On the congruence of modular forms

Sturm, Jacob

doi:10.1007/bfb0072985

Cited by 225 publications

(177 citation statements)

References 3 publications

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“…To see explicitly that 3 | r E , observe that the newform corresponding to E is f = q + q 2 + q 4 − 3q 5 − q 7 + · · · and the newform corresponding to X 0 (27) is g = q − 2q 4 − q 7 + · · · , so g(q) + g(q 2 ) appears to be congruent to f modulo 3. To prove this congruence, we checked it for 18 Fourier coefficients, where the sufficiency of precision to degree 18 was determined using [Stu87].…”

Section: S2(z)mentioning

confidence: 99%

The modular degree, congruence primes, and multiplicity one

Agashe¹,

Ribet

Stein

2011

Number Theory, Analysis and Geometry

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Abstract.The modular degree and congruence number are two fundamental invariants of an elliptic curve over the rational field. Frey and Müller have asked whether these invariants coincide. We find that the question has a negative answer, and show that in the counterexamples, multiplicity one (defined below) does not hold. At the same time, we prove a theorem about the relation between the two invariants: the modular degree divides the congruence number, and the ratio is divisible only by primes whose squares divide the conductor of the elliptic curve. We discuss the ratio even in the case where the square of a prime does divide the conductor, and we study analogues of the two invariants for modular abelian varieties of arbitrary dimension.

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Section: S2(z)mentioning

confidence: 99%

The modular degree, congruence primes, and multiplicity one

Agashe¹,

Ribet

Stein

2011

Number Theory, Analysis and Geometry

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“…Then by the properties of l and the similar arguments in the proof of Theorem 2 in [14], which use a theorem of Sturm [16] on the congruence of modular forms, we have that there must be an integer 1 W n W k p l coprime to l for which …”

Section: Proof Of Theorem 13mentioning

confidence: 97%

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Byeon

2001

Compositio Mathematica

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“…We apply Proposition 2 to the base case (1) diag,p = 12 (see [13]), and if p ≥ 5, to the base case (2) diag,p = 10 (see [2]). Example 1.…”

Section: Proposition 2 If (G−1)mentioning

confidence: 99%

“…A celebrated theorem of Sturm [13] implies that an elliptic modular form with p-integral rational Fourier series coefficients is determined by its "first few" Fourier series coefficients modulo p. Sturm's theorem is an important tool in the theory of modular forms (for example, see [7,12] for some of its applications). Poor and Yuen [9] (and later [2] for p ≥ 5) proved a Sturm theorem for Siegel modular forms of degree 2.…”

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confidence: 99%

Sturm bounds for Siegel modular forms

Richter

Raum

2015

Res. number theory

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We establish Sturm bounds for degree g Siegel modular forms modulo a prime p, which are vital for explicit computations. Our inductive proof exploits Fourier-Jacobi expansions of Siegel modular forms and properties of specializations of Jacobi forms to torsion points. In particular, our approach is completely different from the proofs of the previously known cases g = 1, 2, which do not extend to the case of general g. MSC 2010: Primary 11F46; Secondary 11F33Let p be a prime. A celebrated theorem of Sturm [13] implies that an elliptic modular form with p-integral rational Fourier series coefficients is determined by its "first few" Fourier series coefficients modulo p. Sturm's theorem is an important tool in the theory of modular forms (for example, see [7,12] for some of its applications). Poor and Yuen [9] (and later [2] for p ≥ 5) proved a Sturm theorem for Siegel modular forms of degree 2. Their work has been applied in different contexts, and for example, it allowed [3,4] to confirm Ramanujan-type congruences for specific Siegel modular forms of degree 2. In [10], we gave a characterization of U(p) congruences of Siegel modular forms of arbitrary degree, but (lacking a Sturm theorem) we could only discuss one explicit example that occurred as a Duke-Imamoglu-Ikeda lift. If a Siegel modular form does not arise as a lift, then one needs a Sturm theorem to justify its U(p) congruences.In this paper, we provide such a Sturm theorem for Siegel modular forms of degree g ≥ 2. Our proof is totally different from the proofs of the cases g = 1, 2 in [2,9,13], which do not have visible extensions to the case g > 2. More precisely, we perform an induction on the degree g. As in [1], we employ Fourier-Jacobi expansions of Siegel modular forms, and we study vanishing orders of Jacobi forms. However, in contrast to [1] we consider restrictions of Jacobi forms to torsion points (instead of their theta decompositions), which allow us to relate mod p diagonal vanishing orders (defined in the first Section) of Jacobi forms and Siegel modular forms. We deduce the following theorem. Theorem I. Let F be a Siegel modular form of degree g ≥ 2, weight k, and with p-integral rational Fourier series coefficients c(T).Suppose that c(T) ≡ 0 (mod p) for all T = (t ij ) with t ii ≤ 4 3 g k 16 .Then c(T) ≡ 0 (mod p) for all T.

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On the congruence of modular forms

Cited by 225 publications

References 3 publications

The modular degree, congruence primes, and multiplicity one

The modular degree, congruence primes, and multiplicity one

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Sturm bounds for Siegel modular forms

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