The Manin constant and the modular degree. (English) Zbl 07815222
The Manin constant \(c\) of an elliptic curve \(E\) over \(\mathbb Q\) is the nonzero integer that scales the differential \(\omega_f\) determined by the normalized newform \(f\) associated to \(E\) into the pullback of a Néron differential under a minimal parametrization \(\phi: X_0(N)_{\mathbb Q} \to E\). Manin conjectured that \(c = \pm 1\) for optimal parametrizations.
“The initial theoretical results on the Manin conjecture were based on exactness properties of Néron models and showed that \(p \not | c_{\phi}\) for those \(p > 2\) at which \(E\) has semistable reduction (see [B. Mazur, Invent. Math. 44, 129–162 (1978; Zbl 0386.14009); A. Abbes and E. Ullmo, Compos. Math. 103, No. 3, 269–286 (1996; Zbl 0865.11049); A. Agashe et al., Pure Appl. Math. Q. 2, No. 2, 617–636 (2006; Zbl 1109.11032)] for some sharpenings). Note that the conclusion \(p \not | c_{\phi}\) was recently established in [K. Česnavičius, Compos. Math. 154, No. 9, 1889–1920 (2018; Zbl 1430.11086)] for all primes \(p\) of semistable reduction for \(E\) by a different method.”
The authors prove (Theorems 1.1 and 1.2) that in general \(c\) divides \(\text{deg}(\phi)\) under a minor assumption at \(2\) and \(3\) that is not needed for cube-free \(N\) or for parametrizations by \(X_1(N)_{\mathbb Q}\).
“The initial theoretical results on the Manin conjecture were based on exactness properties of Néron models and showed that \(p \not | c_{\phi}\) for those \(p > 2\) at which \(E\) has semistable reduction (see [B. Mazur, Invent. Math. 44, 129–162 (1978; Zbl 0386.14009); A. Abbes and E. Ullmo, Compos. Math. 103, No. 3, 269–286 (1996; Zbl 0865.11049); A. Agashe et al., Pure Appl. Math. Q. 2, No. 2, 617–636 (2006; Zbl 1109.11032)] for some sharpenings). Note that the conclusion \(p \not | c_{\phi}\) was recently established in [K. Česnavičius, Compos. Math. 154, No. 9, 1889–1920 (2018; Zbl 1430.11086)] for all primes \(p\) of semistable reduction for \(E\) by a different method.”
The authors prove (Theorems 1.1 and 1.2) that in general \(c\) divides \(\text{deg}(\phi)\) under a minor assumption at \(2\) and \(3\) that is not needed for cube-free \(N\) or for parametrizations by \(X_1(N)_{\mathbb Q}\).
Reviewer: Andrzej Dąbrowski (Szczecin)
MSC:
| 11G05 | Elliptic curves over global fields |
| 11F11 | Holomorphic modular forms of integral weight |
| 11F30 | Fourier coefficients of automorphic forms |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 11F85 | \(p\)-adic theory, local fields |
| 11G18 | Arithmetic aspects of modular and Shimura varieties |
| 11L05 | Gauss and Kloosterman sums; generalizations |
Keywords:
admissible representation; cusp; elliptic curve; \(\varepsilon\)-factor; Fourier coefficient; Gauss sum; Manin constant; modular degree; modular parametrization; newform; rational singularity; Whittaker modelSoftware:
LMFDBReferences:
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