ0 ≤ 49 max{1, g} max,,
degins j(E2 ) degins j(E1 )where degins j(E1 ), degins j(E2 ) are the inseparability degrees of j(E1 ), j(E2 ) respectively (see §1.4 for thedefinition); if K has characteristic 0, they should be interpreted as 1.
Isogeny estimates have attracted a lot of attention in the number field case since the groundbreaking
series of papers of Masser and Wüstholtz [MW90, MW93]: Masser’s and Wüstholz’s results were later
refined by Pellarin [Pel01], and more recently by Gaudron and Rémond [GR14]. Each of these papers
provides the existence of a “small” isogeny between a pair of isogenous elliptic curves (in fact [MW93, GR14]
even deal with general Abelian varieties). Here “small” means that the degree of the isogeny is bounded(at worst) in terms of the height of the elliptic curves and simple invariants of the base field. Theorem B
has the same flavour as these statements.
The proofs in [MW90, MW93, Pel01, GR14] heavily rely on transcendence methods, and uniformisation
at an Archimedean place. In the context of function fields, all places are non-Archimedean and, as far
as the authors know, only few transcendence results are available in positive characteristic. Our proof
therefore follows a different strategy as the above mentioned works: we essentially rely on properties ofmodular curves X0 (N ), the Riemann–Hurwitz formula and an estimate of the genus of X0 (N ). Note that
one could also follow a similar approach relying instead on the gonality of X0 (N ) in place of its genus.
We also note that David and Denis have proved an analogous statement for isogenies between Drinfeld
modules (see [DD99, Théorème 1.3]).
In Theorem B too, the case where K has characteristic 0 offers the most striking result: in this situation,
the degree of ϕ0 can be bounded independently of E1 , E2 . In other words, Theorem B provides a uniform
isogeny estimate for elliptic curves over a function field of characteristic 0. In positive characteristic p,
one can easily see that such an isogeny estimate cannot be uniform: the smallest isogeny between a nonisotrivial elliptic curve E and its p-th Frobenius twist E (p) is indeed the p-th power Frobenius isogeny
Frp : E → E (p) , which has degree p. In that situation, the dependency of Theorem B on the curves E1 , E2
(through the inseparability degree of their j-invariants) is actually best possible.
Theorem B is proved in Section 6 of the paper, where we also establish a few corollaries of this isogeny
estimate. We give, in particular, the following explicit bound on the number of K-isomorphism classes of
elliptic curves in a given K-isogeny class (see Theorem 6.9).
Corollary C. Let E be a non-isotrivial elliptic curve over a function field K of genus g. For any M ≥ 1,
consider the set

EK (E, M ) := {E 0 /K is K-isogenous to E, with degins j(E 0 ) ≤ M } K-isomorphism.
The set EK (E, M ) is finite, and we have
• EK (E, M ) ≤ 74 max{1, g}2 if K has characteristic 0,


log M
4
2
+ 1 if K has characteristic p > 0.
• EK (E, M ) ≤ 7 max{1, g}log p

1. Preliminaries about function fields
Let k be a perfect field of characteristic p ≥ 0, and let C be a smooth projective and geometricallyirreducible curve over k. We let K := k(C) denote the function field of C over k. The field k is thenalgebraically closed in K, and we call it the constant field of K. It is well-known that the (isomorphismclass of the) field K characterises C up to birational equivalence over k (see [Sil09] Chapter II, Remark 2.5
for instance). Any smooth projective curve over k whose function field is K will be called a model for K.
The genus of a model C for K will be called the genus of K.
Any finite extension L of K is also a function field in the above sense. Precisely, there is a finite
extension k 0 /k and a smooth projective and geometrically irreducible curve C 0 /k 0 such that L = k 0 (C 0 ).
The inclusion K ⊂ L induces a morphism C 0 → C between the underlying curves.
We refer to [Ros02] for more details about function fields, in particular to [Ros02, Chapter V].
2

1.1. Absolute values on K. – For a function field K as above, we let MK denote the set of places of K
i.e., equivalence classes of discrete valuations on K. Once a model C of K has been chosen, there is abijection between MK and the set of closed points on C. Given a place v ∈ MK , the residue field kv of K
at v is a finite extension of k: the degree deg v := [kv : k] of this extension will be called the degree of v.
To each place v ∈ MK , one associates an absolute value |.|v on K, defined by |x|v := e−v(x) deg v forall x ∈ K, where v(x) is the order of x at v (so normalised that v(K ∗ ) = Z). It is classical that K thensatisfies a product formula:XYv(x) deg v = 0;|x|v = 1, i.e.,∀x ∈ K ∗ ,
v∈MK

v∈MK

(see [Lan83, Chapter I, §1]). In terms of a model C for K, this identity is a reformulation of the fact thata rational function on C has as many poles as zeros (with multiplicities), see [Ros02, Proposition 5.1].
1.2. Absolute values on finite extensions of K. – Let K be as in the previous subsection, andlet L/K be a finite field extension. The constant field k 0 of L is then a finite extension of k. We let ML
denote the set of places of L. Given a place w ∈ ML , we write v for the place of K lying under w (i.e., v
0is the restriction of w to K ⊂ L). The residue field kw
of L at w is then a finite extension of k 0 , itself a0
finite extension of k; and we let deg w := [kw : k]. (This choice might not be the most common one, butwill avoid some notational complications later on).
We associate to w ∈ ML a normalised absolute value |.|w on L: the normalisation is the one such that|x|w = |x|v for all x ∈ K. The image of L∗ under |.|w is then a subgroup of R>0 which contains as a
subgroup of finite indexthe image of K ∗ under |.|v . That index is the ramification index of w, we denote it0: kv ] the residual degree at w. With our normalisations,by ew := |L∗ |w : |K ∗ |v . We denote by fw := [kw00
notice that deg w = [kw : k] = [kw : kv ] [kv : k] = fw deg v. Finally, write Lw and Kv for the completionsof L at w and of K and v, respectively, and let nw := [Lw : Kv ] denote the local degree. By [Lan83,
Chapter I, Proposition 2.4], we have nw = ew fw . With these definitions at hand, one shows thatnw log |x|w = −w(x) deg w.

∀x ∈ L∗ ,Thus endowed with these absolute values, one can prove that L too satisfies a product formula:YX∀x ∈ L∗ ,|x|nww = 1, i.e.,w(x) deg w = 0.
w∈ML

w∈ML

Q
The key part of the proof is to show that, for any x ∈ L and any v ∈ MK , we have |NL/K (x)|v = w|v |x|w .
The latter identity is usually proved under the assumption that L/K be separable but, as noted in [Lan83],
it remains true without this assumption, provided that v is “well-behaved” in the terminology of Lang. A
general proof may be found in [DGS94, Chapter I, Theorem 5.3], or deduced from [Ros02, Theorem 7.6].
1.3. Weil height on K. – We use the notation introduced above and we also fix an algebraic closure K
of K. Algebraic extensions of K will be viewed as sub-extensions of K. Let P = [x0 : x1 ] ∈ P1 (K) be apoint, and pick a finite extension L/K over which P is rational. We define the relative height of P byX
hL (P ) =nw log max{|x0 |w , |x1 |w },w∈ML

and the absolute logarithmic Weil height of P by the formula:h(P ) :=X
hL (P )
1=
nw log max{|x0 |w , |x1 |w }.
[L : K][L : K]

(1.1)

w∈ML

One may check that this last definition does not depend on the choice of an extension L containing P , noron the choice of homogeneous coordinates for P (see [Lan83, Chapter III, §1] or [Ros02, Proposition 7.7]).
This construction thus defines a height function on P1 (K), which takes values in Q≥0 .
For any f ∈ K, we write h(f ) for the absolute logarithmic Weil height of the point [f : 1] ∈ P1 (K).
Explicitly, for any f ∈ K, we haveh(f ) =

XX
1
1
nw max{0, log |f |w } =deg w max{0, −w(f )},[L : K][L : K]
w∈ML

w∈ML

3

where L is any finite extension of K containing f . Choosing a model C 0 /k 0 for L, one may view an elementf ∈ L∗ as a morphism f : C 0 → P1 defined over k 0 . Write div∞ (f ) ∈ Div(C 0 ) for the divisor of poles ofthat function. By the right-most expression of the previous display, we havedeg div∞ (f )h(f ) =,
[L : K]
which means that hL ([f : 1]) = [L : K]h(f ) equals the degree of f , viewed as a morphism C 0 → P1 .
For a given f ∈ K, note that h(f ) = 0 if and only if f , viewed as a morphism C 0 → P1 , is constant (i.e.,f belongs to an algebraic extension of the constant field k). Indeed, a non-constant morphism C 0 → P1 issurjective so that, in particular, its divisor of poles is a non-zero effective divisor. Hence the height of thismorphism must be positive.
Note that the height h : K → Q≥0 does not necessarily satisfy the Northcott property: unless theconstant field k is finite, the set {f ∈ K : h(f ) = 0} (which equals k) is not finite.
1.4. Inseparability degree. – Let K be a function field with constant field k. We assume in thissubsection that K has positive characteristic p. Recall that the inseparability degree of an element f ∈ K ∗is defined by(1
if f ∈ k,
degins (f ) := / k,
K : k(f ) i if f ∈where K : k(f ) i denotes the inseparability degree of the extension K/k(f ) (which is finite under theassumption that f be non-constant). The inseparability degree of f is a non-negative power of p. If f ise
non-constant, one can prove that degins (f ) = pe where e ≥ 0 is the maximal integer such that f ∈ K p .
Fixing a model C of K, we view f as a morphism f : C → P1 . We may then factor f as the composition
fs ◦ Fq of a (purely inseparable) Frobenius map Fq : C → C (q) for some power q of p, with a separable
map fs : C (q) → P1 . The inseparability degree of f then equals q = deg Fq . We refer to [Sil09, Chapter II,
Corollary 2.12] for a proof.
For completeness, we also note the following easily proved fact. Given a finite extension K 0 /K with
inseparability degree [K 0 : K]i , the inseparability degree of an element f ∈ K viewed as an element of K 0
is equal todegins (f ∈ K 0 ) = degins (f ∈ K) [K 0 : K]i .

2. Preliminaries on reduction of elliptic curves
Let K be a function field (in the sense of section 1) with constant field k. An elliptic curve over K is called
isotrivial if its j-invariant is constant (i.e. is an element of k). After a finite extension of K, an isotrivial
elliptic curve becomes isomorphic to (the base change of) an elliptic curve defined over a finite extensionof k. We will be mostly interested in elliptic curves which are not isotrivial.
For more complete overviews of the arithmetic of elliptic curves over function fields, the reader isreferred to [Ulm11] and [Sil94, Chapter III].
2.1. Minimal discriminant and conductor. – Let E be an elliptic curve over K. Let v ∈ MK be aplace of K. After a suitable change of coordinates, the curve E admits Weierstrass models
y 2 + a1 xy + a3 y = x3 + a2 x2 + a4 x + a6 ,(2.1)

where the coefficients a1 , a2 , a3 , a4 , a6 ∈ K are integral at v i.e., models with v(ai ) ≥ 0 for i ∈ {1, 2, 3, 4, 6}.
Each of these models has a discriminant ∆(a1 , a2 , a3 , a4 , a6 ) ∈ K which, being a polynomial in the ai ’s, isalso integral at v. We write δv ∈ Z≥0 for the minimal value of v(∆(a1 , a2 , a3 , a4 , a6 )) among all modelsof E which are integral at v. A v-integral Weierstrass model (2.1) of E is called minimal integral at v if
the valuation at v of its discriminant equals δv .
Given this collection of local data, we define
Pa global invariant of E: the minimal discriminant of E/K
is the divisor on K defined by ∆min (E/K) = v∈MK δv v.
P
We also recall that the conductor of E/K is the divisor given by N (E/K) =v∈MK fv v, wherefv ∈ Z≥0 is the local conductor of E at v. We refer the reader to [Sil94, Chapter IV, §10] for a detailed
definition of fv .
4

2.2. Good and semi-stable places. – Let E/K be an elliptic curve and v ∈ MK be a place of K. We
say that E has good reduction at v if and only if the reduction modulo v of one/any integral minimal modelof E at v is a smooth curve over the residue field kv of K at v.
More generally, we say that E has semi-stable reduction at v if there exists an integral minimal modelof E at v whose reduction modulo v has at most one double point. This is equivalent to requiring that the
reduction of E at v is either good or multiplicative. The theorem of Kodaira-Néron (see [Sil09, Chapter VII,
§6, Theorem 6.1] for instance) implies that for any elliptic curve E/K and any place v ∈ MK of semi-stablereduction for E, one has v(∆v ) = −v(j), where j denotes the j-invariant of E, and ∆v the discriminantof a minimal integral model of E at v.
An elliptic curve over K is called semi-stable if it has semi-stable reduction at every place of K. For
any elliptic curve E over K, there exists a finite extension K 0 /K such that E ×K K 0 /K 0 is semi-stable.
This statement is the famous semi-stable reduction theorem (see [MB85, Chapter XI]).
2.3. Tate’s uniformisation of elliptic curves with non-integral j-invariants. – In this subsection,
we work in the following setting. Consider a field F which is complete for a non-trivial non-Archimedean
valuation |.|. We review some aspects of Tate’s uniformisation of elliptic curves over F : the reader isreferred to Tate’s beautiful survey [Tat93] for a more in-depth presentation, or to [Sil94, Chapter V, §3-§5]
for an overview in the case where F is a finite extension of Qp .
Let t ∈ F be such that |t| < 1. Consider the curve defined over F byEt :Y 2 + XY = X 3 + c4 (t)X + c6 (t),where c4 , c6 : F → F are certain power series which converge on the disc {z ∈ F : |z| < 1}. Tate provedthat Et is an elliptic curve, whose j-invariant j(Et ) ∈ F is given by a convergent power series in t andsatisfies |j(Et )| = |t|−1 > 1. Furthermore, he has shown that there is an analytic group isomorphism
Et → Gm /tZ (the uniformisation map) which is Galois equivariant. In other words, for any algebraic
extension F 0 of F , there is a group isomorphism Et (F 0 ) ' (F 0 )∗ /tZ . Note that Et has split multiplicative
reduction at the maximal ideal of |.|.
Conversely, let E be an elliptic curve over F whose j-invariant satisfies |j(E)| > 1. Then, there exists
a unique t ∈ F with |t| < 1 such that E is isomorphic to Et , the isomorphism being defined over an atmost quadratic extension of F . See [Sil94, Chapter V, Theorem 5.3].
For t1 , t2 ∈ F such that |t1 | < 1 and |t2 | < 1, let Hom(Et1 , Et2 ) denote the Z-module consisting ofisogenies Et1 → Et2 (defined over F ) together with the constant morphism equal to 0Et2 . The theorempage 16 of [Tat93] states that Hom(Et1 , Et2 ) is in bijection with the set (n1 , n2 ) ∈ Z2 : tn1 1 = tn2 2 . In
particular, for any t ∈ F with |t| < 1, the ring of F -endomorphisms of Et is isomorphic to Z. Indeed, theabove implies that we have
Hom(Et , Et ) ' (n, n), n ∈ Z ,(2.2)
where the correspondence associates the multiplication-by-n map Et → Et to the pair (n, n) ∈ Z2 .
2.4. CM and isotriviality. – Over a number field, it is well known that an elliptic curve with nonintegral j-invariant does not have CM (see [Sil94, Chapter 5, §6, Theorem 6.3]).
The analogue statement also holds over function fields:
Lemma 2.1. Let K be a function field as above, and let E be an elliptic curve over K whose j-invariant
is not constant. Then, the ring End(E) of K-endomorphisms of E is isomorphic to Z. In other words, thecurve E has “no complex multiplication”.
By the work of Serre–Tate [ST68], we know that an elliptic curve defined over a local field and withcomplex multiplication has integral j-invariant, hence has potentially good reduction. From this we inferthat an elliptic curve defined over a function field K, with complex multiplication and with potentiallygood reduction everywhere must have integral j-invariant at all places of K. In particular its j-invariant,having no poles, must be a constant rational map, which means that the curve E is isotrivial. In positive characteristic, the isotriviality is also implied by Deuring’s Theorem (see [Hus04, Chapter 13, §6,
Theorem 6.4]). Let us give an ad hoc proof of this lemma, which is independent of the characteristic of K.
Proof. The j-invariant j(E) being non-constant, there exists a place v of K at which j(E) has a pole (i.e.,
for which v(j(E)) < 0). Let Kv denote the completion of K at v; the field Kv is of the type considered insubsection §2.3. Since v(j(E)) < 0, we know from the results recalled there that there is a unique t ∈ Kv
with v(t) > 0 such that E/Kv becomes isomorphic to the Tate curve Et over a finite extension of Kv .
Through this isomorphism, an endomorphism ψ : E → E induces an endomorphism Ψ : Et → Et . By (2.2),
5

there exists an integer n such that Ψ is the multiplication-by-n map of Et . The original endomorphism
ψ : E → E is thus nothing else but the multiplication map [n] : E → E. Hence the result.

3. Heights of elliptic curves
Let k be a perfect field and C be a smooth projective and geometrically irreducible curve over k. We let
K := k(C) be its function field. We let p ≥ 0 denote the characteristic of K (p is then either 0 or a prime).
3.1. Differential and stable heights. – Let L be a finite field extension of K; we choose a model C 0 of L
and write k 0 for the field of constants of L. For an elliptic curve E over L, its minimal regular model E is
the unique (up to isomorphism) smooth projective and geometrically irreducible surface over k 0 , equippedwith a minimal surjective morphism π : E → C 0 whose generic fiber is E. We denote by π : E → C 0 theminimal regular model of E and s0 : C 0 → E its zero section. We refer to [Ulm11, Lecture 3, §1-§2] formore details about the construction of this model. Let Ω1E/C 0 be the sheaf of relative differential 1-forms
on E. Pulling-back Ω1E/C 0 along the zero section s0 results in a line bundle on C 0 , which will be denotedby ωE/L := s∗0 Ω1E/C 0 . One then defines the differential height of E/L byhdiff (E/L) :=deg ωE/L,[L : K]

where deg here means degree of a line bundle on C 0 . The following identity is well-known:
Lemma 3.1. In the above setting, denote by ∆min (E/L) ∈ Div(C 0 ) the minimal discriminant divisorof E. Then one hasdeg ∆min (E/L)
12 hdiff (E/L) =.
[L : K]
This may be proved just as Proposition 1.1 in [Sil86]: the main point is that ∆min (E/L) provides a⊗12
section of ωE/L
. In contrast to the number field setting (treated in [Sil86]) however, no “Archimedean
terms” come into play in the computation of the degree of the line bundle ωE/L .
One checks that the differential height does not increase in finite extensions, by which we mean that,
if L0 /L is a further finite extension, one has hdiff (E ×L L0 /L0 ) ≤ hdiff (E/L). We refer the reader to [MB85,
Proposition 2.3, page 228] for a proof of this fact for general Abelian varieties.
Given an elliptic curve E over L, as was recalled in §2.2 above, we may find a finite extension L0 /Lsuch that the base-changed curve E ×L L0 /L0 is semi-stable. We then define the stable height of E to be
hst (E/L) := hdiff (E ×L L0 /L0 ) =deg(ωE×L L0 /L0 ).
[L0 : K]

This definition makes sense: it is indeed shown in [MB85, Proposition 2.3, page 228] that the quantity hst (E/L) does not depend on the choice of a particular extension L0 /L over which E attains semi-stablereduction.
3.2. Modular height. – Let E be an elliptic curve defined over K. Its j-invariant j(E) lies in K: we
can then define the modular height of E to behmod (E) := h j(E) ∈ Q≥0 ,where h is the Weil height on K defined in §1.3. This modular height is closely related to the height called
hauteur modulaire numérique defined by Moret-Bailly in [MB85, page 226].
If we fix a finite extension L of K containing j(E), and write L = k 0 (C 0 ) (where k 0 /k is a finite extension
and C 0 is a smooth projective geometrically irreducible curve over k 0 ), we may view j(E) as a morphism
j(E) : C 0 → P1 defined over k 0 . Denoting by div∞ (j(E)) ∈ Div(C 0 ) the divisor of poles of this map, wethen havedeg(div∞ (j(E)))
hmod (E) =.
[L : K](See the discussion in §1.3). It follows that the modular height hmod (E) vanishes if and only if j(E) is aconstant element of K i.e., if and only if E is isotrivial.
6

3.3. Comparison of heights. – We now compare the various notions of heights of an elliptic curve
over K that were just introduced. Let L/K be a finite extension, and let E be an elliptic curve over L.
Proposition 3.2. In the above setting, one has
0 ≤ hdiff (E/L) −X
hmod (E)1≤deg w,
12[L : K] w not s.s.

(3.1)

where the sum is over the (finite) set of places of L where E does not have semi-stable reduction.
The reader might want to compare this estimate with the one in [Sil86, Proposition 2.1] where a similar
comparison is carried out between differential and modular heights of an elliptic curve over a number field.
The proof in our setting is simplified by the absence of Archimedean places.
Proof. For any place w of L, we let δw ≥ 0 denote the valuation at w of the discriminant of a minimal
integral model of E at w, and we let θw = w(j(E)) denote the order of the pole/zero of j(E) at w.
By Lemma 3.1 and the definition of the modular height, we have
1 X
deg ∆min (E/L)δw deg w,=[L : K] hdiff (E/L) =
12
12
w∈ML
X
X
max{0, −θw } deg w,
nw max{0, log |j(E)|w } =[L : K] hmod (E) =
w∈ML

w∈ML

where the sums are supported on the places of bad reduction of E. Note indeed that the j-invariant has
no poles outside places of bad reduction. (Actually, as the table page 365 of [Sil94] shows, poles of j only
occur at places of potentially multiplicative reduction.) Hence we haveX
12[L : K] hdiff (E/L) − [L : K] hmod (E) =min{δw , δw + θw } deg w,
w∈ML

the sum being supported on places where E has bad reduction. If w is a place of semi-stable reduction
for E, then the Kodaira–Néron theorem (see §2.2) implies that δw = −θw . If w is a place of additive
reduction, then a quick perusal at the above mentioned table in [Sil94] shows that δw + θw is contained in{0, . . . , 12}. Therefore, for any place w of bad reduction, the integer min{δw , θw + δw } lies in [0, 12]; andit equals 0 if E has semi-stable reduction at w. We have thus proved thatXX
0 ≤ 12[L : K] hdiff (E/L) − [L : K] hmod (E) =min{δw , δw + θw } deg w ≤ 12
deg w.
w bad red.

w not s.s.

This entails the desired bounds.
It is clear that an elliptic curve E/L is semi-stable if and only if the sum on the right-hand side of (3.1)
vanishes. Hence, the above proposition directly implies:
Corollary 3.3. Let E be an elliptic curve defined over a finite extension L of K. Then E/L is semi-stable
if and only if hmod (E) = 12 hdiff (E/L). In particular, we have hmod (E) = 12 hst (E/L).
3.4. Differential height and conductor. – For any elliptic curve E over a finite extension L of K,
Ogg’s formula (see formula (11.1) in [Sil94, Chapter IV, §11]), implies that w(N (E/L)) ≤ w(∆min (E/L))
for all places w ∈ ML . Hence we have deg N (E/L) ≤ deg ∆min (E/L), and it follows thatdeg N (E/L) ≤ 12[L : K] hdiff (E/L).
Obtaining an inequality in the other direction is much harder. The result is as follows:
Theorem 3.4 (Szpiro’s inequality). Let E be an elliptic curve defined over a function field L of genus g(L).
We havedeg ∆min (E/L) ≤ 6 degins j(E) 2g(L) − 2 + deg N (E/L) ,(3.2)
where N (E/L) denotes the conductor of E, and degins j(E) the inseparability degree of its j-invariant.
The proof in the semi-stable case can be found in [Szp90], and the general case is treated in [PS00]. In
particular, for an elliptic curve E/K, Szpiro’s inequality reads:
deg N (E/K)
deg N (E/K)≤ hdiff (E/K) ≤ degins j(E)+ g(K) − 1 .
(3.3)
12
2
Hence, for elliptic curves over K whose j-invariant is separable, the “numerical conductor” deg N (E/K)
and the differential height hdiff (E/K) have the same order of magnitude (up to constants depending at
most on g(K)).
7

4. Isogenies between elliptic curves
The goal of this section is to recall a few standard facts about isogenies between elliptic curves, as wellas prove decomposition results for them. The reader is referred to [Sil09, Chapter III, §4] for further
details about isogenies. We work in the same setting as before: we let k be a perfect field, C be a smoothprojective and geometrically irreducible curve over k, and K be the function field k(C). We denote thecharacteristic of K by p (whether it is a prime or 0), and fix an algebraic closure K of K. All algebraic
extensions of K are viewed as sub-extensions of K.
4.1. Preliminaries on isogenies. – Let L ⊂ K be a finite extension of K. Let E1 and E2 be two elliptic
curves defined over L. We denote by 0E1 ∈ E1 (L) and 0E2 ∈ E2 (L) the respective neutral elements of thegroups E1 and E2 . An isogeny ϕ : E1 → E2 is a surjective morphism of varieties satisfying ϕ(0E1 ) = 0E2 .
It can then be shown that ϕ is a group morphism E1 → E2 . (Unless otherwise specified, we do not assume
that isogenies are defined over L.) As a morphism between algebraic varieties, ϕ induces a finite embeddingof function fields
ϕ∗ : K(E2 ) ,→ K(E1 ).
The degree of ϕ, denoted by deg ϕ, is defined to be the degree of the finite field extension K(E1 )/ϕ∗ (K(E2 )).
If the characteristic of K is positive, the extension K(E1 )/ϕ∗ (K(E2 )) may be split into two subextensions, as follows: we let Mϕ be the separable closure of ϕ∗ (K(E2 )) in K(E1 ). Then Mϕ /ϕ∗ (K(E2 )) is a
finite separable extension, whose degree is denoted by degsep ϕ and called the separable degree of ϕ. The
degree of the extension K(E1 )/Mϕ is denoted by degins ϕ and is called the inseparable degree of ϕ. It isclear that degins ϕ is a power of the characteristic of K, and that we have deg ϕ = degsep ϕ degins ϕ.
Coming back to the general case, and viewing ϕ : E1 → E2 as a homomorphism of group varieties, we
may define its kernel ker ϕ = ϕ−1 ({0E2 }) ⊂ E1 as a a finite group variety over L. As a group scheme, thekernel of ϕ is reduced if and only if ϕ is separable (in which case we do not lose much by identifying thekernel with its set of closed points). In general, one can check that the finite Abelian group (ker ϕ)(K)
has order degsep ϕ. A separable isogeny is called cyclic if its kernel is a cyclic Abelian group.
If ϕ : E1 → E2 is an isogeny, recall that there is a dual isogeny E2 → E1 , which we denote by ϕ.
b Thecompositions ϕ ◦ ϕb and ϕb ◦ ϕ are equal to the multiplication by deg ϕ on E2 , respectively E1 . The degreesof ϕ and ϕ
b are equal.
We denote the set of all isogenies E1 → E2 together with the constant morphism 0E2 : E1 → E2 by
Hom(E1 , E2 ). This set is naturally endowed with the structure of a Z-module. If E1 and E2 are nonisotrivial and isogenous, one can show that the Z-module Hom(E1 , E2 ) is torsion-free of rank 1 (since they
have no CM, by §2.4); thus, there exists an isogeny ϕ0 : E1 → E2 so that Hom(E1 , E2 ) = Z ϕ0 . Such a ϕ0
is unique up to multiplication by [±1], and is characterised (up to sign) by the fact that it has minimal
degree among all non-zero elements of Hom(E1 , E2 ).
Let us now assume that K has positive characteristic p, and let E be an elliptic curve over L. For
any power q of p, we write E (q) for the q-th power Frobenius twist of E: if E is defined by a Weierstrassmodel with coefficients a1 , a2 , a3 , a4 , a6 ∈ L, then E (q) is the elliptic curve given by the Weierstrass modelwith coefficients aq1 , aq2 , aq3 , aq4 , aq6 (note that j(E (q) ) = j(E)q ). Recall that there is a q-th power Frobenius
morphism Frq : E → E (q) which is defined over L, and that this morphism is an isogeny of degree q. As
such, it admits a dual isogeny Vq : E (q) → E which is called the q-th power Verschiebung isogeny, and isalso defined over L. The multiplication-by-q map [q] : E → E then decomposes as [q] = Vq ◦ Frq .
If E is non-isotrivial, it is known that Frq : E → E (q) is purely inseparable of degree q (that is, wehave degsep Frq = 1 and degins Frq = q) and that its dual Vq : E (q) → E is separable of degree q. Note that,since E is non-isotrivial, it is ordinary in the sense that: E[q] ' Z/qZ for any power q of p, where E[q]denotes the subgroup of q-torsion elements of E(K). These facts follow from [Sil09, Chapter V, §3].
Finally, we recall three results concerning isogenies between elliptic curves.
Proposition 4.1. Let E1 , E2 be two elliptic curves defined over a field L, whose characteristic p is positive,
and let ϕ : E1 → E2 be an isogeny between them. If one writes q = degins ϕ, then the isogeny ϕ factors as(q)(q)
ϕ = ψ ◦ Frq where Frq : E1 → E1 is the q-th power Frobenius isogeny and ψ : E1 → E2 is a separable
isogeny.

8

Proposition 4.2. Let E1 , E2 , E3 be elliptic curves defined over a field L. Let ϕ : E1 → E2 be a separable
isogeny, and ψ : E1 → E3 be an isogeny. If ker ϕ ⊆ ker ψ, then ψ factors uniquely through ϕ i.e., there isa unique isogeny λ : E2 → E3 such that ψ = λ ◦ ϕ.
Proposition 4.3. Let E be an elliptic curve defined over a field L, and let G be a finite subgroup of E.
Then there exist a unique elliptic curve E 0 defined over L and a separable isogeny π : E → E 0 such thatker π = G. The curve E 0 is usually denoted by E/G.
These three statements and their proofs can be found in [Sil09]: see Corollary 2.12 in Chapter II,
Corollary 4.11 in Chapter III, and Proposition 4.12 in Chapter III, respectively.
4.2. Biseparable isogenies. – It will be convenient to consider the following class of isogenies separately:
Definition 4.4. Let ϕ : E → E 0 be an isogeny between two elliptic curves E, E 0 defined over K. We will
say that ϕ is biseparable if both ϕ and its dual ϕb are separable. We will say that two elliptic curves E, E 0
over K are biseparably isogenous if there exists a biseparable isogeny between them.
If K has characteristic 0, all isogenies between elliptic curves over K are of course biseparable. In
positive characteristic, such isogenies may be characterised as follows:
Lemma 4.5. Let E and E 0 be two elliptic curves over K and let ϕ : E → E 0 be an isogeny. Then ϕ isbiseparable if and only if deg ϕ is coprime to the characteristic of K.
Proof. Let p > 0 denote the characteristic of K. We factor the degree d := deg ϕ as d = pr d0 , withr ≥ 0 and d0 ∈ Z coprime to p. By construction of the dual isogeny, we have ϕb ◦ ϕ = [d] = [d0 ] ◦ [pr ],where [n] : E → E denotes the multiplication-by-n map on E. Now, the multiplication-by-pr map [pr ] on
E is inseparable of degree p2r (with inseparability degree degins [pr ] ∈ {pr , p2r } depending on whether E
is ordinary or not); and, since d0 is coprime to p, the map [d0 ] : E → E is separable. All in all, theinseparability degree of the map ϕb ◦ ϕ = [d] : E → E satisfies degins [d] ∈ {pr , p2r }.
Now assume that ϕ is biseparable of degree d = pr d0 . Since both ϕ and ϕ
b are separable, theircomposition is separable. Hence the multiplication-by-d map on E is separable. By the previous paragraph,
we must have r = 0. Therefore, the degree of ϕ is coprime to p.
Conversely, assume that the degree d of ϕ is coprime to p. Then ϕ must be separable because themap [d] : E → E is separable and factors through ϕ. Since the dual of ϕ has degree deg ϕb = d, the same
argument shows that ϕb is also separable.
We also note the following:
Lemma 4.6. Let K be a function field as above, and E1 , E2 be two non-isotrivial elliptic curves overa finite extension L of K. Let ϕ : E1 → E2 be a biseparable isogeny between them. Then the kernel
H = (ker ϕ)(K) of ϕ is defined over a finite separable extension of L, that is, the extension L(H)/L isseparable.
Proof. Let d be the degree of ϕ, [d] : E1 → E1 denote the multiplication-by-[d] map on E1 , and E1 [d] bethe subgroup of E1 (K) formed by d-torsion points. We have H = (ker ϕ)(K) ⊂ (ker[d])(K) = E1 [d], sothat L(H) is a subfield of L(E1 [d]). It thus suffices to prove that the extension L(E1 [d])/L is separable.
We know by Lemma 4.5 that d is coprime to the characteristic of L. It is then known (see [ST68, §1] or[BLV09, Proposition 3.8] for instance) that E1 [d] is contained in E1 (Lsep ), where Lsep denotes the separable
closure of L in K. The extension L(E1 [d])/L is therefore separable, and the lemma is proved.
4.3. Useful decompositions of isogenies. – Let K be a function field in the above sense. In thissubsection, we assume that the characteristic p of K is positive.
Let E1 , E2 be two non-isotrivial elliptic curves defined over K, and let ϕ : E1 → E2 be an isogeny
between them. By Proposition 4.1, one can decompose ϕ as
Frpe(pe ) ψE1 −−−→ E1−→ E2 ,

where Frpe denotes the pe -th power Frobenius isogeny, and ψ is a separable isogeny. Since ψ is separable,(pe )we see that degins ϕ = degins Frpe = pe . Let us now consider the dual isogeny ψb : E2 → E1 to ψ. Using
the same Proposition 4.1, we obtain that ψb factors as
Frpf0(pf ) ψE2 −−−→ E2

(pe )

−→ E1

9

,

where ψ 0 is a separable isogeny, and where pf = degins ϕ.
b We thus have ψb = ψ 0 ◦ Frpf . By contravariancebb0dof taking duals, we deduce that ψ = ψb = Fr
pf ◦ ψ . By definition, the dual of Frpf is the Verschiebung(pf )isogeny Vpf : E2 → E2 . Since E2 is non-isotrivial, Vpf is a separable isogeny of degree pf . The isogeny
ϕs := ψb0 is separable, because both ψ = Vpf ◦ ϕs and Vpf are (by multiplicativity of the inseparability
degree in compositions). In particular, ϕs is a separable isogeny whose dual ϕ
cs = ψ 0 is also separable.
We have therefore decomposed our original isogeny ϕ : E1 → E2 as a compositionFrpe(pf ) Vpf

(pe ) ϕsE1 −−−→ E1−→ E2−−→ E2 ,where pe = degins ϕ and pf = degins ϕ,b and where ϕs is biseparable.
This discussion proves the following decomposition result:
Proposition 4.7. Let E1 , E2 be two non-isotrivial isogenous elliptic curves over K, and let ϕ : E1 → E2
be an isogeny between them. The isogeny ϕ factors as
Frpe

(pe )

E1 −−−−→ E1

(pf )

ψ

−−→ E2

Vpf−−−−→ E2 ,where ψ is a biseparable isogeny, pe = degins ϕ and pf = degins ϕ.
b
The above statement will be used repeatedly in the remainder of the article. Let us also prove now thefollowing factorisation result:
Proposition 4.8. Let ϕ : E1 → E2 be an isogeny between two non-isotrivial elliptic curves over K, andwrite q := min{degins ϕ, degins ϕ}.
b Then the isogeny ϕ factors through the multiplication-by-q map [q].
Proof. Up to replacing ϕ by its dual and exchanging the roles of E1 , E2 , we may assume that q = degins ϕ.
It suffices to treat this case because ϕ factors through [q] if and only if ϕb does. We begin by decomposing ϕ(pe )as in Proposition 4.7 above: there is a biseparable isogeny ψ : E1

(pf )

→ E2

such thatϕ = Vpf ◦ ψ ◦ Frpe ,where degins ϕ = pe and degins ϕb = pf , and where the degree m of ψ is coprime to p. By assumption weefhave q = p ≤ p . Note that deg ϕ = pe mpf = q 2 pd m where d := f − e ≥ 0, and that degsep ϕ = mpf .
(pd )We note that it is enough to treat the case where e = f . Consider indeed the isogeny ϕ0 : E1 → E2
defined by Vpe ◦ ψ ◦ Frpe : if ϕ0 factors through [q] then so does ϕ, because ϕ = Vpd ◦ ϕ0 . We furthermore
note that the result is immediate if f = 0, we thus assume that f ≥ 1. This reduces the problem to thecase where e = f ≥ 1 (i.e. d = 0 and ϕ = ϕ0 ), which is now assumed for the rest of the proof.
The set G := (ker ϕ)(K) of K-rational points in the kernel of ϕ is a finite Abelian group of orderdegsep ϕ = mpf . (Note that, as a group scheme, the kernel ker ϕ is not reduced.) Since pf divides |G|,
G contains a unique subgroup H of order pf (which consists of its elements of order a power of p, since m
is coprime to p). By Proposition 4.3, there exist a unique elliptic curve E10 over K and a separable isogeny
π : E1 → E10 with kernel H. The isogeny π has degree deg π = degsep π = pf .
By construction, H = (ker π)(K) is a subgroup of G and, π being separable, we deduce from Proposition 4.2 that ϕ : E1 → E2 factors through π: there exists an isogeny λ : E10 → E2 such that ϕ = λ ◦ π. It
follows from the multiplicativity of degrees that degsep λ = m and degins λ = pe . We use Proposition 4.7 todecompose π. Since π is separable of degree a power of p, we obtain that the diagram
E1

π

ι

E10
Vpf

f
(E10 )(p )is commutative, where ι is a biseparable isogeny. Comparing degrees, we observe that ι has degree 1 andfmust therefore be an isomorphism E1 ' (E10 )(p ) .
Let us apply Proposition 4.7 once more, this time to factor λ. Since the separability degree of λ isb is separable. Therefore, there is a biseparable isogeny γ : (E 0 )(pe ) → E2
coprime to p, we deduce that λ1such that λ = γ ◦ Frpe . Here is a diagram summarising the various isogenies considered here:

10

E1
'

ι
f

(E10 )(p )

Vpf

ϕ

E2
φ

π

λ

γ

E10

Frpe

(E10 )(p )

e

'
ι

E1 .

[pe ]

In this diagram, the triangles are commutative (recall that we assume e = f ).
We now set φ := γ ◦ ι. The composition φ : E1 → E2 is an isogeny between E1 and E2 , and we haveϕ = φ ◦bι ◦ Frpe ◦ Vpe ◦ ι. But we know that Frpe ◦ Vpe = [pe ] = [q], and we have proved that bι ◦ ι = id. Thisshows that ϕ = φ ◦ [q] = [q] ◦ φ. Hence, ϕ factors through [q] in the case at hand, and thus, in general.
Remark 4.9. Let E1 , E2 be two non-isotrivial isogenous elliptic curves. As was mentioned in §4.1, theZ-module Hom(E1 , E2 ) is free of rank 1: we may thus fix an isogeny ϕ0 : E1 → E2 which generates
Hom(E1 , E2 ). This generator ϕ0 has minimal degree among all non-zero elements in Hom(E1 , E2 ). By theabove proposition and the minimality of deg ϕ0 , one has min{degins ϕ0 , degins ϕc0 } = 1.
This means that at least one of ϕ0 or ϕc0 is separable.

5. Isogenies and heights
Let k be a perfect field of characteristic p ≥ 0. Let C be a smooth projective geometrically irreducible
curve over k, and K = k(C) denote its function field. We fix an algebraic closure K of K.
The goal of this section is to describe the effect of an isogeny between elliptic curves over K on theheight of their j-invariants: we prove Theorem A (see §5.3), as well as state a few consequences thereof.
The general idea is that “biseparable isogenies preserve the modular height”. If K has characteristic 0,
this will directly lead to the desired result. In positive characteristic p, more work is required.
5.1. Biseparable isogenies preserve the differential height. – We begin by recalling the following
result due to Parshin (see [Par70, Par73] for instance) as well as giving a proof thereof, for the convenienceof the reader. The reader is referred to [BLR90] (especially §7.3 there) and [Ray85] for more details about
Néron models and isogenies.
Theorem 5.1. Let E1 , E2 be non-isotrivial elliptic curves over a finite extension L of K. Assume that
there exists a biseparable isogeny ϕ : E1 → E2 . Then, we have
hdiff (E1 /L) = hdiff (E2 /L).
Proof. We fix a model C 0 /k 0 of L. For i = 1, 2, we denote the Néron model of Ei /L by πi : Ei → C 0 ,and we write si : C 0 → Ei for its zero-section. The surface Ei is smooth over C 0 : let Ω1Ei /C 0 denote thesheaf of relative differential 1-forms on Ei /C 0 , and consider the line bundle ωi := s∗i Ω1Ei /C 0 on C 0 . Recall
from §3.1 that the differential height of Ei /L essentially equals deg ωi . To prove the theorem, it is (morethan) sufficient to show that ω1 ' ω2 as line bundles on C 0 . The given isogeny ϕ : E1 → E2 extends intoa group morphism Φ : E1 → E2 which is still an isogeny, which means that Φ is, fiber by fiber, finite andsurjective on the identity components. The morphism Φ induces a map Φ∗ Ω1E2 /C 0 → Ω1E1 /C 0 , which we
may restrict to the zero-section. Using that Φ satisfies Φ ◦ s1 = s2 , we obtain a map of OC 0 -modules
F : ω2 = s∗2 Ω1E2 /C 0 ' s∗1 Φ∗ Ω1E2 /C 0 −→ s∗1 Ω1E1 /C 0 = ω1 .
Is suffices to show that F is an isomorphism: to do so, we may argue locally on C 0 . We thus set out toshow that, for any closed point w of C 0 , the restriction of F to the fibers ω1,w and ω2,w of ω1 and ω2
above w is an isomorphism.
Let w be a closed point of C 0 . Write Ow for the local ring of C 0 at w and S := Spec Ow . We also let kw
denote the residue field at w (note that kw is a finite extension of k 0 , and thus has the same characteristic
as k 0 ). Denoting the fiber of Ei at w by Ei,w := Ei ×C 0 Spec Ow , the restriction ϕw : E1,w → E2,w of Φ is an
isogeny. It is then known (see [BLR90, §7.3, Lemma 5]) that there exists a dual isogeny ϕ
bw : E2,w → E1,w ,such that ϕ
bw ◦ ϕw = [d] with d = deg ϕ. Since d is coprime to the characteristic of kw , all three of ϕw , ϕbw
and [d] are étale by [Ray85, §1.1, Exemples 1.1.2], or [BLR90, §7.3, Lemma 2].
The isogeny ϕw induces a canonical map ϕ∗w Ω1E2,w /S → Ω1E1,w /S which, since ϕw is étale on E1,w/S , isactually an isomorphism (see [BLR90, §2.2, Corollary 10]). Moreover, ϕw being a group morphism, we
11have ϕw ◦ s1 |E1,w = s2 |E2,w . Pulling back the above isomorphism along (the restriction of) s1 , we obtainan isomorphism of Ow -modules'

ω2,w = s∗2 Ω1E2,w /S ' s∗1 ϕ∗w Ω1E2,w /S −−−→ s∗1 Ω1E1,w /S = ω1,w ,which is the restriction of F to the fibers above w. This concludes the proof.
5.2. Biseparable isogenies preserve the inseparability degree of the j-invariants. – In thissubsection, we assume that the function field K has positive characteristic p. We describe the effect of abiseparable isogeny on the inseparability degree of the j-invariants.
Proposition 5.2. Let E1 , E2 be two non-isotrivial elliptic curves over a finite extension L of K. If there
exists a biseparable isogeny E1 → E2 , then the inseparability degrees of j(E1 ) and j(E2 ) are equal.
The proof below is adapted from [BLV09, 2.2] where it is proven that, if E1 → E2 is a biseparable
isogeny and if j(E1 ) ∈ K is separable, then j(E2 ) is separable too.
Proof. Fix a model C 0 /k 0 for L. We write j1 , j2 to denote the j-invariants of E1 and E2 respectively, andlet ϕ : E1 → E2 be a biseparable isogeny, whose degree is denoted by d (by Lemma 4.5, d is thus coprimeto p). Choose an elliptic curve E20 defined over k 0 (j2 ) whose j-invariant is j2 : by construction, E2 and E20
are isomorphic over K. Actually, E2 and E20 become isomorphic over a finite separable extension L0 /L ofdegree ≤ 24, see [Sil09, Chapter III, Proposition 1.4]. Moreover one can assume that the field of constantsof L0 is k 0 . We may and do base change the situation to L0 , so that E2 ' E20 over L0 = L.
For the convenience of the reader, here is a diagram of the field extensions we are going to consider:
all these extensions are finite, we have indicated some of the inseparability degrees, and the dashed lines
denote extensions which are separable (see below).
L
L1 (H)

di =degins (j1 )di =degins (j2 )k 0 (j1 , j2 )L1 = k 0 (j1 )k 0 (j2 )Let L1 = k 0 (j1 ) ⊂ L. We view ϕ as an isogeny ϕ : E1 → E20 . By Lemma 4.6, the group H := (ker ϕ)(K)of K-rational points in the kernel of ϕ is then defined over a separable extension of L: the (finite) extension
L1 (H)/L1 obtained by adjoining to L1 the coordinates of points of H is therefore separable.
Because the curves E1 and E20 are linked by the isogeny ϕ, we have E20 ' E1 /H. In particular, E2 is isomorphic to an elliptic curve defined over L1 (H), so that j2 ∈ L1 (H). This implies that L1 (j2 ) = k 0 (j1 , j2 ),being a field sub-extension of L1 (H)/L1 , is a finite separable extension of L1 . Hence the extension L/L1
has the same inseparability degree as the extension L/k 0 (j1 , j2 ):degins (j1 ) = [L : L1 ]i = [L : k 0 (j1 , j2 )]i .
On the other hand, we have a chain of finite extensions k 0 (j2 ) ⊂ k 0 (j1 , j2 ) ⊂ L. Therefore, [L : k 0 (j1 , j2 )]i
divides [L : k 0 (j2 )]i = degins (j2 ). We thus conclude that [L : k 0 (j1 )]i = degins (j1 ) divides degins (j2 ).
Applying the same argument to the dual isogeny ϕb : E20 → E1 (which is also biseparable), we conclude
that degins (j2 ) divides degins (j1 ). These two quantities are therefore equal.
Making use of Proposition 4.7, we now describe the effect of an arbitrary isogeny on the inseparability
degrees of the j-invariants.
Corollary 5.3. Let E1 , E2 be two non-isotrivial elliptic curves defined over a finite extension L of K.
Assume that there exists an isogeny ϕ : E1 → E2 . Then we havedegins ϕb degins j(E2 ) = degins ϕ degins j(E1 ).
Proof. By Proposition 4.7, one may decompose ϕ : E1 → E2 as a composition
Frpe

(pe )

E1 −−−−→ E1

ψ

(pf )

−−→ E2

Vpf−−−→ E2 ,where ψ is biseparable, pe = degins ϕ, and pf = degins ϕ.
b By the previous proposition, the inseparability
(pe )

(pf )

(pa )

degrees of j(E1 ) and j(E2 ) are equal. For i ∈ {1, 2} and any a ∈ Z≥0 , since j(Ei(pa )have degins j(Ei ) = pa degins (j(Ei )). The claimed identity is then clear.
12a) = j(Ei )p , we

5.3. Isogenies and modular heights. – We can now conclude our study of the effect of isogenies
between elliptic curves on their modular height. We fix a function field K as above. In case K has
characteristic 0, all the inseparability degrees should be interpreted as being 1. The following result wasannounced in the introduction as Theorem A:
Theorem 5.4. Let E1 and E2 be two non-isotrivial elliptic curves defined over K. Assume that there is
an isogeny ϕ : E1 → E2 between them. Then one has
hmod (E2 ) =degins ϕhmod (E1 ).
degins ϕb

In particular, biseparable isogenies preserve the modular height.
Proof. Fix a finite extension L of K over which both E1 , E2 are defined. We may and will also assume
that L is chosen so that E1 and E2 are semi-stable over L. As above (see Proposition 4.7), we factor the(pe )isogeny ϕ : E1 → E2 as a composition of Frpe : E1 → E1

(pe ), ψ : E1

(pf )

→ E2

(pf )

and Vpf : E2

→ E2
(pe )
(pf )
0
For simplicity, we denote E1 by E1 and E2

where ψ is biseparable and p = degins ϕ, p = degins ϕ.
bby E20 . We write that
hmod (E2 ) hmod (E20 ) hmod (E10 )hmod (E2 )=.
hmod (E1 )hmod (E20 ) hmod (E10 ) hmod (E1 )
e

f

(pe )

e

By definition, we have j(E10 ) = j(E1 ) = j(E1 )p , so that hmod (E10 ) = pe hmod (E1 ) and, similarly, we have
hmod (E20 ) = pf hmod (E2 ). We now use the “semi-stable case” of Proposition 3.2 and obtain that
(i)

(ii)

hmod (E20 ) − hmod (E10 ) = 12 hdiff (E20 /L) − 12 hdiff (E10 /L) = 0.
Here, equality (i) follows from Proposition 3.2: the “error term” in the comparison of heights vanishes
because the curves are semi-stable over L; equality (ii) comes from the fact that ψ : E10 → E20 , being
biseparable, preserves the differential height (see Theorem 5.1).
Therefore hmod (E10 ) = hmod (E20 ), and the result is proved.
Remark 5.5. An alternative (and rather natural) approach to proving Theorem 5.4 would be to rely on
considering the modular polynomial relations satisfied by the j-invariants of isogenous elliptic curves. We
now explain why the most immediate properties of these modular relations are insufficient to conclude.
The crucial step towards Theorem 5.4 consists in showing that biseparable cyclic isogenies preserve themodular height. Let, thus, j1 , j2 ∈ K be the j-invariants of two non-isotrivial elliptic curves linked by abiseparable cyclic isogeny. Fix a finite extension L = k 0 (C 0 ) of K containing j1 and j2 , and a biseparable
cyclic isogeny ϕ between two elliptic curves over L with respective j-invariants j1 and j2 .
Writing N := deg ϕ for the degree of ϕ, we then have ΦN (j1 , j2 ) = 0 in L, where ΦN (X, Y ) ∈ Z[X, Y ]denotes the N -th modular polynomial. Recall from (see [Lan87, Chapter V, §2] that ΦN (X, Y ) has
coefficients in Z, is symmetric in X, Y , and that it is monic of degree ψ(N ) in each variable.
In this situation, using integrality and symmetry of the modular relation ΦN (j1 , j2 ) = 0, one can showthat the divisors of poles div∞ (j1 ), div∞ (j2 ) have the same support. One cannot, however, conclude thath(j1 ) = h(j2 ) from these elementary results on ΦN (X, Y ) alone, as illustrated by the following example.
For any integer d ≥ 2, consider the polynomial Pd (X, Y ) := X d+1 + Y d+1 − (XY )d − XY ∈ Z[X, Y ]: it
too is symmetric in X, Y , and is monic of degree d+1 in each variable. For any y1 ∈ Lrk 0 , we let y2 := y1d .
We then have Pd (y1 , y2 ) = 0, even though y2 has height h(y2 ) = d h(y1 ) > h(y1 ).
Therefore, any attempt at proving Theorem 5.4 through this route must make use of finer resultson ΦN (X, Y ), such as irreducibility or arithmetic facts about its coefficients (see [Coh84] or [Paz19, §4]). We
think it unlikely that these considerations lead to a simpler proof of Theorem 5.4 in arbitrary characteristic.
5.4. Isogenies and differential heights. – As we saw in §5.1, biseparable isogenies preserve the differential height (Theorem 5.1). If K has characteristic 0, all isogenies are biseparable so that Theorem 5.1
completely solves the problem of describing the effect of an arbitrary isogeny on the differential height.
In this subsection, we thus assume that the function field K has positive characteristic p and attempt todescribe this effect. In view of the decomposition given by Proposition 4.7, it is enough to focus on theeffect of the Frobenius isogeny on hdiff : this is what we elucidate now.
Lemma 5.6. Let L be a finite extension of K. For any non-isotrivial elliptic curve E over L, there existsα(E/L) ≥ 1 such that, for any power q of p, we haveα(E/L)−1 q hdiff (E/L) ≤ hdiff (E (q) /L) ≤ q hdiff (E/L).
If, moreover, E is semi-stable over L, we have α(E/L) = 1, so that hdiff (E (q) /L) = q hdiff (E/L).
13

(5.1)

Proof. To lighten notation, we write E 0 := E (q) . For any place w of L, we denote the ring of integers at w
by Ow ⊂ L. We may pick a minimal w-integral Weierstrass model for E of the form:E:y 2 + a1,w xy + a3,w y = x3 + a2,w x2 + a4,w x + a6,w ,with a1,w , a2,w , a3,w , a4,w , a6,w ∈ Ow . By definition, the discriminant∆E,w = ∆(a1,w , a2,w , a3,w , a4,w , a6,w ) ∈ Ow
of this model has minimal valuation w(∆E,w ) ∈ Z≥0 among all choices of w-integral Weierstrass coefficients
a1,w , a2,w , a3,w , a4,w , a6,w for E. A Weierstrass model for the Frobenius twist E 0 of E is then given by:
E0 :y 2 + aq1,w xy + aq3,w y = x3 + aq2,w x2 + aq4,w x + aq6,w .

(5.2)

We note that aq1,w , aq2,w , aq3,w , aq4,w , aq6,w all lie in Ow , so that (5.2) is a w-integral model for E 0 . The
discriminant ∆0w of this model is clearly equal to ∆qE,w . Let ∆E 0 ,w be the discriminant of a minimal
w-integral Weierstrass model of E 0 . The difference w(∆0w ) − w(∆E 0 ,w ) is then a non-negative integral
multiple of 12, for ∆0w and ∆E 0 ,w differ by the 12-th power of an element of Ow . In particular, we obtainthat q w(∆E,w ) ≥ w(∆E 0 ,w ). Multiplying this inequality by deg w, and summing over all places w of L
yields thatq deg ∆min (E/L) ≥ deg ∆min (E 0 /L).
The right-most inequality in (5.1) ensues immediately. To prove the other inequality in (5.1), we argue asfollows. The proof of Proposition 3.2 implies the bounds:
0 ≤ deg ∆min (E/L) − deg div∞ (j(E)) ≤ 12 deg A(E/L),(5.3)

where A(E/L) = w not s.s. w is the divisor whose support consists in the places w of L where E does nothave semi-stable reduction. In particular, we havePdeg ∆min (E 0 /L) ≥ deg div∞ (j(E 0 ))anddeg ∆min (E/L) ≤ deg div∞ (j(E)) + 12 deg A(E/L).

Therefore, since j(E 0 ) = j(E)q , we deduce thatq deg ∆min (E/L)
q deg ∆min (E/L)
q deg ∆min (E/L)≤=
deg ∆min (E 0 /L)
deg div∞ (j(E 0 ))q deg div∞ (j(E))
deg div∞ (j(E)) + 12 deg A(E/L)≤= α(E/L),deg div∞ (j(E))
where we have set α(E/L) := 1 + 12 deg A(E/L)/(deg div∞ j(E)). From which we obtain thatq hdiff (E/L) ≤ α(E/L) hdiff (E 0 /L).
It is clear that α(E/L) ≥ 1, and that α(E/L) = 1 if and only if E/L is semi-stable. The above proves
both the left-most inequality in (5.1) and the last assertion of the Lemma.
Remark 5.7. We assume here that K has characteristic p 6= 2, 3. Let E be a non-isotrivial elliptic curve
over L, and E 0 := E (q) be its q-th power Frobenius twist. Since E and E 0 are L-isogenous, they have thesame reduction behaviour at all places of L.
If w is a place where E has multiplicative reduction of type In , for some n ≥ 1, then E 0 also has
multiplicative reduction at w, and its fiber at w is of type In q . Therefore we havew(∆min (E/L)) ≤ q w(∆min (E/L)) = q n = w(∆min (E 0 /L)).
If w is a place where E has additive reduction, then by inspection of the possible Kodaira–Néron reduction
types of E 0 at w (see the table page 365 of [Sil94], which is only valid in characteristic p 6= 2, 3), we
find that w(∆min (E/L)) ≤ 12 w(∆min (E 0 /L)). Therefore, for any place w where either of E and E 0 havebad reduction, we have w(∆min (E/L)) ≤ 12 w(∆min (E 0 /L)). Multiplying this inequality by deg w andsumming over all places w of L, we obtain that1
hdiff (E/L) ≤ hdiff (E (q) /L).
12This may be viewed as a weak but uniform version of the lower bound in (5.1).
14

We can now give the final estimate of this subsection.
Proposition 5.8. Let E1 , E2 be a pair of non-isotrivial elliptic curves over a finite extension L of K.
Assume that there exists an isogeny ϕ : E1 → E2 . If both E1 and E2 are semi-stable over L, we have
hdiff (E2 /L) =In general, we haveα(E2 /L)−1 ≤degins ϕhdiff (E1 /L).
degins ϕbdegins ϕ hdiff (E1 /L)≤ α(E1 /L),degins ϕb hdiff (E2 /L)

where α(E1 /L), α(E2 /L) ≥ 1 are the same as in Lemma 5.6.

Proof. The semi-stable case is a direct consequence of Theorem 5.4 and Corollary 3.3. In the general case,by Proposition 4.7, the isogeny ϕ : E1 → E2 decomposes as
Frpe

(pe )

ϕ : E1 −−−−→ E1

ψ

Vpf

(pf )

−−→ E2−−−→ E2 ,where ψ is a biseparable isogeny, pe = degins ϕ, and pf = degins ϕ.
b We then write that
(pe )

(pf )

hdiff (E1 /L) hdiff (E1 /L) hdiff (E2 /L)
hdiff (E1 /L)
=
.
f
(pe )
hdiff (E2 /L)
hdiff (E/L) hdiff (E (p ) /L) hdiff (E2 /L)
1

Since the isogeny ψ

(pe )
: E1

(pf )
→ E2

2

(pe )

is biseparable, Theorem 5.1 yields that ratio of hdiff (E1/L) to(pf )hdiff (E2 /L) is 1. To conclude, one applies inequality (5.1) in Lemma 5.6 to the other two ratios.

Remark 5.9. If K has characteristic p 6= 2, 3, we may carry out the same argument using the bound in
Remark 5.7 instead of inequality (5.1); we would then obtain the following bound. If ϕ : E1 → E2 is an
isogeny between two non-isotrivial elliptic curves E1 , E2 over L, we have(12)−1 ≤degins ϕ hdiff (E1 /L)≤ 12.
degins ϕb hdiff (E2 /L)

5.5. A surprising consequence on isogeny classes. – Let E be a non CM elliptic curve over Q. For
any B ≥ 0, consider the set
JQ (E, B) = E 0 /Q : E 0 is isogenous to E and ht(j(E 0 )) ≤ B Q-isomorphism,
where ht : Q → R is the standard logarithmic absolute Weil height on Q. It is known that JQ (E, B) is a
finite set (see Lemma 5.9 in [Hab13]); the main input in the proof of that statement is the estimate fromThéorème 1.1 in [SU99] which asserts thatht(j(E 0 )) ≥ ht(j(E)) +1
log deg ϕ − o(log deg ϕ)2

if there is a cyclic isogeny ϕ : E → E 0 (where the implicit constants in the error term depend on E). Using
our results in §5.3, we study the analogue to JQ (E, B) in the context of function fields.
Let K be a function field as in section 1, with characteristic p ≥ 0. Fix a non-isotrivial elliptic curve E
over K, and write Ibs (E) for the set of elliptic curves over K which are biseparably isogenous to E (i.e.,such that there is a biseparable isogeny E → E 0 ). When the characteristic of K is 0, all isogenies arebiseparable, so that Ibs (E) is the whole isogeny class of E. For any real number B ≥ 0, consider the set
JK (E, B) = E 0 ∈ Ibs (E) : hmod (E 0 ) ≤ B K-isomorphism= j(E 0 ), E 0 ∈ Ibs (E) with hmod (E 0 ) ≤ B .
The biseparability condition in Ibs (E) is added in order to avoid trivial situations in positive characteristic: without this condition, if K has characteristic p, we would indeed directly obtain from Theorem 5.4
infinitely many (isomorphism classes of) elliptic curves over K which are isogenous to E and have bounded
modular height by considering the sequence
Vp

Vp

Vp

E −−−→ E (1/p) −−−→ . . . −−−→ E (1/p

n−1

)

Vp

n

Vp

−−−→ E (1/p ) −−−→ . . . .

If B < hmod (E), Theorem 5.4 implies that JK (E, B) = ∅. If B ≥ hmod (E), in stark contrast to theabove mentioned result of Habegger, we prove:15

Proposition 5.10. If B ≥ hmod (E), the set JK (E, B) is infinite.
Proof. Given an integer n ≥ 1 which is coprime to the characteristic p of K, we may pick a point Pn ∈ E(K)of exact order n. We then let πn denote the quotient isogeny πn : E → E/hPn i from E to its quotient En
by the subgroup generated by Pn . The isogeny πn has degree n (see Proposition 4.3), and is therefore
biseparable (see Lemma 4.5).
This construction provides a sequence (En )n of elliptic curves over K indexed by prime-to-p integers. It
is clear that the curve En is biseparably isogenous to E. Applying Theorem 5.4 to the isogeny πn : E → En
yields that hmod (En ) = hmod (E). Therefore, the isomorphism class of En does belong to JK (E, B), thanks
to our assumption that hmod (E) ≤ B.
In order to conclude the proof, it now suffices to show that the above-constructed sequence (En )nprovides infinitely many isomorphism classes. Let `1 , `2 be two prime numbers, both coprime to p. Assume
that E`1 ' E`2 and denote the isomorphism by ι : E`1 → E`2 . The composition ι◦π`1 is an isogeny E → E`2of degree `1 . Hence, the isogeny ψ : E → E defined by ψ = πc`2 ◦ ι ◦ π`1 is a biseparable endomorphismof E of degree `1 `2 . Since E is non-isotrivial, its endomorphism ring is trivial (see §2.4); hence, there isan integer d such that ψ = [d]. Taking degrees, we obtain that d2 = `1 `2 . Since `1 and `2 are primes, we
deduce that `1 = `2 = d. We thereby conclude that the sequence (E` )` indexed by prime numbers ` 6= p
yields infinitely many elements in JK (E, B).
The main difference between the number field and the function field cases lies in how much the modular
height varies along an isogeny class. Over number fields, the above-mentioned Théorème 1.1 in [SU99]
shows that ht(j(E 0 )) does vary as E 0 /Q runs through the isogeny class of E/Q, provided that E has
no CM. Therefore, bounding ht(j(E 0 )) sufficiently constrains the degree of possible isogenies E → E 0 thatthe set JQ (E, B) is finite. Over a function field K, on the contrary, our Theorem 5.4 shows that hmod (E 0 )remains constant as E 0 runs through Ibs (E), provided that E is non-isotrivial. The absence of constraints
on the degree of isogenies E → E 0 other than coprimality with the characteristic, allows the set JK (E, B)
to be infinite. In this situation we will, in §6.5, recover a finiteness statement by further imposing a boundon [K(j(E 0 )) = K] for E 0 ∈ JK (E, B).

6. An isogeny estimate
The primary goal of this last section is to prove the second main result of the paper (Theorem B), whichis the following isogeny estimate:
Theorem 6.1. Let K be a function field of genus g. For any pair of non-isotrivial isogenous elliptic
curves E1 , E2 defined over K, there exists an isogeny ϕ0 : E1 → E2 with
degins j(E1 ) degins j(E2 )deg ϕ0 ≤ 49 max{1, g} max,,
degins j(E2 ) degins j(E1 )where degins j(E1 ), degins j(E2 ) are the inseparability degrees of j(E1 ), j(E2 ), respectively.
If K has characteristic 0, the inseparability degrees appearing on the right-hand side of the inequality
should be interpreted as 1. Specifically, in that situation, the statement above yields that there is aneffective constant c1 > 0 (depending only on the genus of K) such that: for all pairs E1 , E2 of nonisotrivial K-isogenous elliptic curves, there exists a (K-)isogeny ϕ0 : E1 → E2 with deg ϕ0 ≤ c1 . This
isogeny estimate is thus uniform for a fixed K. This should be compared to the number field case (treatedin [MW90, Pel01, GR14]) where the right-hand side of such isogeny estimates depends on the heights of
the involved elliptic curves.
Let us remark that, in positive characteristic p, the appearance of the inseparability degrees on theright-hand side is unavoidable, because of Theorem 5.4 and the existence of the Frobenius isogeny. Givena non-isotrivial elliptic curve E/K, the smallest isogeny between E and its Frobenius twist E (p) is indeed
the p-th power Frobenius, which has degree p. Similar considerations with the Verschiebung show (withCorollary 5.3) that the dependency in degins j(E1 ) and degins j(E2 ) on the right-hand side of the bound in
Theorem 6.1 is optimal.
On the other hand, the dependency of Theorem 6.1 in the genus of K can sometimes be improved. For
instance, if K has genus 0 one can prove (see Remark 6.4) that there exists an isogeny ϕ0 : E1 → E2 with
degins j(E1 ) degins j(E2 )(6.1)
deg ϕ0 ≤ 25 max,
degins j(E2 ) degins j(E1 )between two isogenous non-isotrivial elliptic curves E1 , E2 defined over K.
16

Remark 6.2. If one assumes that E1 , E2 are linked by an isogeny which is defined over K, then theisogeny ϕ0 : E1 → E2 whose existence is asserted in Theorem 6.1 is also defined over K. More generally,
all isogenies E1 → E2 are then defined over K. See Lemma 6.10 below.
6.1. Preliminaries about modular curves. – Let us briefly recall some facts about modular curves.
Given an integer N ≥ 1, there is a smooth scheme Y0 (N ) of relative dimension 1 over Spec Z[1/N ] which isa coarse moduli scheme for elliptic curves endowed with a cyclic subgroup of order N . In other words, thecurve Y0 (N ) enjoys the following property: For any field F whose characteristic is coprime to N , there isa bijection (which is functorial in F ) between the set Y0 (N )(F ) and the set of equivalence classes of pairs(E, H) where E is an elliptic curve over F , and H is a cyclic subgroup of E of order N which is stable under
the action of the absolute Galois group of F . Two such pairs are called equivalent if they are isomorphic
over the algebraic closure F of F . These properties of Y0 (N ) are explained in more details in [DI95, §8],and the construction is carried out comprehensively in [KM85] (in particular, Chapters III, VI and VIII
there). Adding a finite number of points (called cusps) to Y0 (N ), one obtains the usual compactification,
denoted by X0 (N ), of Y0 (N ). The smooth projective curve X0 (N ) is also defined over Spec Z[1/N ] (and,further, has an interpretation as a moduli space, in terms of generalized elliptic curves, see [DI95, §9]).
In particular, the curve X0 (1) is nothing but the “j-line” over Spec Z i.e., X0 (1) ' P1/Z , where theisomorphism is given on isomorphism classes of elliptic curves by E 7→ j(E).
For any divisor n of N , there is a “degeneration” morphism X0 (N ) → X0 (n), which extends the map
on pairs (E, H) as above, defined by (E, H) 7→ (E, Nn H). In the special case n = 1, we obtain a map
fN : X0 (N ) → X0 (1) = P1 , which extends the map (E, H) 7→ E 7→ j(E).
Let Γ0 (N ) ⊂ SL2 (Z) be the subgroup formed by 2×2 integral matrices of determinant 1 whose reduction
modulo N is upper triangular. The fiber X0 (N ) := X0 (N ) × C is isomorphic (as a Riemann surface) tothe compactification of the quotient {τ ∈ C : Im τ > 0}/Γ0 (N ). One can show (see [Shi94, Chapter I]
for instance) that the degree of the degeneration morphism fN : X0 (N ) → X0 (1) = P1/C is equal to theindex [SL2 (Z) : Γ0 (N )]. By Propositions 1.40 and 1.43 in [Shi94], we haveY1deg fN = [SL2 (Z) : Γ0 (N )] = N
1+:= ψ(N ).
(6.2)
`
`|N

A detailed study of the ramification behaviour of fN , combined with the Riemann–Hurwitz formula thenallows to compute the genus of X0 (N ). Specifically, Proposition 1.40 and Proposition 1.43 in [Shi94] yieldthat the modular curve X0 (N ) has genus
ψ(N ) ν2 (N ) ν3 (N ) ν∞ (N )
−
−
−
,
12
4
3
2
P
where ψ(N ) is as defined in (6.2), ν∞ (N ) = d|N ϕ(gcd(d, N/d)), and((00
if 4 | N,and ν3 (N ) = Q
ν2 (N ) = Q−1
if 4 - N,`|N 1 +`|N 1 +`g(X0 (N )) = 1 +

(6.3)

−3
`



if 9 | N,
if 9 - N.

Here ϕ denotes Euler’s totient function, and (·/`) is the Legendre symbol.
We end this brief summary by providing explicit bounds on the genus g(X0 (N )) of X0 (N ) as N grows.
We thank Gaël Rémond for the following argument, which yields better bounds than our original proof.
Lemma 6.3. For any integer N ≥ 300, we have g(X0 (N )) ≥ N/49. Moreover, for all N ≥ 1, we haveN ≤ 49 max{1, g(X0 (N ))}.
Proof. We estimate separately the terms on the right-hand side of equation (6.3). We start by noticingthat ν∞ , ν2 and ν3 are all multiplicative functions. For any prime `, and any integer e ≥ 1, one has(√`e 1 + `−1
if 2 | e,eν∞ (` ) = √ e −1/2
` 2`
if 2 - e.
Since, for all `, we have 1 + `−1 ≥ 2`−1/2 , the multiplicativity of ν∞ implies that, for all N ≥ 1,


ψ(N )
ν∞ (N ) Y
1
√
≤
1+
=
.
`
N
N
`|N

17

√
Hence we deduce that ν∞ (N ) ≤ ψ(N )/ N .
Similarly, we notice that max{ν2 (`e ), ν3 (`e )} ≤ ν∞ (`e ) for all primes ` and integers e ≥ 1. We directly
infer that, for any integer N ≥ 1, one has max{ν2 (N ), ν3 (N )} ≤ ν∞ (N ).
Plugging these bounds into equation (6.3) yields that
7
1
ψ(N )
−
max{ν2 (N ), ν3 (N )} − ν∞ (N )
12
12
 2

ψ(N ) 13
ψ(N )
13
≥1+
− ν∞ (N ) ≥ 1 +
1− √
.
12
12
12
N
√
By definition of ψ, we have ψ(N ) ≥ N . Therefore, g(X0 (N )) ≥ 1 + (N − 13 N )/12 for N ≥ 132 .
Furthermore, as a quick computation shows, for any N ≥ 297, we have


N
13
N
g(X0 (N )) >
1− √
≥
.
12
49
N
g(X0 (N )) ≥ 1 +This proves the first assertion of the lemma. Direct calculations with formula (6.3) using a computer showthat the bound N ≤ 49 max{1, g(X0 (N ))} also holds for all N ∈ {1, . . . , 296}. (The worst case is N = 49
for which g(X0 (N )) = 1.)
Remark 6.4. Depending on the situation at hand,√ the√bounds of Lemma 6.3 can be somewhat optimized.
For instance, one can deduce from 12g(X0 (N )) > N ( N − 13) that N ≤ 13g(X0 (N )) for all N ≥ 28561,
and check with the help of a computer that, for all N ≥ 1,
N ≤ max{3721, 13g(X0 (N ))} ≤ 3721 max {1, 0.004 g(X0 (N ))} .

(6.4)

The worst case happens for N = 3721 when g(X0 (N ))√= 284.
√
theinequality 12(g(X0 (N )) − 1) ≥ N ( N − 13) proved above,p
p one also deduces that
√ Starting from( N − 13/2)2 ≤ ( 12g(X0 (N )) + 11/2)2 , which leads to N ≤ 12g(X0 (N )) + 2b 12g(X0 (N )) + b2 withb = 13/2 + 11/2 = 12. We obtain thatpN ≤ 12g(X0 (N )) + 24 12g(X0 (N )) + 144.
(6.5)
The last two displayed inequalities are more precise than Lemma 6.3 for larger genera.
For small genera, on the other hand, explicit computations using formula (6.3) show that the modular
curve X0 (N ) has genus zero if and only if N ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25}.
6.2. Bounding biseparable isogenies. – Let K be a function field with field of constants k, and fix amodel C/k of K (see §1). We write g(C) for the genus of C (which we also call the genus of K), and we
let GK denote the absolute Galois group of K.
Let d ≥ 1 be an integer which is coprime to the characteristic of K. Assume that we are given a
non-isotrivial elliptic curve E over K, equipped with a cyclic, GK -stable subgroup H ⊂ E of order d.
Consider the modular curve X0 (d), defined over Spec Z[1/d], and write X0 (d)/F for its base change to a
field F whose characteristic is prime to d. By the coarse moduli space interpretation of Y0 (d), there isa canonical map sending the K-isomorphism class of the pair (E, H) to a K-rational point P on Y0 (d).
From the given data (E, H), we thus deduce a non-cuspidal K-rational point P ∈ X0 (d).
By definition, the point P is given by a morphism Spec K → X0 (d) over Spec Z[1/d], which factors asa morphism Spec K → X0 (d)/k over Spec k. Since Spec K is the generic point of C, the latter induces arational map C 99K X0 (d)/k . Since both C and X0 (d)/k are smooth projective curves over k, this rational
map extends to a morphism sP : C → X0 (d)/k over k.
Let fd : X0 (d)/k → X0 (1)/k = P1/k denote the degeneration morphism, which extends the map sending
a pair (E 0 , H 0 ) to j(E 0 ). By construction, the morphism jE : C → P1/k deduced from the j-invariant of E
factors through sP and fd . In other words, the diagramsP

C

X0 (d)/k

jE

fd

P1/k
is commutative. We may now prove the following:
18

(6.6)

Proposition 6.5. Let E be a non-isotrivial elliptic curve over K. Let H ⊂ E be a cyclic, GK -stable
subgroup of order d. If d is coprime to the characteristic of K, we haved ≤ 49 max{1, g(C)}.
Proof. By the discussion above, the data (E, H) of the proposition yields a morphism sP : C → X0 (d)/ksuch that the diagram (6.6) commutes. The fact that E is non-isotrivial implies that jE : C → P1/k is notconstant which, in particular, ensures that the morphism sP : C → X0 (d)/k is non-constant.
A weak version of the Riemann–Hurwitz formula (see [Ros02, Theorem 7.16]) then entails that thegenus of X0 (d)/k is no greater than that of C. Since X0 (d) is a smooth curve over Spec Z[1/d], the genusof X0 (d)/k is equal to the genus of the complex Riemann surface X0 (d)/C . Hence g(C) ≥ g(X0 (d)/C ). We
then make use of the lower bound of Lemma 6.3, and conclude that d ≤ 49 max{1, g(C)}.
We now state and prove a variant of the previous proposition, which is significantly weaker (it is not
uniform in the elliptic curve E) but which might prove more flexible. This version may indeed be of interest
for applications where one studies elliptic curves defined over a varying field L whose degree over K isbounded (in which case a bound depending on the genus of L may be too crude). We provide an instanceof such an application in §6.5.
Proposition 6.6. Let K be a function field and let L/K be a finite extension. Let E be a non-isotrivial
elliptic curve over L, and H ⊂ E be a cyclic, GL -stable subgroup of order d. Assuming that d is coprime
to the characteristic of K, we haved ≤ [L : K] hmod (E).
Proof. We fix a model C 0 of L, and denote the constant field of L by k 0 . Carrying out the argument
preceding the previous proposition with L instead of K, we deduce from the input (E, H) an L-rational
point Q on the modular curve Y0 (d), and a morphism sQ : C 0 → X0 (d)/k0 . As in (6.6), the j-invariant
jE : C 0 → P1/k0 factors through sQ ; we thus have jE = fd ◦ sQ , where fd : X0 (d)/k0 → P1/k0 is thedegeneration map. Since jE = fd ◦ sQ , it is clear that deg(jE ) ≥ deg(fd ).
As was recalled in the previous subsection (see (6.2)), the degree of fd equals [SL2 (Z) : Γ0 (d)] = ψ(d).
It is clear that ψ(d) ≥ d, so thatdeg(jE ) ≥ deg(fd ) = [SL2 (Z) : Γ0 (d)] = ψ(d) ≥ d.
Now, the degree of jE : C 0 → P1/k0 is the degree of its divisor of poles div∞ (jE ) ∈ Div(C 0 ). As was
remarked in §3.2, the modular height of E satisfies hmod (E) = [L : K]−1 deg(div∞ (jE )). Rearranging theterms in the inequality above then yields that d ≤ [L : K] hmod (E), which concludes the proof.
6.3. Proof of Theorem 6.1. – We first prove a special case of Theorem 6.1, using the tools introducedin the previous two subsections:
Proposition 6.7. Let E1 , E2 be two non-isotrivial elliptic curves over K. Assume that there exists a
biseparable isogeny ϕ : E1 → E2 . Then there exists a biseparable isogeny ϕ0 : E1 → E2 withdeg ϕ0 ≤ 49 max{1, g(K)}.
If K has characteristic 0, the proof of this statement will conclude that of Theorem 6.1. The following
proof uses arguments inspired by the ones of [Ulm11, Lecture I, Proposition 7.1], which proves a uniform
upper bound for the order of the prime-to-p K-rational torsion on an elliptic curve over K.
Proof. Fix a biseparable isogeny ϕ : E1 → E2 , and consider the set of positive integers:
D := {deg φ, φ : E1 → E2 biseparable isogeny} ⊂ Z≥1 .
By assumption, D contains a minimal element d0 , and there exists a biseparable isogeny ϕ0 : E1 → E2
with deg ϕ0 = d0 . Note that the degree d0 of ϕ0 is coprime to p, for ϕ0 is biseparable (see Lemma 4.5).
Let us now prove that ϕ0 is cyclic i.e., that the group H0 := (ker ϕ0 )(K) is cyclic. By the structure
theorem for finite Abelian groups, and by the description of finite subgroups of E1 , we know that H0 is
isomorphic to Z/mZ × Z/nZ for some integers m, n ≥ 1 with m | n and mn = d0 . In particular, thekernel of the multiplication-by-m map [m] : E1 → E1 is a subgroup of H0 which is isomorphic to (Z/mZ)2 .
Proposition 4.2 then implies that the isogeny ϕ0 can be factored as ϕ0 = ϕ1 ◦ [m] for a unique isogeny
ϕ1 : E1 → E2 . Taking degrees yields that deg ϕ0 = d0 = deg ϕ1 m2 . Being a divisor of d0 , deg ϕ1 must be19

coprime to p so that, by Lemma 4.5, the isogeny ϕ1 is biseparable. On the other hand, the degree of ϕ0 is
minimal among all degrees of biseparable isogenies E1 → E2 . We thus have m = 1, and the kernel of ϕ0
is isomorphic to Z/nZ. Hence H0 is cyclic, as claimed.
Since the degree of ϕ0 is coprime to the characteristic of K, Lemma 4.6 shows that the group H0 isdefined over a separable extension of K i.e., that the extension K(H0 )/K is separable. Let GK denotethe absolute Galois group of K. We now prove that H0 is stable under the action of GK on E1 (in otherwords, we show that H0 is “defined over K”).
To do so, we first prove that σ ϕ0 = ±ϕ0 for all σ ∈ GK . Let σ ∈ GK be an arbitrary automorphism.
Since both E1 and E2 are defined over K, we get an isogeny σ ϕ0 : E1 → E2 of degree d0 . The compositionof the dual ϕc0 : E2 → E1 with σ ϕ0 : E1 → E2 yields an endomorphism ϕc0 ◦ σ ϕ0 of E1 . The curve E1
being non-isotrivial, it has no non-trivial endomorphisms (see §2.4), so that there exists an integer n suchthat ϕc0 ◦ σ ϕ0 = [n] is the multiplication-by-n map. Comparing degrees, we get that n = ±d0 , whenceσϕc0 ◦ ϕ0 = [±d0 ] = [±1] ◦ ϕ
c0 ◦ ϕ0 . Therefore ϕc0 ◦ (σ ϕ0 − [±1] ◦ ϕ0 ) = 0E1 , and we deduce that the imageσof the isogeny ϕ0 − [±1] ◦ ϕ0 is contained in the kernel of ϕc0 . Since the latter is finite, σ ϕ0 − [±1] ◦ ϕ0σmust be constant equal to 0E2 . Thus ϕ0 = ±ϕ0 , as claimed.
It then formally follows from the previous paragraph that H0 is stable under the action of GK . We may
now apply Proposition 6.5 to the pair (E1 , H0 ), and we infer that d0 = deg ϕ0 ≤ 49 max{1, g(K)}.
To conclude the proof of the general case of Theorem 6.1, we require the following lemma:
Lemma 6.8. Let E1 , E2 be two non-isotrivial elliptic curves over K. We assume that j(E1 ) and j(E2 )have the same inseparability degree, and that there exists an isogeny ϕ : E1 → E2 . Then there exists a
biseparable isogeny ψ : E1 → E2 with deg ψ | deg ϕ.
Proof. We write degins ϕ = pa and degins ϕb = pb . By assumption we have degins j(E2 ) = degins j(E1 ), henceaCorollary 5.3 implies that p = degins ϕ = degins ϕb = pb , so that a = b. Using the decomposition of ϕprovided by Proposition 4.7, one deduces that the integer m := p−2a deg ϕ is coprime to p.
By Proposition 4.8, the isogeny ϕ factors through the multiplication-by-pa map on E1 : i.e., there exists
an isogeny ψ : E1 → E2 so that ϕ = ψ ◦ [pa ]. By multiplicativity of degrees, ψ has degree m dividing deg ϕ.
Since m is coprime to p, Lemma 4.5 ensures that ψ is biseparable.
Proof of Theorem 6.1. Let E1 and E2 be non-isotrivial isogenous elliptic curves over K. If K has characteristic 0, Proposition 6.7 proves an assertion which is equivalent to Theorem 6.1. We may therefore
assume that the characteristic p of K is positive.
Let ϕ : E1 → E2 be a K-isogeny. As the sought statement is symmetric in E1 and E2 we may, up to
permuting the roles of E1 and E2 and using duals, assume that degins j(E1 ) ≥ degins j(E2 ).
We then let f be the non-negative integer such that degins j(E1 )/degins j(E2 ) = pf and, to lighten(pf )notation, we write E20 for E2 .
The composition of ϕ : E1 → E2 with Frpf : E2 → E20 yields an isogeny E1 → E20 . By constructionof f we have degins j(E1 ) = degins j(E20 ), so that Lemma 6.8 ensures the existence of a biseparable isogeny
ψ : E1 → E20 . By Proposition 6.7, there exists an isogeny ψ0 : E1 → E20 with deg ψ0 ≤ 49 max{1, g(K)}.
Composing ψ0 : E1 → E20 with Vpf : E20 → E2 , we obtain an isogeny ϕ0 : E1 → E2 of degree
deg ϕ0 = pf deg ψ0 ≤ 49 max{1, g(K)}degins j(E1 ),degins j(E2 )which concludes the proof of Theorem 6.1.
6.4. Number of K-isomorphism classes in a K-isogeny class. – Let K be a function field as above,
and E be a non-isotrivial elliptic curve over K. For any M ≥ 1, let us introduce the set

EK (E, M ) := {E 0 /K : E 0 is K-isogenous to E, and degins j(E 0 ) ≤ M } K-isomorphism.
(As before, if K has characteristic 0, one may ignore the condition on degins j(E 0 ).)By construction, the elliptic curves E 0 /K (whose K-isomorphism classes are) in EK (E, M ) have thesame conductor as E. In particular, they all have good reduction outside the set S of places of K where E
has bad reduction. By a version of Shafarevich’s theorem, there are finitely many K-isomorphism classes of
elliptic curves over K with good reduction outside S and with bounded inseparability degree of j-invariant.
The set EK (E, M ) is thus finite. We prove an effective form of that statement (announced as Corollary C
in the introduction), as follows:
Theorem 6.9. In the above setting, we have20• EK (E, M ) ≤ 74 max{1, g(K)}2 if K has characteristic 0,


log M
4
2
• EK (E, M ) ≤ 7 max{1, g(K)}+ 1 if K has characteristic p > 0.
log p
We are grateful to one of the referees for remarks that led to an improvement on our original bound.
To prove the proposition, we take inspiration from section 2 of [MW89], which proves the analogue ofTheorem 6.9 for elliptic curves over number fields. We need the following two lemmas:
Lemma 6.10. Let E1 , E2 be two non-isotrivial elliptic curves over K. We assume that there exists an
isogeny ϕ : E1 → E2 which is defined over K. Then all isogenies E1 → E2 are defined over K.
Proof. Since E1 is non-isotrivial and since E1 , E2 are isogenous, the Z-module Hom(E1 , E2 ) is free ofrank 1; we choose an isogeny ϕ0 : E1 → E2 such that Hom(E1 , E2 ) = Z ϕ0 . To conclude, it suffices tocheck that ϕ0 is defined over K.
We may write ϕ as ϕ = ϕ0 ◦ [m] for some integer m 6= 0. Let H := ker ϕ denote the schematic kernelof ϕ, and E1,m := ker[m] denote the subgroup scheme of E1 formed by m-torsion points. Both H and E1,m
are (non-necessarily reduced) group schemes over K. Since ϕ = ϕ0 ◦ [m], E1,m is a subgroup scheme of H.
We then use Theorem 1 in [Mum08, §12] (which is a scheme-theoretic version of Proposition 4.2): thisshows that the isogeny ϕ factors uniquely (up to K-isomorphism) through [m] as ϕ = ψ ◦ [m], whereψ : E1 → E2 is a K-isogeny. The uniqueness of ψ implies that ϕ0 = ψ, which is defined over K.
Lemma 6.11. Let E1 be a non-isotrivial elliptic curve over K. For any real number X ≥ 1, we denote
by C (E1 , X) the number of cyclic subgroups H of E1 with |H| ≤ X and |H| coprime to the characteristicof K. Then we have C (E1 , X) ≤ X 2 .
Proof. Let d ≥ 1 be an integer coprime to the characteristic p of K. Given the structure of the d-torsion
subgroup of E1 , the number of cyclic subgroups of E of order d equals ψ(d). (Recall indeed that the group(Z/dZ)2 contains ψ(d) cyclic subgroups of order d.) For any n ≥ 1, an elementary computation showsthat ψ(n) ≤ σ(n), where σ(n) denotes the sum of divisors of n (both functions are multiplicative, and theinequality holds for prime powers). One easily sees that, for X ≥ 1,
 
X
X X
X
X
X
X
σ(n) ≤d≤d=d≤ X 2.
d1≤n≤X

1≤n≤X d|n1≤d≤X 1≤n≤X
s.t. nd≤X

1≤d≤X

We therefore obtain the desired upper bound:X
X
X
C (E1 , X) =ψ(d) ≤
ψ(n) ≤
σ(n) ≤ X 2 .
1≤d≤X(d,p)=1

1≤n≤X

1≤n≤X

Proof of Theorem 6.9. For clarity, we first prove the assertion in the case when K has characteristic 0.
Let E 0 be an elliptic curve over K whose K-isomorphism class lies in EK (E, M ), and choose an isogeny
ϕ0 ∈ Hom(E, E 0 ) with minimal degree. Proposition 6.7 and its proof ensure that H0 := (ker ψ0 )(K) is
cyclic of order |H0 | = deg ψ0 ≤ cK , where we have set cK := 49 max{1, g(K)}. The elliptic curve E 0
is then K-isomorphic to the quotient E/H0 , because ψ0 is defined over K by Lemma 6.10. This entails that the cardinality of EK (E, M ) is no greater than C (E, cK ). With Lemma 6.11, we conclude that
|EK (E, M )| ≤ c2K , which proves the theorem in the characteristic 0 case.
We now assume that the characteristic p of K is positive. For any non-negative integer u, we let
E (u) := {E 0 /K : E 0 is K-isogenous to E, and degins j(E 0 ) = pu } K-isomorphism.
S
We can write the set EK (E, M ) as a disjoint union EK (E, M ) = 0≤u≤m E (u) where m = log M/ log p.
Let us prove that for any power pu of p, |E (u)| has cardinality at most c2K . Given an integer u ≥ 0such that the set E (u) is non-empty, let E1 be an elliptic curve over K whose K-isomorphism class liesin E (u). Any elliptic curve E2 over K whose isomorphism class lies in E (u) is then K-isogenous to E1 .
Furthermore, since the inseparability degrees of the j-invariants of E1 and E2 are the same, we know from
Lemma 6.8 and Lemma 6.10 that E1 and E2 are linked by a biseparable isogeny which is defined over K.
We then carry out, as we may, the same argument as in the characteristic 0 case above: this entails thatthe cardinality of E (u) is no greater than C (E1 , cK ). Lemma 6.11 then yields that |E (u)| ≤ c2K .
21

If p > 0, we therefore conclude that
|EK (E, M )| ≤X

|E (u)| ≤ (m + 1) c2K = c2K

0≤u≤m




log M
+1 ,
log p

as was to be shown.
6.5. Back to isogeny classes. – We close this paper by going back to the situation studied in §5.5, andby recovering a finiteness result. Let K be a function field as above, and fix a non-isotrivial elliptic curve E
over K. Recall that Ibs (E) denotes the set of elliptic curves E 0 /K which are biseparably isogenous to E.
For any B ≥ 0 and D ≥ 1, consider the set
JK0 (E, B, D) = E 0 ∈ Ibs (E) : hmod (E 0 ) ≤ B and [K(j(E 0 )) : K] ≤ D K-isomorphism.
By Theorem 5.4, all elliptic curves E 0 ∈ Ibs (E) satisfy hmod (E 0 ) = hmod (E). The set JK0 (E, B, D) is hence
empty if B < hmod (E) and, for B ≥ hmod (E), we have
JK0 (E, B, D) = E 0 ∈ Ibs (E) : [K(j(E 0 )) : K] ≤ D K-isomorphism.
S
Note that the set JK (E, B) studied in §5.5 is the union D≥1 JK0 (E, B, D). We prove the following bound:
Proposition 6.12. For any B ≥ 0 and D ≥ 1, the set JK0 (E, B, D) is finite. Moreover, we have
|JK0 (E, B, D)| ≤ D2 hmod (E)2 .
Proof. Let E 0 ∈ Ibs (E) be an elliptic curve which is biseparably isogenous to E. Let j(E 0 ) ∈ K denote its
j-invariant and K 0 := K(j(E 0 )). If necessary, we replace E 0 by a K-isomorphic elliptic curve E2 definedover K 0 . We write E1 for the base-change E1 := E ×K K 0 .
By assumption, there exists a biseparable isogeny ϕ : E1 → E2 . We may assume (see the proof of
Proposition 6.7) that ϕ = ϕ0 has minimal degree among all biseparable isogenies E1 → E2 . By the same
arguments as in the proof of Proposition 6.7, the group H0 := (ker ϕ0 )(K) is then cyclic and stable under
the action of the absolute Galois group GK 0 = Gal(K/K 0 ). Proposition 6.6 then yields the bound|H0 | = deg ϕ0 ≤ [K 0 : K] hmod (E1 ) ≤ D hmod (E).
The elliptic curve E 0 ' E2 is then K-isomorphic to the quotient E1 /H0 . Hence, the cardinality of theset JK0 (E, B, D) does not exceed the number C (E1 , D hmod (E)), introduced in the previous subsection, of
cyclic subgroups H of E1 with order at most D hmod (E) and with |H| coprime to the characteristic of K.
By Lemma 6.11, we thus have |JK0 (E, B, D)| ≤ (D hmod (E))2 = D2 hmod (E)2 .
This proves the finiteness of JK0 (E, B, D) and the asserted upper bound on its cardinality.

Acknowledgements RG is supported by the Swiss National Science Foundation, through the SNSF
Professorship #170565 awarded to Pierre Le Boudec, and received additional funding from ANR projectANR-17-CE40-0012 (FLAIR). He wishes to thank Universität Basel for providing excellent working conditions. FP thanks the ANR project ANR-17-CE40-0012 (FLAIR) and the IRN GANDA (CNRS).
Both authors would like to thank Gabriel Dill, Marc Hindry, Pierre Le Boudec, Samuel Le Fourn, Harry
Schmidt and Douglas Ulmer for stimulating conversations about the topic of this work. They also express
their deepest gratitude to Gaël Rémond for his valuable comments and suggestions on an earlier versionof the article. Finally, the authors thank the referee for their careful reading and beneficial input.

References[BLR90] Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud. Néron models, volume 21 of Ergebnisseder Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, Berlin, 1990. ↑ 11
[BLV09] Andrea Bandini, Ignazio Longhi, and Stefano Vigni. Torsion points on elliptic curves over function fields
and a theorem of Igusa. Expo. Math., 27(3):175–209, 2009. ↑ 9, 12[BPR21] Florian Breuer, Fabien Pazuki, and Mahefason H. Razafinjatovo. Heights and isogenies of Drinfeld
modules. Acta Arithmetica, 197(2):111–128, 2021. ↑ 1

22

[Coh84]

Paula Cohen. On the coefficients of the transformation polynomials for the elliptic modular function.
Math. Proc. Cambridge Philos. Soc., 95(3):389–402, 1984. ↑ 13

[DD99]

Sinnou David and Laurent Denis. Isogénie minimale entre modules de Drinfel’d. Math. Ann., 315(1):97–
140, 1999. ↑ 2[DGS94] Bernard Dwork, Giovanni Gerotto, and Francis J. Sullivan. An introduction to G-functions, volume 133of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1994. ↑ 3[DI95]

Fred Diamond and John Im. Modular forms and modular curves. In Seminar on Fermat’s Last Theorem(Toronto, ON, 1993–1994), volume 17 of CMS Conf. Proc., pages 39–133. Amer. Math. Soc., Providence,
RI, 1995. ↑ 17

[GR14]

Éric Gaudron and Gaël Rémond. Théorème des périodes et degrés minimaux d’isogénies. Comment.
Math. Helv., 89(2):343–403, 2014. ↑ 2, 16[Hab13]

Philipp Habegger. Special points on fibered powers of elliptic surfaces. J. Reine Angew. Math., 685:143–
179, 2013. ↑ 15

[Hus04]

Dale Husemöller. Elliptic curves, volume 111 of Graduate Texts in Mathematics. Springer-Verlag, New
York, second edition, 2004. With appendices by Otto Forster, Ruth Lawrence and Stefan Theisen. ↑ 5[KM85]

Nicholas M. Katz and Barry Mazur. Arithmetic moduli of elliptic curves, volume 108 of Annals of
Mathematics Studies. Princeton University Press, Princeton, NJ, 1985. ↑ 17

[Lan83]Serge Lang. Fundamentals of Diophantine geometry. Springer-Verlag, New York, 1983. ↑ 3[Lan87]Serge Lang. Elliptic functions, volume 112 of Graduate Texts in Mathematics. Springer-Verlag, New
York, second edition, 1987. With an appendix by J. Tate. ↑ 13

[MB85]

Laurent Moret-Bailly. Pinceaux de variétés abéliennes. Astérisque, (129):266, 1985. ↑ 5, 6

[Mum08] David Mumford. Abelian varieties, volume 5 of Tata Institute of Fundamental Research Studies in Mathematics. 2008. Corrected reprint of the second (1974) edition. ↑ 21
[MW89] David W. Masser and Gisbert Wüstholz. Some effective estimates for elliptic curves. In Arithmetic ofcomplex manifolds (Erlangen, 1988), volume 1399 of Lecture Notes in Math., pages 103–109. Springer,
Berlin, 1989. ↑ 21
[MW90] David W. Masser and Gisbert Wüstholz. Estimating isogenies on elliptic curves. Invent. Math., 100(1):1–
24, 1990. ↑ 2, 16[MW93] David Masser and Gisbert Wüstholz. Isogeny estimates for abelian varieties, and finiteness theorems.
Ann. of Math. (2), 137(3):459–472, 1993. ↑ 2[Par70]

Aleksei N. Paršin. Isogenies and torsion of elliptic curves. Izv. Akad. Nauk SSSR Ser. Mat., 34:409–424,
1970. ↑ 11

[Par73]

Aleksei N. Paršin. Modular correspondences, heights and isogenies of abelian varieties. Trudy Mat. Inst.
Steklov., 132:211–236, 266, 1973. ↑ 11

[Paz19]Fabien Pazuki. Modular invariants and isogenies. Int. J. Number Theory, 15(3):569–584, 2019. ↑ 1, 13[Pel01]

Federico Pellarin. Sur une majoration explicite pour un degré d’isogénie liant deux courbes elliptiques.
Acta Arith., 100(3):203–243, 2001. ↑ 2, 16[PS00]Jérôme Pesenti and Lucien Szpiro. Inégalité du discriminant pour les pinceaux elliptiques à réductions
quelconques. Compositio Math., 120(1):83–117, 2000. ↑ 7[Ray85]Michel Raynaud. Hauteurs et isogénies. Number 127, pages 199–234. 1985. Seminar on arithmetic
bundles: the Mordell conjecture (Paris, 1983/84). ↑ 11

[Ros02]

Michael Rosen. Number theory in function fields, volume 210 of Graduate Texts in Mathematics. SpringerVerlag, New York, 2002. ↑ 2, 3, 19[Shi94]Goro Shimura. Introduction to the arithmetic theory of automorphic functions, volume 11 of Publicationsof the Mathematical Society of Japan. Princeton University Press, Princeton, NJ, 1994. ↑ 17

[Sil86]

Joseph H. Silverman. Heights and elliptic curves. In Arithmetic geometry (Storrs, Conn., 1984), pages
253–265. Springer, New York, 1986. ↑ 6, 7

[Sil94]

Joseph H. Silverman. Advanced topics in the arithmetic of elliptic curves, volume 151 of Graduate Texts
in Mathematics. Springer-Verlag, New York, 1994. ↑ 4, 5, 7, 14[Sil09]Joseph H. Silverman. The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics.
Springer, Dordrecht, 2nd edition, 2009. ↑ 2, 4, 5, 8, 9, 12[ST68]

Jean-Pierre Serre and John Tate. Good reduction of abelian varieties. Ann. of Math. (2), 88:492–517,
1968. ↑ 5, 9

[SU99]

Lucien Szpiro and Emmanuel Ullmo. Variation de la hauteur de Faltings dans une classe de Q-isogénie
de courbe elliptique. Duke Math. J., 97(1):81–97, 1999. ↑ 1, 15, 16

23

[Szp90]

Lucien Szpiro. Discriminant et conducteur des courbes elliptiques. Astérisque, (183):7–18, 1990. Séminaire
sur les Pinceaux de Courbes Elliptiques (Paris, 1988). ↑ 7[Tat93]

John Tate. A review of non-Archimedean elliptic functions. In Int. Press, editor, Elliptic curves, modular
forms, and Fermat’s last theorem (Hong Kong, 1993), pages 162–184, 1993. ↑ 5[Ulm11]

Douglas Ulmer. Elliptic curves over function fields. In Arithmetic of L-functions, volume 18 of IAS/Park
City Math. Ser., pages 211–280. Amer. Math. Soc., Providence, RI, 2011. ↑ 4, 6, 19

Richard Griffon (richard.griffon@unibas.ch) – Departement Mathematik, Universität Basel, Spiegelgasse 1, 4051 Basel, Switzerland.
Fabien Pazuki (fpazuki@math.ku.dk) – Department of Mathematical Sciences, University of
Copenhagen, Universitetsparken 5, 2100 Copenhagen ø, Denmark.
24