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Search: All articles in the CJM digital archive with keyword modular forms

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1. CJM Online first

Martin, Kimball
Congruences for modular forms mod 2 and quaternionic $S$-ideal classes
We prove many simultaneous congruences mod 2 for elliptic and Hilbert modular forms among forms with different Atkin--Lehner eigenvalues. The proofs involve the notion of quaternionic $S$-ideal classes and the distribution of Atkin--Lehner signs among newforms.

Keywords:modular forms, congruences, quaternion algebras
Categories:11F33, 11R52

2. CJM Online first

Böcherer, Siegfried; Kikuta, Toshiyuki; Takemori, Sho
Weights of the mod $p$ kernel of the theta operators
Let $\Theta ^{[j]}$ be an analogue of the Ramanujan theta operator for Siegel modular forms. For a given prime $p$, we give the weights of elements of mod $p$ kernel of $\Theta ^{[j]}$, where the mod $p$ kernel of $\Theta ^{[j]}$ is the set of all Siegel modular forms $F$ such that $\Theta ^{[j]}(F)$ is congruent to zero modulo $p$. In order to construct examples of the mod $p$ kernel of $\Theta ^{[j]}$ from any Siegel modular form, we introduce new operators $A^{(j)}(M)$ and show the modularity of $F|A^{(j)}(M)$ when $F$ is a Siegel modular form. Finally, we give some examples of the mod $p$ kernel of $\Theta ^{[j]}$ and the filtrations of some of them.

Keywords:Siegel modular form, congruences for modular forms, Fourier coefficients, Ramanujan's operator, filtration
Categories:11F33, 11F46

3. CJM 2016 (vol 68 pp. 961)

Greenberg, Matthew; Seveso, Marco
$p$-adic Families of Cohomological Modular Forms for Indefinite Quaternion Algebras and the Jacquet-Langlands Correspondence
We use the method of Ash and Stevens to prove the existence of small slope $p$-adic families of cohomological modular forms for an indefinite quaternion algebra $B$. We prove that the Jacquet-Langlands correspondence relating modular forms on $\textbf{GL}_2/\mathbb{Q}$ and cohomomological modular forms for $B$ is compatible with the formation of $p$-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras.

Keywords:modular forms, p-adic families, Jacquet-Langlands correspondence, Shimura curves, eigencurves
Categories:11F11, 11F67, 11F85

4. CJM 2016 (vol 68 pp. 1227)

Brasca, Riccardo
Eigenvarieties for Cuspforms over PEL Type Shimura Varieties with Dense Ordinary locus
Let $p \gt 2$ be a prime and let $X$ be a compactified PEL Shimura variety of type (A) or (C) such that $p$ is an unramified prime for the PEL datum and such that the ordinary locus is dense in the reduction of $X$. Using the geometric approach of Andreatta, Iovita, Pilloni, and Stevens we define the notion of families of overconvergent locally analytic $p$-adic modular forms of Iwahoric level for $X$. We show that the system of eigenvalues of any finite slope cuspidal eigenform of Iwahoric level can be deformed to a family of systems of eigenvalues living over an open subset of the weight space. To prove these results, we actually construct eigenvarieties of the expected dimension that parameterize finite slope systems of eigenvalues appearing in the space of families of cuspidal forms.

Keywords:$p$-adic modular forms, eigenvarieties, PEL-type Shimura varieties
Categories:11F55, 11F33

5. CJM 2012 (vol 66 pp. 170)

Guitart, Xavier; Quer, Jordi
Modular Abelian Varieties Over Number Fields
The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the $L$-function $L(B/K;s)$ is a product of $L$-functions of non-CM newforms over $\mathbb Q$ for congruence subgroups of the form $\Gamma_1(N)$. The characterization involves the structure of $\operatorname{End}(B)$, isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$. We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.

Keywords:Modular abelian varieties, $GL_2$-type varieties, modular forms
Categories:11G10, 11G18, 11F11

6. CJM 2012 (vol 65 pp. 403)

Van Order, Jeanine
On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms
We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban.

Keywords:Iwasawa theory, Hilbert modular forms, abelian varieties
Categories:11G10, 11G18, 11G40

7. CJM 2011 (vol 64 pp. 588)

Nekovář, Jan
Level Raising and Anticyclotomic Selmer Groups for Hilbert Modular Forms of Weight Two
In this article we refine the method of Bertolini and Darmon and prove several finiteness results for anticyclotomic Selmer groups of Hilbert modular forms of parallel weight two.

Keywords:Hilbert modular forms, Selmer groups, Shimura curves
Categories:11G40, 11F41, 11G18

8. CJM 2011 (vol 64 pp. 282)

Dahmen, Sander R.; Yazdani, Soroosh
Level Lowering Modulo Prime Powers and Twisted Fermat Equations
We discuss a clean level lowering theorem modulo prime powers for weight $2$ cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations $ax^n+by^n+cz^n=0$.

Keywords:modular forms, level lowering, Diophantine equations
Categories:11D41, 11F33, 11F11, 11F80, 11G05

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