Counting modular forms by rationality field

We are especially grateful to Noam Elkies for many insights and suggestions. We have also benefited from conversations with Eran Assaf, Armand Brumer, Bjorn Poonen, Ari Shnidman, and John Voight. Computations were performed at the OU Supercomputing Center for Education & Research (OSCER) at the University of Oklahoma (OU).

Here we present a random model to estimate the distribution of degree $d$ newforms that will lead us to 1.1. This is based on ideas for heuristics suggested in [33] and [28].

Consider a newspace $S_{2k}^{\mathrm{new}}(N)$. One can further decompose this space into $2^{\omega(N)}$ joint eigenspaces of the Atkin–Lehner operators $W_{p}$ for $p\mid N$, which we call the Atkin–Lehner eigenspaces. Each Atkin–Lehner eigenspace is Galois invariant. For non-squarefree levels, one can further decompose each Atkin–Lehner eigenspace into smaller Galois invariant subspaces according to local inertia types of non-CM forms (see [11]) and the subspace of CM forms.

For simplicity, assume $N$ is squarefree. Then there are no CM forms of trivial nebentypus and there is only one local inertial type. Let $S$ be an Atkin–Lehner eigenspace in $S_{2k}^{\mathrm{new}}(N)$. For a newform $f\in S$, the single Fourier coefficient $a_{p}(f)$ generates $K_{f}$ for 100% of $p$ [22], and it is conjectured that this is true for all but finitely many $p$ if $[K_{f}:\mathbb{Q}]>4$ [32]. Hence, for fixed $p\nmid N$, the factorization type of the characteristic polynomial $c_{T_{p}}(x)\in\mathbb{Z}[x]$ of the Hecke operator $T_{p}$ will usually tell us the degrees of the newforms in $S$. In fact, it will always give us lower bounds.

Let $n=\dim S$. As in [33] and [28], we can model $c_{T_{p}}(x)$ as a random polynomial in the set $H_{n}=H_{n}(k,p)$ of degree $n$ monic integral polynomials whose roots $\alpha$ satisfy $|\alpha|\leq 2p^{k-1/2}$. Alternatively, one can consider the set of Weil $q$-polynomials of degree $2n$ where $q=p^{k}$, or the isogeny classes of $n$-dimensional abelian varieties over $\mathbb{F}_{p^{k}}$.

Set $h(n)=\#H_{n}$. As discussed in [28, §2.1], the number of polynomials in $H_{n}$ with a degree $d<\tfrac{n}{2}$ factor is approximately $h(d)h(n-d)$. Thus, if we select polynomials in $H_{n}$ uniformly at random, then

In this section, by approximately ($\approx$), we mean that for fixed $d$ both sides have the same growth rate in $n$ as $n\to\infty$.

For fixed $q$, no good asymptotics are known for $h(n)$ to directly estimate this probability. There is an asymptotic for $h(n)$ when $n$ is fixed and $q$ varies. Instead, [33] and [28, §2.1] analyzed how this probability behaves if one uses a known asymptotic for $\#H_{n}(k,p)$ in $p^{k}$ when $n$ is fixed. (The result is certainly too small, as it would predict only finitely many degree 1 forms.)

To circumvent this lack of precise asymptotics for $h(n)$ as $n\to\infty$, we rewrite the right hand side of (3) as

One should have that $\frac{h(n-2)}{h(n-1)}\approx\frac{h(n-1)}{h(n)}$, so applying this a small fixed number of times for a given $d$ yields

Combining (3) and (4) suggests that the probability of a degree $d$ factor of $c_{T_{p}}$ should approximately be the $d$-th power of the probability of a degree $1$ factor of $c_{T_{p}}$. The latter typically corresponds to a degree 1 form, and so we can model it using well-known expectations about counts of elliptic curves.

We will also use the following lemma.

Lemma 2.1 combines with (3) and (4) to produce the following heuristic.

Our reasoning for this heuristic is as follows. Under the hypothetical bound $O(X^{1-\alpha})$, the lemma indicates that the probability of an Atkin–Lehner space of dimension $n$ having a size 1 Galois orbit is approximately $n^{-\alpha}$. Assuming uniform distribution of Hecke polynomials in $H(n)$, the probability of a size $d$ Galois orbit is approximately the probability of a degree $d$ factor of $c_{T_{p}}$, which by (3) and (4) is approximately $n^{-d\alpha}$. Applying the lemma again leads to the stated heuristic.

Combining the Brumer–McGuinness and Watkins heuristics for $d=1$ with 2.2 now suggests 1.1 from the introduction.

The random model for Hecke polynomials in Section 2.1 uses the counting measure on $H_{n}$, i.e., all polynomials in $H_{n}$ are equally likely. If this were the case, the heuristic reasoning above would suggest both upper and lower bounds: $X^{1-\frac{d}{6}-\varepsilon}\ll\mathcal{C}_{d}(X)\ll X^{1-\frac{d}{6}+\varepsilon}$, so the upper bound in 1.1 would be essentially optimal.

However, there are other factors controlling the distribution of Hecke polynomials in $H_{n}$. For instance, trace formulas place arithmetic conditions on the roots of Hecke polynomials. Moreover, there are vertical and horizontal equidistribution results about convergence of the roots to Plancherel and Sato–Tate measures. For this reason, we view our random model as a first approximation to counting degree $d$ forms.

In Section 4.1, we will present data which suggests this heuristic does give an upper bound, but possibly not an optimal one. It is really the data that lends credence to 1.1.

An alternative perspective, which we will explore in Section 3, is that it is more natural to model the distribution of degree $d$ forms by modeling $d$-dimensional modular abelian varieties. The analysis we do there for $d=2$ is compatible with the notion that the random Hecke polynomial heuristic gives a valid upper bound which might not be optimal.

Related to the question of asymptotics are several questions about finiteness. We do not investigate them here, but suggest them for future consideration.

1. (1)

   We can ask: for what $d$ are there infinitely many weight 2 newforms of squarefree level? 1.1 asserts such $d$ must be at most 6, but also suggests the answer could be negative for $d=6$. We know a positive answer for $d=1$, and expect a positive answer for $d=2$. Section 3.4 and our data suggest the answers may be positive for $d=3,4$ also.

2. (2)

   More generally, one can ask the same question in weight $2k$. Roberts’ conjecture [33] implies that there are only finitely many quadratic twist classes of non-CM rational newforms in weight $2k\geq 6$. So 2.2 suggests that there are only finitely many newforms of squarefree level of fixed weight $2k\geq 6$ and any fixed degree $d\geq 1$. One might similarly expect to have finitely many quadratic twist classes of non-CM newforms of fixed degree $d$ and weight $2k\geq 6$.

3. (3)

   One can also ask whether there should be a uniform version of the finiteness part of 1.1, i.e., whether for sufficiently large squarefree $N$ and some fixed $d_{0}$ (possibly $d_{0}=6$), each Atkin–Lehner eigenspace has a unique Galois orbit of size $d\geq d_{0}$. This seems plausible based on 2.2. See 4.4 for a more precise question in prime level.

First recall the connection between weight 2 modular forms and abelian varieties.

Let $N\geq 1$. To a newform $f\in S_{2}(N)$ with $[K_{f}:\mathbb{Q}]=d$, Shimura constructed a $d$-dimensional simple abelian variety $A_{f}/\mathbb{Q}$ satisfying the following properties. First, $A_{f}$ is a quotient of $J_{0}(N)$. Moreover $A_{f}$ is isogenous to $A_{g}$ if and only if $f$ and $g$ are Galois conjugates. The endomorphism algebra $\mathrm{End}^{0}(A_{f})\coloneqq\mathrm{End}(A_{f})\otimes\mathbb{Q}\simeq K_{f}$. The conductor of $A_{f}$ is $N^{d}$. Finally, $L(s,A_{f})=\prod_{\sigma}L(s,f^{\sigma})$, where $f^{\sigma}$ ranges over the Galois conjugates of $f$.

In general, the center of the endomorphism algebra of a $d$-dimensional abelian variety $A$ has degree $\leq d$. If $\mathrm{End}^{0}(A)$ contains a totally real field $K$ of degree $d=\dim A$, then we say $A$ has maximal real multiplication (RM). Any $A_{f}$ as above has maximal RM, and conversely if $A/\mathbb{Q}$ is a simple abelian variety with maximal RM, then it is isogenous to some $A_{f}$ [26, Lemma 3.1]. Hence the correspondence $f\mapsto A_{f}$ yields a bijection between degree $d$ newforms $f$ of weight $2$ and isogeny classes of $d$-dimensional simple abelian varieties $A/\mathbb{Q}$ with maximal RM.

We propose a heuristic approach to predicting coarse asymptotic counts of such objects. Let $K$ be a totally real number field of degree $d$, and $\mathfrak{a}$ be an ideal in $\mathcal{O}_{K}$. The quotient $\mathfrak{H}^{d}/\mathrm{SL}(\mathcal{O}_{K}\oplus\mathfrak{a})$ parametrizes $d$-dimensional complex abelian varieties with RM by $\mathcal{O}_{K}$ together with a polarization structure corresponding to $\mathfrak{a}$ (see [15] for a precise statement). Compactifying this quotient and desingularizing gives a Hilbert modular variety $Y(\mathcal{O}_{K}\oplus\mathfrak{a})$.

Now consider a newform $f\in S_{2}(N)$ with $K_{f}=K$. The abelian variety $A_{f}$ has endomorphism ring an order in $\mathcal{O}_{K}$. Typically we expect it is all of $\mathcal{O}_{K}$, but if not, one can replace $A_{f}$ by an isogenous variety with RM by $\mathcal{O}_{K}$. Thus $f$ corresponds to a rational point $y$ on $Y(\mathcal{O}_{K}\oplus\mathfrak{a})$ for some $\mathfrak{a}$, which we can take to be in a given set of representatives for $\mathrm{Cl}^{+}(K)$.

This correspondence is far from one-to-one. First, replacing $A_{f}$ by an isogenous variety, or modifying the polarization structure, may give a different point $y$ on $Y(\mathcal{O}_{K}\oplus\mathfrak{a})$. Second, if $g$ is another weight 2 newform and $A_{g}$ is $\mathbb{C}$-isogenous to $A_{f}$, then both $f$ and $g$ correspond to the same rational points. Third, this is not a fine moduli space, so not all rational points on $Y(\mathcal{O}_{K}\oplus\mathfrak{a})$ will correspond to abelian varieties defined over $\mathbb{Q}$, and of those that do, some will correspond to non-simple abelian varieties.

That said, it seems reasonable to expect that, generically, quadratic twist classes of Galois orbits of weight 2 newforms correspond to finite sets of rational points on $Y=\bigcup_{a\in\mathrm{Cl}^{+}(K)}Y(\mathcal{O}_{K}\oplus\mathfrak{a})$. Thus one can attempt estimate the number of quadratic twist classes by estimating counts of rational points on $Y$. A priori, it is not clear how different orderings of classes of newforms (e.g., by minimal level) will correlate with different orderings of sets of rational points (e.g., by minimal height, for some choice of height function), and we will speculate more on this for $d=2$ anon.

Note that typically (each component of) $Y$ will be of general type, and one might expect that it has finitely many (and often no) rational points. Hence, for a given $d$, to estimate counts of degree $d$ newforms, it should in principle suffice to consider finitely many $Y$. Moreover, this philosophy suggests that some totally real degree $d$ rationality fields will be more common than others, roughly according to whether the moduli spaces $Y$ have many or few rational points. Of course this is not the only consideration, due to various complications of the correspondence between newforms and rational points mentioned above.

This philosophy is in line with Coleman’s conjecture (e.g., see [5]), which predicts there are only finitely many isomorphism classes of endomorphism algebras for $d$-dimensional abelian varieties over $\mathbb{Q}$. Hence Coleman’s conjecture implies that, for a fixed $d$, only finitely many degree $d$ rationality fields $K_{f}$ occur as $f$ varies over weight 2 newforms.

Now we estimate point counts on certain Hilbert modular surfaces, and pursue the ideas of the previous section for $d=2$.

Let $D>0$ be a fundamental discriminant and $\mathcal{O}_{D}$ be the ring of integers of $\mathbb{Q}(\sqrt{D})$. Let $Y_{-}(D)$ be the Hilbert modular surface constructed from the quotient $\mathfrak{H}^{2}/\mathrm{SL}(\mathcal{O}_{D}\oplus\sqrt{D}\mathcal{O}_{D})$. This parametrizes principally polarized abelian surfaces with RM by $\mathcal{O}_{D}$ (together with a polarization structure). See [38], [15] for details. For brevity, we will write RM $D$ for RM by $\mathcal{O}_{D}$.

For our heuristic point counts, we will use explicit models for Hilbert modular surfaces. For $D<100$, Elkies and Kumar [14] computed models for $Y_{-}(D)$. By work of Hirzebruch and Zagier [19], $Y_{-}(D)$ is rational (i.e., birational to $\mathbb{P}^{2}$) if and only if $D\in\{5,8,12,13,17\}$.

We expect that 100% of degree 2 weight 2 newforms correspond to rational points on Hilbert modular surfaces with the most rational points, i.e., the rational surfaces. While the Hilbert modular surfaces parametrizing non-principally polarizable surfaces with RM $D$ are rational over $\mathbb{C}$ for $D=12,21,24,28,33,60$ (see [38, Theorem VII.3.3]), we at least expect that the 5 rational $Y_{-}(D)$’s should account for a positive proportion of degree 2 weight 2 newforms, and this is supported by data.

To be more precise, the polarization classes of abelian surfaces with RM $D$ are in bijection with the narrow ideal classes $\mathrm{Cl}^{+}(\mathbb{Q}(\sqrt{D}))$. In particular, if $D=5,8,13,17$, the abelian surface is automatically principally polarizable. For $D=12$, the narrow class number is 2, and the moduli spaces for each polarization type are rational (at least over $\mathbb{C}$), so it is not clear whether a positive proportion of abelian surfaces with RM $12$ should be principally polarizable. We cannot yet analyze counts for the non-prinicipal polarization types as we do not know models for the corresponding moduli spaces together with appropriate invariants. However, our data suggest that, at least for prime level, most degree 2 weight 2 newforms have rationality field $\mathbb{Q}(\sqrt{D})$ with $D=5$ or $8$, and thus correspond to points on $Y_{-}(5)$ and $Y_{-}(8)$.

For the remainder of the section, assume $D\in\{5,8,12,13,17\}$. Then $Y_{-}(D)$ is birational to $\mathbb{P}^{2}_{m,n}$. Let $\mathcal{A}_{2}$ be the moduli space for principally polarized abelian surfaces. Forgetting the RM action yields a map $Y_{-}(D)\to\mathcal{A}_{2}$.

Let $\mathcal{M}_{2}$ be the moduli space of genus 2 curves. To a genus 2 curve $C:y^{2}=h(x)$, one associates Igusa–Clebsch invariants $I_{2j}(C)$ for $j=1,2,3,5$. Here $I_{2j}(C)$ can be regarded as a degree $2j$-polynomial in the coefficients of $h(x)$, and $I_{10}(C)$ is the discriminant of $h$. One can realize $\mathcal{A}_{2}$ as weighted projective space $\mathbb{P}^{3}_{1,2,3,5}$ with coordinates $(I_{2}:I_{4}:I_{6}:I_{10})$. The Torelli map $\mathcal{M}_{2}\to\mathcal{A}_{2}$ sends the moduli of $C$ to $(I_{2}(C):I_{4}(C):I_{6}(C):I_{10}(C))$, and the image is the complement of the hyperplane $I_{10}=0$. We note that Igusa–Clebsch invariants are only isomorphism invariants of $C$ up to weighted projective scaling.

Elkies and Kumar [14] gave a birational model for $Y_{-}(D)$. In particular, for generic affine coordinates $(m,n)\in\mathbb{A}^{2}$, one has an associated point on $Y_{-}(D)$ and thus weighted projective coordinates $(I_{2}(m,n):I_{4}(m,n):I_{6}(m,n):I_{10}(m,n))\in\mathcal{A}_{2}$, where the $I_{2j}(m,n)$’s are explicit rational functions in $m,n$.

Now we will attempt to estimate the number of rational points $(m,n)$ with bounded Igusa–Clebsch invariants. First we want to scale Igusa–Clebsch invariants (in $\mathbb{P}^{3}_{1,2,3,5}$) to be integral, as will be the case for the $I_{2j}(C)$’s given a curve $C$ over $\mathbb{Z}$. Let us write $(m,n)=(a/c,b/c)$ for $a,b,c\in\mathbb{Z}$ with $\gcd(a,b,c)=1$. Regarding $a,b,c$ as variables, we scale the $I_{2j}(m,n)$’s to get polynomials $I_{2j}(a,b,c)\in\mathbb{Z}[a,b,c]$’s which are minimal integral over $\mathbb{Z}[a,b,c]$. That is, we scale out denominators, and also any factors of the numerators $\pi$ so that $\pi^{j}\mid I_{2j}(a,b,c)$ for all $j\in\{1,2,3,5\}$ implies $\pi$ is a unit in $\mathbb{Z}[a,b,c]$. The resulting $I_{10}(a,b,c)$’s (which are uniquely determined up to $\pm 1$) are given in 3.1.

Specializing $a,b,c$ to integers, we denote by $I_{2j}^{\min}(a,b,c)\in\mathbb{Z}$ scalings which are minimal integral over $\mathbb{Z}$. Note that $I_{2j}^{\min}(a,b,c)\mid I_{2j}(a,b,c)$ but they are often not equal. E.g., when $D=5$, then the invariants $I_{2j}(1,3,2)$’s are $(-2^{4}\cdot 5^{3},2^{8}\cdot 5^{4},-2^{15}\cdot 5\cdot 599,2^{21}\cdot 3^{8})$, whereas the $\mathbb{Z}$-minimal invariants $I_{2j}^{\min}(a,b,c)$ are obtained by scaling out a factor of $2^{4}$, i.e., they are $(-5^{3},5^{4},-2^{3}\cdot 5\cdot 599,2\cdot 3^{8})$.

First we want to estimate, in terms of a real parameter $T$, the growth of the cardinality of

where $U_{D}$ consists of $(a,b,c)$ such that the map $(a/c,b/c)\to\mathcal{A}_{2}$ is either undefined (e.g., $c=0$) or is not finite-to-one (e.g., for $D=8$, all points with $m=a/c=-1$ map to $(1:0:0:0)\in\mathcal{A}_{2}$). Really our interest is just in bounding $I^{\min}_{10}$, but we impose bounds on the other $I_{2j}^{\min}$’s to guarantee finiteness of $Z_{D}(T)$.

Precise estimates are difficult, so we make two simplifications which are sufficient to get lower bounds: (1) We impose the stronger bound $|I^{\min}_{2j}(a,b,c)|\leq|I_{2j}(a,b,c)|<T^{2j}$. (2) We will suppose each monomial in $I_{2j}(a,b,c)$ is bounded by $T^{2j}$. Note that (1) and (2) can respectively be thought of as non-archimedean and archimedean simplifications to monomials.

The lower bounds in the proposition are the optimal ones we could find using the $\mathbb{P}^{2}$ or $\mathbb{P}^{1}\times\mathbb{P}^{1}$ parametrizations for $(m,n)$ by allowing either each of $a,b,c$ or $r,s,t,u$ to vary independently up to some power of $T$ (not necessarily the same power for each variable). We remark that the optimal exponents for lower bounds using the $\mathbb{P}^{1}\times\mathbb{P}^{1}$ parametrization for $D=5,8,13,17$ are $2,4/3,1,1$, respectively. To get these exponents, for $D=5$ one can take each of $|r|,|s|,|t|,|u|\ll T$. For $D=8$, one takes $|r|,|s|\ll T^{2/3}$ and $|t|,|u|\ll 1$. For both $D=13$ and $D=17$, one takes $|r|,|s|\ll 1$ and $|t|,|u|\ll T^{1/2}$.

It is not clear if the simplifications to monomials affect the exponents in our estimates, but if the $I_{2j}$ polynomials are sufficiently general type, one might expect they only account for a multiplicative factor of size $O(1+T^{\varepsilon})$. Note that in the Brumer–McGuinness heuristics, it is believed the analogous archimedean simplification (2) only affects counts by an $O(1)$ factor.

A more serious reason to doubt the exponents in these lower bounds are optimal is that there may exist (i) other rational parametrizations of $Y_{-}(D)$ where the $I_{2j}$ degrees are smaller, or (ii) special curves on $Y_{-}(D)$ which intersect $Z_{D}$ in an especially large number of rational points. Indeed, the proof makes clear that different parametrizations may yield different counts for a given surface.

Now we explain how these estimates for $\#Z_{D}(T)$ are related to counting quadratic twist classes of weight 2 newforms $f$ with rationality field $\mathbb{Q}(\sqrt{D})$. As explained above, a positive proportion of $f$ (at least if $D\neq 12$) should correspond to rational points on $Y_{-}(D)$. Conversely, a rational point on $Y_{-}(D)$ may not come from a simple abelian surface with RM defined over $\mathbb{Q}$—one needs that the Mestre conic has a point, the RM is defined over $\mathbb{Q}$, and the Jacobian is nonsplit. However, we expect that these obstructions will only contribute logarithmic factors to asymptotics. (See [9] for details about when the Mestre obstruction vanishes and when the RM is defined over $\mathbb{Q}$.)

Consider a point $(a,b,c)\in Z_{D}(T)$ which corresponds to a $\bar{\mathbb{Q}}$-isomorphism class of simple abelian surfaces $A/\mathbb{Q}$. Generically this $\bar{\mathbb{Q}}$-isomorphism class should be the family of Jacobians of quadratic twists of a genus 2 curve $C/\mathbb{Q}$ with RM $D$. Suppose this, and assume $C$ is a minimal quadratic twist of conductor $N_{C}$. Then $(a,b,c)$ corresponds to the quadratic twist class of some minimal newform $f$ of level $N_{f}$ where $N_{C}=N_{f}^{2}$. One can write down a minimal integral model for $C$, and the polynomially-defined Igusa–Clebsch invariants $I_{2j}(C)$ are necessarily divisible by $I_{2j}^{\min}(a,b,c)$. Thus $I_{10}^{\min}(a,b,c)$ divides the minimal discriminant $\Delta_{C}=2^{-12}I_{10}(C)$ of $C$. One also knows that the conductor $N_{C}\mid\Delta_{C}$.

Now we would like to understand how $N_{C}$ relates to $I_{10}^{\min}(a,b,c)$ or $I_{10}(a,b,c)$. There are no general upper or lower bounds, and in fact there are competing issues in opposite directions. One is that $\Delta_{C}$ may be much larger than either $I_{10}^{\min}(a,b,c)$ or $I_{10}(a,b,c)$, and numerically this is quite typical. E.g., if $p^{m}\parallel I_{10}^{\min}(a,b,c)$ it often happens that $p^{m+10}\mid\Delta_{C}$ (see [24] for local results). On the other hand, the prime powers occurring in $N_{C}$ are often smaller than those in $\Delta_{C}$. E.g., if $p^{3}\parallel\Delta_{C}$ then necessarily $p^{2}\parallel N_{C}$ or $p\nmid N_{C}$. Some preliminary investigations suggest that the latter issue has more impact, and asymptotic counts of curves by $I^{\min}_{10}$ may essentially be lower bounds (up to logarithmic factors) of counts by conductor. This leads us to ask:

Observe that all of these exponents are less than the exponent of 2/3 from the $d=2$ case of 1.1. We will compare these speculative lower bounds with our prime level data below.

Even if these lower bounds hold, there are several reasons why they may not be sharp. For one, there is the issue of 3.3. Perhaps most serious is the issue of how prime powers in $N_{C}$ relate to prime powers in $\Delta_{C}$ mentioned above.

Another potential issue comes from the way we defined $Z_{D}(T)$: for the comparison with conductors, we are only interested in bounds on $I_{10}^{\min}$, and there may be many points with $I^{\min}_{10}$ small relative to $I_{2}^{\min},I_{4}^{\min},I_{6}^{\min}$. In particular, for $D=12$, if one views $I_{2j}(a,b,c)$ as a polynomial in $b$ with $a=0$ and $c$ fixed, then the degrees are $2,2,4,4$, so one gets at least $T^{5/2}$ points with $I_{10}^{\min}\ll T^{10}$, which is a better than the lower bound $\gg T^{2}$ in Proposition 3.2. These rational points correspond to the curve $m=0$ on $Y_{-}(12)$, and numerical investigations suggests this is a Shimura curve parametrizing abelian surfaces with geometric endomorphism algebra the quaternion algebra of discriminant 6. Consequently, one might be able to take $\alpha=\tfrac{1}{2}$ in 3.4 when $D=12$. However, it is not clear that there is a family of genus 2 curves with RM 12 over $\mathbb{Q}$ that would achieve $\alpha=\tfrac{1}{2}$.

There are several known families of genus 2 curves with RM 5 and RM 8. These can be used to give lower bounds on counting such curves with bounded discriminant, and therefore conductor. For a genus 2 curve $C/\mathbb{Q}$, let $\Delta_{C}$ denote the minimal integral discriminant.