(iv) $p = 2$ and there is a $p'$ $| N$ with $p' \equiv 3 \mod 4$; or

(v) $p = 2$ and $X_0(2^n \cdot 5)_{\mathbb{Z}(2)}$ has rational singularities and $N \neq 2^n$; or

(vi) $p = 3$ (resp., $p = 2$) and for the level $\Gamma_0(p^n)$ universal deformation ring $R$ of $(E, C)$, where $E/\overline{\mathbb{F}}_p$ is the elliptic curve with $j = 0$ and $C \subset E$ the cyclic (in the sense of Drinfeld) subgroup of order $p^n$, and for every subgroup $G' \subset G := \text{Aut}(E)/\{\pm 1\}$ of $p$-power order, $R^{G'}$ has rational singularities (resp., same, but if $N \neq 2^n$, then we may restrict to cyclic $G'$),

then $X_0(N)_{\mathbb{Z}(p)}$ has rational singularities.

*Proof.* Since $X_0(N)_{\mathbb{Z}(p)}$ is regular away from the $\overline{\mathbb{F}}_p$-points $x$ with $j = 0$ or $j = 1728$ (see [Čes17, Theorem 6.7]), we need to show that $\mathcal{O}_{X_0(N),x}$ has rational singularities for every such $x$. By Lipman's [SP, Theorem 0BGP], there exists a proper birational map $Z \to \text{Spec}(\mathcal{O}_{X_0(N),x})$ with $Z$ regular and, by [EGA IV₄, Corollaire 6.4.2, Scholie 7.8.3 (v)] (see also [Gre76, Corollary 5.6]), the $\hat{\mathcal{O}}_{X_0(N),x}^{\text{sh}}$-base change of $Z$ is regular. Thus, by checking the vanishing $H^1(Z, \mathcal{O}_Z) = 0$ after flat base change, $\mathcal{O}_{X_0(N),x}$ has rational singularities if and only if so does $\hat{\mathcal{O}}_{X_0(N),x}^{\text{sh}}$. However, by [DR73, Chapitre I, Section (8.2.1)] (or [Ols06, Theorem 2.12]), we have

$$ \hat{\mathcal{O}}_{X_0(N),x}^{\text{sh}} \cong (\hat{\mathcal{O}}_{\mathcal{X}_0(N),x}^{\text{sh}})^{\text{Aut}(x)/\{\pm 1\}}, \quad (6.10.1) $$

and $\hat{\mathcal{O}}_{\mathcal{X}_0(N),x}^{\text{sh}}$ is regular by [KM85, Theorem 6.6.1]. Thus, (i), (ii), and (iv) follow from Lemmas 6.8 and 6.9.

In (vi), the unique $E$ is supersingular, $C$ is the kernel of the $p^n$-fold relative Frobenius (see [KM85, Lemma 12.2.1]) and hence is preserved by $\text{Aut}(E)$, and $x$ maps to $(E, C)$. Moreover, $E[N/p^n]$ is étale, so its subgroups $C' \subset E[N/p^n]$ deform uniquely, and hence $R \cong \hat{\mathcal{O}}_{\mathcal{X}_0(N),x}^{\text{sh}}$ by the modular interpretation of $\mathcal{X}_0(N)$. Since $G$ injects into (in fact, equals) $SL_2(\overline{\mathbb{F}}_3)/\{\pm 1\}$ if $p = 2$ and $SL_2(\mathbb{F}_2)$ if $p = 3$ (see [KM85, Corollary 2.7.2], also [Del75, Proposition 5.9 (IV)-(V), Section 7.4]), its $p$-Sylow subgroup $G^{(p)} \subset G$ is normal. Thus, the same holds for $H := \text{Aut}(x)/\{\pm 1\} \subset G$, so that $R^H \cong (R^{H^{(p)}})^{H/H^{(p)}}$. The assumption of (vi) ensures that $R^{H^{(p)}}$ has rational singularities, so, by Lemma 6.8, so does $R^H \cong \hat{\mathcal{O}}_{X_0(N),x}^{\text{sh}}$ (see (6.10.1)). To conclude (vi), we note that $H$ is cyclic when $p = 2$ and $N \neq 2^n$; then the preimage of $H$ in $\text{Aut}(E)$ lies in the cyclic group $(\mathbb{Z}/p'\mathbb{Z})^\times$ for an odd prime $p' \mid N$ (see the proof of Lemma 6.9).

To show that (iii) and (v) follow from (vi), we set $\Gamma := \Gamma_0(3^n \cdot 7)$ or $\Gamma := \Gamma_0(3^n) \cap \tilde{C}_3$ in (iii) and $\Gamma := \Gamma_0(2^n \cdot 5)$ in (v) and, in view of the above, especially the analogue of (6.10.1) for $\mathcal{X}_\Gamma$ and the insensitivity of the universal deformation ring $R$ of $(E, C)$ in (vi) to tame level, we need to show that every cyclic subgroup $G' \subset \text{Aut}(E)/\{\pm 1\}$ of $p$-power order is $\text{Aut}(z)/\{\pm 1\}$ for some $z \in \mathcal{X}_\Gamma(\overline{\mathbb{F}}_p)$. For $p = 3$, the unique $G'$ of 3-power order is $\mathbb{Z}/3\mathbb{Z}$ and its preimage $\tilde{G}' \subset \text{Aut}(E)$ is $\mathbb{Z}/6\mathbb{Z}$. Since $\overline{\mathbb{F}}_7$ contains sixth roots of unity, the action of $\tilde{G}'$ on $E[7]$ is diagonalizable and either of the resulting $\tilde{G}'$-stable $\overline{\mathbb{F}}_7$-lines $C' \subset E[7]$ is the 7-primary part of a level structure that determines the desired $z$ for $\Gamma = \Gamma_0(3^n \cdot 7)$. Similarly, the faithful action of $G'$ on $E[2]$ determines a $\tilde{C}_3$-structure, and so a desired $z$ for $\Gamma = \Gamma_0(3^n) \cap \tilde{C}_3$. For $p = 2$, the argument is analogous: now
$G'$ is $\mathbb{Z}/2\mathbb{Z}$ but is no longer unique (the 2-Sylow of $\mathrm{SL}_2(\mathbb{F}_3)/\{\pm 1\}$ is $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$), its preimage $\tilde{G}'$ is $\mathbb{Z}/4\mathbb{Z}$, and one can diagonalize the action of $\tilde{G}'$ on $E[5]$ because $\mathbb{F}_5$ contains fourth roots of unity.

■

**Remark 6.11.** By the preceding proof, if $N \neq 2^n$, then the $p$-Sylow subgroup of the exceptional automorphism group at each $\overline{\mathbb{F}}_p$-point of $\mathcal{X}_0(N)$ is normal and either trivial or $\mathbb{Z}/p\mathbb{Z}$ (the latter can occur only for $p=2$ and $p=3$). In particular, Lemma 6.8 and the preceding proof show that for any proper birational $\pi: Z \to X_0(N)$ with $Z$ normal, the $\mathcal{O}_{X_0(N)}$-module $R^1\pi_*(\mathcal{O}_Z)$ is killed by 6.

A big portion of the following partial positive answer to Question 6.4 appeared in [Ray91, Théorème 2]: our main improvement is the inclusion of the cases $\mathrm{val}_p(N) = 2$ for $p \le 3$.

**Theorem 6.12.** For a prime $p$, the modular curve $(X_0(N))_{\mathbb{Z}(p)}$ has rational singularities whenever

(a) $p \ge 5$; or

(b) $p=3$ and either $\mathrm{val}_p(N) \le 2$ or there is a prime $p' \mid N$ with $p' \equiv 2 \pmod 3$; or

(c) $p=2$ and either $\mathrm{val}_p(N) \le 2$ or there is a prime $p' \mid N$ with $p' \equiv 3 \pmod 4$.

*Proof.* Thanks to Proposition 6.10, it suffices to check that $X_0(7)$, $X_0(21)$, and $X_{\Gamma_0(9)\cap\tilde{C}_3}$, as well as $X_0(5)$, $X_0(10)$, $X_0(20)$, $X_0(1)$, $X_0(2)$, and $X_0(4)$, have rational singularities. We have already done this in Proposition 6.6 (see also Example 6.7).

■

**Remark 6.13.** The method would show that $X_0(N)$ has rational singularities for every $N \neq 2^n$ equal to a conductor of an elliptic curve over $\mathbb{Q}$ if one knew that $X_{\Gamma_0(27)\cap\tilde{C}_3}$, $X_{\Gamma_0(81)\cap\tilde{C}_3}$, and $X_{\Gamma_0(243)\cap\tilde{C}_3}$ (or, if one prefers, $X_0(27 \cdot 7)$, $X_0(81 \cdot 7)$, and $X_0(243 \cdot 7)$), as well as $X_0(8 \cdot 5)$, $X_0(16 \cdot 5)$, $X_0(32 \cdot 5)$, $X_0(64 \cdot 5)$, $X_0(128 \cdot 5)$, $X_0(256 \cdot 5)$, $X_0(64)$, $X_0(128)$, and $X_0(256)$ have rational singularities (for well-known conductor exponent bounds for an elliptic curve over $\mathbb{Q}$, see [Pap93, Corollaire du Théorème 1]).

**Corollary 6.14.** For a normalized newform $f \in H^0(\mathcal{X}_0(N)_{\mathbb{Q}}, \omega^{\otimes 2}(-\text{cusps}))$ (see Section 5.5) and the Néron model $\mathcal{J}_0(N)$ over $\mathbb{Z}$ of the Jacobian $J_0(N)$ of $X_0(N)_{\mathbb{Q}}$,

$$6 \cdot \omega_f \in H^0(\mathcal{J}_0(N), \Omega^1), \quad \text{where } \omega_f \text{ is the differential associated to } f;$$

if $X_0(N)$ has rational singularities, then even $\omega_f \in H^0(\mathcal{J}_0(N), \Omega^1)$.

*Proof.* The Manin conjecture for the quotient $\pi: J_0(N) \to E$ with connected $\mathrm{Ker}(\pi)$ determined by $f$ predicts that $\omega_f$ is the pullback of a Néron differential $\omega_E$ of the elliptic curve $E$. By the functoriality of Néron models, this pullback lies in $H^0(\mathcal{J}_0(N), \Omega^1)$, so, by, for instance, Cremona's [ARS06, Theorem 5.2] that verified the Manin conjecture for small $N$, we may assume that $N \neq 2^n$. By Proposition 6.2, there is an inclusion $H^0(\mathcal{J}_0(N), \Omega^1) \hookrightarrow H^0(X_0(N), \Omega)$ that is an isomorphism if and only if $X_0(N)$ has rational singularities and, by Remark 6.11, in general its cokernel is killed by 6. Thus, it remains to recall from Theorem 5.15 that $\omega_f \in H^0(X_0(N), \Omega)$. ■
## 7. A relation between the Manin constant and the modular degree

Our final goal is to use the work above to establish Theorems 1.1 and 1.2. The following basic fact is the underlying source of the relationship between the modular degree and the Manin constant.

**Lemma 7.1.** For a field $k$, a proper, smooth $k$-curve $X$ with Jacobian $J := \text{Pic}_{X/k}^0$, a $k$-surjection $\phi: X \to E$ onto an elliptic curve, a point $P \in X(k)$ with $\phi(P) = 0$, the closed immersion $i_P: X \hookrightarrow J$ given by $Q \mapsto \mathcal{O}_X(Q-P)$, and the homomorphism $\pi: J \to E$ obtained from $\phi$ by the Albanese functoriality of $J$, the composition $\pi \circ \pi^\vee: E \to J \to E$ is multiplication by $\deg \phi$.

*Proof.* The existence of $\phi$ implies that $X$ has genus $> 0$, and the map $\pi: J \to E$ is characterized by $\mathcal{O}_X(Q-P) \mapsto \phi(Q)$; see [Mil86, Proposition 6.1]. Moreover, by [Mil86, Lemma 6.9 and Remark 6.10 (c)], the map $\text{Pic}^0(i_P)$ is the negative of the inverse of the canonical principal polarization of $J$, and the canonical principal polarization of $E$ sends a $Q \in E(\bar{k})$ to $\mathcal{O}_{E_{\bar{k}}}([0] - [Q])$ (see also [Con04, Example 2.5]). In particular, the map $\text{Pic}^0(\phi) = \text{Pic}^0(i_P) \circ \pi^\vee$ sends such a $Q$ to $\mathcal{O}_{X_{\bar{k}}}([\phi^{-1}(0)] - [\phi^{-1}(Q)])$ and, by taking into account the canonical principal polarization of $J$, we find that $\pi \circ \pi^\vee$ sends $Q$ to $\deg \phi \cdot Q$. ■

**Theorem 7.2.** For an elliptic curve $E$ over $\mathbb{Q}$ of conductor $N$, a Néron differential $\omega_E \in H^0(E, \Omega^1)$, the normalized newform $f$ determined by $E$, its associated differential $\omega_f \in H^0(X_0(N)_{\mathbb{Q}}, \Omega^1)$, a subgroup $\Gamma_1(N) \subset \Gamma \subset \Gamma_0(N)$, and a prime $p$, if for some subgroup $\Gamma \subseteq \Gamma' \subseteq \Gamma_0(N)$ the curve $(X_{\Gamma'})_{\mathbb{Z}(p)}$ has rational singularities (see Theorem 6.12), then every surjection $\phi: (X_\Gamma)_{\mathbb{Q}} \to E$ satisfies

$$ \text{val}_p(c_\phi) \le \text{val}_p(\deg(\phi)) \quad \text{with } c_\phi \in \mathbb{Z} \text{ defined by } \phi^*(\omega_E) = c_\phi \cdot \omega_f. $$

Without the rational singularities assumption, we still have

$$ \text{val}_p(c_\phi) \le \text{val}_p(\deg(\phi)) + \begin{cases} 1 & \text{if } p=2 \text{ with } \text{val}_2(N) \ge 3 \text{ and there is no } p' \mid N \text{ with } p' \equiv 3 \mod 4, \\ 1 & \text{if } p=3 \text{ with } \text{val}_3(N) \ge 3 \text{ and there is no } p' \mid N \text{ with } p' \equiv 2 \mod 3, \\ 0 & \text{otherwise.} \end{cases} $$

*Proof.* By Theorem 5.15, we have $\omega_f \in H^0(X_{\Gamma'}, \Omega)$. Thus, by Proposition 6.2, the rational singularity assumption ensures that $\omega_f \in H^0((\mathcal{J}_{\Gamma'})_{\mathbb{Z}(p)}, \Omega^1)$ where $\mathcal{J}_{\Gamma'}$ is the Néron model of the Jacobian $J_{\Gamma'}$ of $(X_{\Gamma'})_{\mathbb{Q}}$. We choose a $P \in X_\Gamma(\mathbb{Q})$, for instance, a rational cusp, and consider the resulting embeddings $(X_\Gamma)_{\mathbb{Q}} \hookrightarrow J_\Gamma$ and $(X_{\Gamma'})_{\mathbb{Q}} \hookrightarrow J_{\Gamma'}$. By the Albanese functoriality of the Jacobian, the map $X_\Gamma \to X_{\Gamma'}$ induces a morphism $\mathcal{J}_\Gamma \to \mathcal{J}_{\Gamma'}$, and we conclude by pullback that

$$ \omega_f \in H^0((\mathcal{J}_\Gamma)_{\mathbb{Z}(p)}, \Omega^1) \qquad (7.2.1) $$
(here we use the compatibility of the identification $H^0((X_{\Gamma'})_{\mathbb{Q}}, \Omega^1) \cong H^0(J_{\Gamma'}, \Omega^1)$ obtained by pullback along $(X_{\Gamma'})_{\mathbb{Q}} \hookrightarrow J_{\Gamma'}$ with its counterpart obtained by Grothendieck-Serre duality as in (6.2.2); see [Con00, Theorem B.4.1]). By postcomposing with a translation, we may assume that $\phi(P) = 0$, and we then let $\pi: J_{\Gamma} \to E$ be the map that $\phi$ induces via the Albanese functoriality. Lemma 7.1 ensures that $\pi \circ \pi^{\vee}: E \to J_{\Gamma} \to E$ is multiplication by $\deg(\phi)$, so the same holds for the induced $\mathcal{E} \to \mathcal{J}_{\Gamma} \to \mathcal{E}$ on Néron models. Thus, by pullback,

$$ \deg(\phi) \cdot \omega_E = c_{\phi} \cdot (\pi^{\vee})^*(\omega_f). $$

Since $c_{\phi} \in \mathbb{Z}$ by Lemma 6.5 and

$$ (\pi^{\vee})^*(\omega_f) \in H^0(\mathcal{E}_{\mathbb{Z}(p)}, \Omega^1) \cong \mathbb{Z}(p) \cdot \omega_E $$

by (7.2.1), we obtain the sought

$$ \mathrm{val}_p(c_{\phi}) \leq \mathrm{val}_p(\deg(\phi)). $$

Without the rational singularities assumption, by Corollary 6.14 and the Albanese functoriality as above, we still have $6 \cdot \omega_f \in H^0(\mathcal{J}_{\Gamma}, \Omega^1)$, so the same argument gives $\mathrm{val}_p(c_{\phi}) \leq \mathrm{val}_p(\deg(\phi)) + \mathrm{val}_p(6)$. In particular, by also using Theorem 6.12, we obtain the claimed last display in the statement. ■

Since $X_1(N)$ almost always agrees with the regular $\mathcal{X}_1(N)$, we now show that the above minor hypothetical exceptions to the divisibility $c_{\phi} | \deg(\phi)$ cannot occur for parametrizations by $X_1(N)_{\mathbb{Q}}$.

**Corollary 7.3.** For an elliptic curve $E$ over $\mathbb{Q}$ of conductor $N$, a Néron differential $\omega_E \in H^0(E, \Omega^1)$, the normalized newform $f$ determined by $E$, and its associated differential $\omega_f \in H^0(X_1(N)_{\mathbb{Q}}, \Omega^1)$, every surjection $\phi: X_1(N)_{\mathbb{Q}} \to E$ satisfies

$$ c_{\phi} | \deg(\phi) \quad \text{with } c_{\phi} \in \mathbb{Z} \text{ defined by } \phi^*(\omega_E) = c_{\phi} \cdot \omega_f. $$

*Proof.* By Theorem 7.2, we have $\mathrm{val}_p(c_{\phi}) \leq \mathrm{val}_p(\deg(\phi))$ for every prime $p \geq 5$. For the remaining $p = 2$ and $p = 3$, Theorem 7.2 applied with $\Gamma = \Gamma' = \Gamma_1(N)$ gives the same as long as $X_1(N)_{\mathbb{Z}(p)}$ is regular. By [KM85, Corollary 2.7.3, Theorem 5.5.1] and [Čes17, Lemma 4.1.3, Theorem 4.4.4], this happens whenever $p' | N$ for a prime $p' \geq 5$. Thus, we may assume that $N = 2^a \cdot 3^b$, in fact, by the last statement of Theorem 7.2, even that $N = 2^a$ or $N = 3^b$ (so $a \leq 8$ and $b \leq 5$; see [Pap93, Corollaire du Théorème 1]). For any isogeny $\psi: E' \to E$, since the composition with the dual isogeny is multiplication by $\deg(\psi)$, we have $\psi^*(\omega_E) = c_{\psi} \cdot \omega_{E'}$ for some $c_{\psi} \in \mathbb{Z}$ with $c_{\psi} | \deg(\psi)$. Thus, we may assume that $\phi$ does not factor through any such $\psi$. For low conductor curves, by Cremona's [ARS06, Theorem 5.2], the Manin constant of such optimal parametrizations by $X_0(N)_{\mathbb{Q}}$ is $\pm 1$. Thus, Lemma 6.5 allows us to conclude the same for parametrizations by $X_1(N)_{\mathbb{Q}}$ with $N = 2^a$ and $N = 3^b$, so that indeed $\mathrm{val}_p(c_{\phi}) \leq \mathrm{val}_p(\deg(\phi))$. ■

*Acknowledgments.* The first-named author thanks Naoki Imai for numerous discussions about the Manin constant during 2017–2018. We thank the referee for helpful comments. We thank the Tech-
nische Universität Darmstadt and the Queen Mary University of London for hospitality during visits between the authors.

*Funding.* The first-named author acknowledges the support of the ANR-19-CE40-0015-02 COLOSS. The second-named author was partially funded by the DFG research group 1920 and the LOEWE research unit “Uniformized Structures in Arithmetic and Geometry.” The third-named author acknowledges the support of the Leverhulme Trust Research Project Grant RPG-2018-401. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 851146).

## References

[AU96] Abbes, A., Ullmo, E.: À propos de la conjecture de Manin pour les courbes elliptiques modulaires. Compos. Math. 103, 269–286 (1996) Zbl 0865.11049 MR 1414591

[ARS06] Agashe, A., Ribet, K., Stein, W. A.: The Manin constant. Pure Appl. Math. Quart. 2, 617–636 (2006) Zbl 1109.11032 MR 2251484

[AT09] Artin, E., Tate, J.: Class field theory. AMS Chelsea Publishing, Providence, RI (2009) Zbl 1179.11040 MR 2467155

[Ass19] Assing, E.: On the size of *p*-adic Whittaker functions. Trans. Amer. Math. Soc. 372, 5287–5340 (2019) Zbl 1471.11168 MR 4014277

[Ass19e] Assing, E.: Errata to [Ass19]. http://www.math.uni-bonn.de/people/assing/Notes/Errata.pdf

[AL78] Atkin, A. O. L., Li, W. C. W.: Twists of newforms and pseudo-eigenvalues of W-operators. Invent. Math. 48, 221–243 (1978) Zbl 0369.10016 MR 508986

[BR99] Bayer, P., Rio, A.: Dyadic exercises for octahedral extensions. J. Reine Angew. Math. 517, 1–17 (1999) Zbl 0941.12001 MR 1728550

[BDP17] Bertolini, M., Darmon, H., Prasanna, K.: *p*-adic *L*-functions and the coniveau filtration on Chow groups. J. Reine Angew. Math. 731, 21–86 (2017) Zbl 1436.11141 MR 3709060

[Bla06] Blasius, D.: Hilbert modular forms and the Ramanujan conjecture. In: Noncommutative geometry and number theory, Aspects of Mathematics E37, Vieweg, Wiesbaden, 35–56 (2006) Zbl 1183.11023 MR 2327298

[BLR90] Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 21, Springer, Berlin (1990) Zbl 0705.14001 MR 1045822

[BCDT01] Breuil, C., Conrad, B., Diamond, F., Taylor, R.: On the modularity of elliptic curves over $\mathbb{Q}$: wild 3-adic exercises. J. Amer. Math. Soc. 14, 843–939 (2001) Zbl 0982.11033 MR 1839918

[BN19] Brunault, F., Neururer, M.: Fourier expansions at cusps. Ramanujan J. 53, 423–437 (2020) Zbl 1468.11099 MR 4161938

[Bum97] Bump, D.: Automorphic forms and representations. Cambridge Studies in Advanced Mathematics 55, Cambridge University Press, Cambridge (1997) Zbl 0911.11022 MR 1431508

[BH06] Bushnell, C. J., Henniart, G.: The local Langlands conjecture for GL(2). Grundlehren der mathematischen Wissenschaften 335, Springer, Berlin (2006) Zbl 1100.11041 MR 2234120

[Cal78] Callahan, T.: Some orbital integrals and a technique for counting representations of $GL_2(F)$. Canad. J. Math. 30, 431–448 (1978) Zbl 0385.22010 MR 499002

[Cas73] Casselman, W.: On some results of Atkin and Lehner. Math. Ann. 201, 301–314 (1973) Zbl 0239.10015 MR 337789
[Čes17] Česnavičius, K.: A modular description of $\mathcal{X}_0(n)$. Algebra Number Theory **11**, 2001–2089 (2017) Zbl **1429.14010** MR **3735461**

[Čes18] Česnavičius, K.: The Manin–Stevens constant in the semistable case. arXiv:1604.02165v2 (2018)

[Čes18] Česnavičius, K.: The Manin constant in the semistable case. Compos. Math. **154**, 1889–1920 (2018) Zbl **1430.11086** MR **3867287**

[Coh19] Cohen, H.: Expansions at cusps and Petersson products in Pari/GP. In: Elliptic integrals, elliptic functions and modular forms in quantum field theory, Texts and Monographs in Symbolic Computation, Springer, Cham, 161–181 (2019) MR **3889557**

[Con00] Conrad, B.: Grothendieck duality and base change. Lecture Notes in Mathematics **1750**, Springer, Berlin (2000) Zbl **0992.14001** MR **1804902**

[Con04] Conrad, B.: Polarizations. Unpublished notes. http://virtualmath1.stanford.edu/~conrad/vigregroup/vigre04/polarization.pdf (2004)

[CS18] Corbett, A., Saha, A.: On the order of vanishing of newforms at cusps. Math. Res. Lett. **25**, 1771–1804 (2018) Zbl **1442.11092** MR **3934844**

[Cre22] Cremona, J.: Manin constants and optimal curves. https://johncremona.github.io/eddata/manin.txt (2022)

[Del71] Deligne, P.: Formes modulaires et représentations $l$-adiques. In: Séminaire Bourbaki. Vol. 1968/69: Exposés 347–363, Lecture Notes in Mathematics 175, Springer, Berlin, exp. 355, 139–172 (1971) Zbl **0206.49901** MR **3077124**

[Del73a] Deligne, P.: Formes modulaires et représentations de GL(2). In: Modular functions of one variable, II (Antwerp, 1972), Lecture Notes in Mathematics 349, Springer, Berlin, 55–105 (1973) Zbl **0271.10032** MR **0347738**

[Del73b] Deligne, P.: Les constantes des équations fonctionnelles des fonctions $L$. In: Modular functions of one variable, II (Antwerp, 1972), Lecture Notes in Mathematics 349, Springer, Berlin, 501–597 (1973) Zbl **0271.14011** MR **0349635**

[Del75] Deligne, P.: Courbes elliptiques: formulaire d'après J. Tate. In: Modular functions of one variable, IV (Antwerp, 1972), Lecture Notes in Mathematics 476, Springer, Berlin, 53–73 (1975) Zbl **1214.11075** MR **0387292**

[DR73] Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Modular functions of one variable, II (Antwerp, 1972), Lecture Notes in Mathematics 349, Springer, Berlin, 143–316 (1973) Zbl **0281.14010** MR **0337993**

[DI95] Diamond, F., Im, J.: Modular forms and modular curves. In: Seminar on Fermat's Last Theorem (Toronto, ON, 1993–1994), CMS Conference Proceedings 17, American Mathematical Society, Providence, RI, 39–133 (1995) Zbl **0853.11032** MR **1357209**

[DS05] Diamond, F., Shurman, J.: A first course in modular forms. Graduate Texts in Mathematics 228, Springer, New York (2005) Zbl **1062.11022** MR **2112196**

[DN18] Dickson, M., Neururer, M.: Products of Eisenstein series and Fourier expansions of modular forms at cusps. J. Number Theory **188**, 137–164 (2018) Zbl **1435.11078** MR **3778627**

[Edi91] Edixhoven, B.: On the Manin constants of modular elliptic curves. In: Arithmetic algebraic geometry (Texel, 1989), Progress in Mathematics 89, Birkhäuser Boston, Boston, MA, 25–39 (1991) Zbl **0749.14025** MR **1085254**

[Edi01] Edixhoven, B.: Modular parametrizations at primes of bad reduction (2001)

[Fla79] Flath, D.: Decomposition of representations into tensor products. In: Automorphic forms, representations and $L$-functions (Corvallis, OR, 1977), Proceedings of Symposia in Pure Mathematics 33, Part 1, American Mathematical Society, Providence, RI, 179–183 (1979) Zbl **0414.22019** MR **546596**
[Ge175] Gelbart, S. S.: *Automorphic forms on adèle groups*. Annals of Mathematics Studies 83, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo (1975) Zbl 0329.10018 MR 0379375

[Gre76] Greco, S.: *Two theorems on excellent rings*. Nagoya Math. J. **60**, 139–149 (1976) Zbl 0308.13010 MR 409452

[EGA III₁] Grothendieck, A.: *Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérentes*. I. Inst. Hautes Études Sci. Publ. Math. **11** (1961) Zbl 0118.36206 MR 217085

[EGA IV₂] Grothendieck, A.: *Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas*. II. Inst. Hautes Études Sci. Publ. Math. **24** (1965) Zbl 0153.22301 MR 199181

[EGA IV₄] Grothendieck, A.: *Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas*. IV. Inst. Hautes Études Sci. Publ. Math. **32** (1967) Zbl 0153.22301 MR 0238860

[Har66] Hartshorne, R.: *Residues and duality*. Lecture Notes in Mathematics 20, Springer, Berlin (1966) Zbl 0212.26101 MR 0222093

[Hen79] Henniart, G.: *Représentations du groupe de Weil d'un corps local. Publications Mathématiques d'Orsay 2*, Université de Paris-Sud, Département de Mathématiques, Orsay (1979) Zbl 0476.12015 MR 551495

[HNS19] Hu, Y., Nelson, P. D., Saha, A.: *Some analytic aspects of automorphic forms on GL(2) of minimal type*. Comment. Math. Helv. **94**, 767–801 (2019) Zbl 1443.11062 MR 4046004

[JL70] Jacquet, H., Langlands, R. P.: *Automorphic forms on GL(2)*. Lecture Notes in Mathematics 114, Springer, Berlin (1970) Zbl 0236.12010 MR 0401654

[SP] de Jong, A. J., et al.: *The Stacks Project*. http://stacks.math.columbia.edu

[Kat73] Katz, N. M.: *p-adic properties of modular schemes and modular forms*. In: *Modular functions of one variable, III (Antwerp, 1972)*, Lecture Notes in Math. 350, Springer, Berlin, 69–190 (1973) Zbl 0271.10033 MR 0447119

[KM85] Katz, N. M., Mazur, B.: *Arithmetic moduli of elliptic curves*. Annals of Mathematics Studies 108, Princeton University Press, Princeton, NJ (1985) Zbl 0576.14026 MR 772569

[Koc77] Koch, H.: *Classification of the primitive representations of the Galois group of local fields*. Invent. Math. **40**, 195–216 (1977) Zbl 0376.12003 MR 450244

[LMB00] Laumon, G., Moret-Bailly, L.: *Champs algébriques. Ergebnisse der Mathematik und ihrer Grenzgebiete (3)* 39, Springer, Berlin (2000) Zbl 0945.14005 MR 1771927

[Li75] Li, W. C. W.: *Newforms and functional equations*. Math. Ann. **212**, 285–315 (1975) Zbl 0278.10026 MR 369263

[Lip69] Lipman, J.: *Rational singularities, with applications to algebraic surfaces and unique factorization*. Inst. Hautes Études Sci. Publ. Math. **36**, 195–279 (1969) Zbl 0181.48903 MR 276239

[Liu02] Liu, Q.: *Algebraic geometry and arithmetic curves*. Oxford Graduate Texts in Mathematics 6, Oxford University Press, Oxford (2002) Zbl 0996.14005 MR 1917232

[LMFDB] The LMFDB Collaboration. The *L*-functions and modular forms database. https://www.lmfdb.org

[Man71] Manin, Y. I.: *Cyclotomic fields and modular curves*. Uspekhi Mat. Nauk **26**, no. 6, 7–71 (1971) (in Russian) Zbl 0241.14014 MR 0401653

[Maz78] Mazur, B.: *Rational isogenies of prime degree (with an appendix by D. Goldfeld)*. Invent. Math. **44**, 129–162 (1978) Zbl 0386.14009 MR 482230

[Mil86] Milne, J. S.: *Jacobian varieties*. In: *Arithmetic geometry (Storrs, CN, 1984)*, Springer, New York, 167–212 (1986) Zbl 0604.14018 MR 861976
[Mon80] Montgomery, S.: Fixed rings of finite automorphism groups of associative rings. Lecture Notes in Mathematics 818, Springer, Berlin (1980) Zbl 0449.16001 MR 590245

[Nek75] Neklyudova, E. A.: A description of the extraordinary representations of the group PGL(2, Q₂). Funktsional. Anal. i Priložhen. 9, no. 1, 78–80 (1975) (in Russian) Zbl 0336.20032 MR 0367119

[Nob78] Nobs, A.: Les représentations exceptionnelles de GL₂(Q₂) et PGL₂(Q₂). C. R. Acad. Sci. Paris Sér. A-B 286, A767–A769 (1978) Zbl 0425.20035 MR 579178

[Ols06] Olsson, M. C.: Hom-stacks and restriction of scalars. Duke Math. J. 134, 139–164 (2006) Zbl 1114.14002 MR 2239345

[Pap93] Papadopoulos, I.: Sur la classification de Néron des courbes elliptiques en caractéristique résiduelle 2 et 3. J. Number Theory 44, 119–152 (1993) Zbl 0786.14020 MR 1225948

[PSS14] Pitale, A., Saha, A., Schmidt, R.: Transfer of Siegel cusp forms of degree 2. Mem. Amer. Math. Soc. 232, no. 1090, vi+107 pp. (2014) Zbl 1303.11063 MR 3243731

[Ray70] Raynaud, M.: Spécialisation du foncteur de Picard. Inst. Hautes Études Sci. Publ. Math. 38, 27–76 (1970) Zbl 0207.51602 MR 282993

[Ray74] Raynaud, M.: Schémas en groupes de type (p, ..., p). Bull. Soc. Math. France 102, 241–280 (1974) Zbl 0325.14020 MR 419467

[Ray91] Raynaud, M.: Jacobienne des courbes modulaires et opérateurs de Hecke. Astérisque 196-197, 9-25 (1991) Zbl 0781.14020 MR 1141454

[Roh97] Rohrlich, D. E., Rohrlich, G. F.: Modular curves, Hecke correspondence, and L-functions. In: Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, New York, 41–100 (1997) Zbl 0897.11019 MR 1638476

[Sah16] Saha, A.: Large values of newforms on GL(2) with highly ramified central character. Int. Math. Res. Notices 2016, 4103–4131 Zbl 1404.11055 MR 3544630

[Sah17] Saha, A.: Hybrid sup-norm bounds for Maass newforms of powerful level. Algebra Number Theory 11, 1009–1045 (2017) Zbl 1432.11044 MR 3671430

[Sch02] Schmidt, R.: Some remarks on local newforms for GL(2). J. Ramanujan Math. Soc. 17, 115–147 (2002) Zbl 0997.11040 MR 1913897

[Sch09] Schütt, M.: CM newforms with rational coefficients. Ramanujan J. 19, 187–205 (2009) Zbl 1226.11057 MR 2511671

[Ser79] Serre, J.-P.: Local fields. Graduate Texts in Mathematics 67, Springer, New York (1979) Zbl 0423.12016 MR 554237

[Ste89] Stevens, G.: Stickelberger elements and modular parametrizations of elliptic curves. Invent. Math. 98, 75–106 (1989) Zbl 0697.14023 MR 1010156

[Tat79] Tate, J.: Number theoretic background. In: Automorphic forms, representations and L-functions (Corvallis, OR, 1977), Proceedings of Symposia in Pure Mathematics 33, Part 2, American Mathematical Society, Providence, RI, 3–26 (1979) Zbl 0422.12007 MR 546607

[TW95] Taylor, R., Wiles, A.: Ring-theoretic properties of certain Hecke algebras. Ann. of Math. (2) 141, 553–572 (1995) Zbl 0823.11030 MR 1333036

[Tun78] Tunnell, J. B.: On the local Langlands conjecture for GL(2). Invent. Math. 46, 179–200 (1978) Zbl 0385.12006 MR 476703

[Was97] Washington, L. C.: Introduction to cyclotomic fields. 2nd ed., Graduate Texts in Mathematics 83, Springer, New York (1997) Zbl 0966.11047 MR 1421575

[Wei74] Weil, A.: Exercices dyadiques. Invent. Math. 27, 1–22 (1974) Zbl 0307.12017 MR 379445

[Wil95] Wiles, A.: Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141, 443–551 (1995) Zbl 0823.11029 MR 1333035

[Zin79] Zink, E.-W.: Ergänzungen zu Weils Exercises dyadiques. Math. Nachr. 92, 163–183 (1979) Zbl 0438.12006 MR 563583