Computing Classical Modular Forms

Best, Alex J.; Bober, Jonathan; Booker, Andrew R.; Costa, Edgar; Cremona, John; Derickx, Maarten; Lowry-Duda, David; Lee, Min; Roe, David; Sutherland, Andrew V.; Voight, John

doi:10.1007/978-3-030-80914-0_4

Cited by 13 publications

(6 citation statements)

References 64 publications

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“…In this work we apply the algorithm of [3] to make a database of modular forms on noncongruence subgroups. This complements the existing related databases such as the database of classical modular forms on congruence subgroups [5], the database of Hilbert modular forms [11], and the database of Belyi maps [20].…”

Section: Introductionmentioning

confidence: 83%

A Database of Modular Forms on Noncongruence Subgroups

Berghaus¹,

Monien²,

Radchenko³

2023

Preprint

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Section: Introductionmentioning

confidence: 83%

A Database of Modular Forms on Noncongruence Subgroups

Berghaus¹,

Monien²,

Radchenko³

2023

Preprint

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“…This labeling system can be extended to arbitrary open using labels of the form , where uniquely identifies according to its level , index and an ordinal m that distinguishes among the open subgroups of level M and index d 3 . To define m , we order the index d subgroups lexicographically according to the lists of labels of Conrey characters of modulus M whose kernels contain D ; see [BBB+21, Section 3.2] for the definition of . We then redefine N and i to be the relative level and index of H as a subgroup of the preimage G of in : this means N is the least positive integer for which , where is the reduction map and .…”

Section: The Modular Curvesmentioning

confidence: 99%

“…We may decompose this space of modular forms as where denotes the Galois orbit of the Dirichlet character and the direct sum varies over Galois orbits of Dirichlet characters of modulus whose conductor divides N . The Galois orbit of a normalised eigenform has an associated (see [BBB+21, Section 4.5]), which has an integral q -expansion with ; and for each prime p , the coefficient is the Frobenius trace of the abelian variety of dimension associated to f via the Eichler–Shimura correspondence; equivalently, is the p th (arithmetically normalised) Dirichlet coefficient of the L -function .…”

Section: Computing the Isogeny Decomposition And Analytic Rank Ofmentioning

confidence: 99%

“…For the 210 arithmetically maximal 𝐻 arising for ℓ ≤ 37, the value of 𝐵 never exceeds 3000, and the matrix 𝑇 (𝐵) with linear independent columns typically has fewer than 100 rows. To compute 𝑇 (𝐵), we use precomputed trace form data stored in the LMFDB along with additional data available in the file cmfdata.txt in [RSZB21] that we computed to handle cases where the set 𝑆(𝐻) contains Galois orbits of eigenforms that are not currently stored in the LMFDB, using the methods described in [BBB+21]. Remark 6.1.…”

Section: Computing the Isogeny Decomposition And Analytic Rank Of 𝐽 𝐻mentioning

confidence: 99%

“… 6 The sequences and are not equal in general (although we do have ), but they uniquely determine each other, which is all we are using; see [BBB+21, Section 8.6]. …”

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confidence: 99%

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-adic images of Galois for elliptic curves over (and an appendix with John Voight)

Rouse

Sutherland

Zureick-Brown

2022

Forum of Mathematics, Sigma

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We discuss the $\ell $ -adic case of Mazur’s ‘Program B’ over $\mathbb {Q}$ : the problem of classifying the possible images of $\ell $ -adic Galois representations attached to elliptic curves E over $\mathbb {Q}$ , equivalently, classifying the rational points on the corresponding modular curves. The primes $\ell =2$ and $\ell \ge 13$ are addressed by prior work, so we focus on the remaining primes $\ell = 3, 5, 7, 11$ . For each of these $\ell $ , we compute the directed graph of arithmetically maximal $\ell $ -power level modular curves $X_H$ , compute explicit equations for all but three of them and classify the rational points on all of them except $X_{\mathrm {ns}}^{+}(N)$ , for $N = 27, 25, 49, 121$ and two-level $49$ curves of genus $9$ whose Jacobians have analytic rank $9$ . Aside from the $\ell $ -adic images that are known to arise for infinitely many ${\overline {\mathbb {Q}}}$ -isomorphism classes of elliptic curves $E/\mathbb {Q}$ , we find only 22 exceptional images that arise for any prime $\ell $ and any $E/\mathbb {Q}$ without complex multiplication; these exceptional images are realised by 20 non-CM rational j-invariants. We conjecture that this list of 22 exceptional images is complete and show that any counterexamples must arise from unexpected rational points on $X_{\mathrm {ns}}^+(\ell )$ with $\ell \ge 19$ , or one of the six modular curves noted above. This yields a very efficient algorithm to compute the $\ell $ -adic images of Galois for any elliptic curve over $\mathbb {Q}$ . In an appendix with John Voight, we generalise Ribet’s observation that simple abelian varieties attached to newforms on $\Gamma _1(N)$ are of $\operatorname {GL}_2$ -type; this extends Kolyvagin’s theorem that analytic rank zero implies algebraic rank zero to isogeny factors of the Jacobian of $X_H$ .

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A Database of Hilbert Modular Forms

Donnelly

Voight

2021

Arithmetic Geometry, Number Theory, and Computation

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Computing Classical Modular Forms

Cited by 13 publications

References 64 publications

A Database of Modular Forms on Noncongruence Subgroups

A Database of Modular Forms on Noncongruence Subgroups

-adic images of Galois for elliptic curves over (and an appendix with John Voight)

A Database of Hilbert Modular Forms

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