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Search: MSC category 11F33 ( Congruences for modular and $p$-adic modular forms [See also 14G20, 22E50] )

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1. 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, filtrationCategories:11F33, 11F46

2. 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 varietiesCategories:11F55, 11F33

3. 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 equationsCategories:11D41, 11F33, 11F11, 11F80, 11G05

4. CJM 2011 (vol 63 pp. 1284)

Dewar, Michael
 Non-Existence of Ramanujan Congruences in Modular Forms of Level Four Ramanujan famously found congruences like $p(5n+4)\equiv 0 \operatorname{mod} 5$ for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on $\Gamma_{1}(4)$ that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored $F$-partitions. Keywords:modular form, Ramanujan congruence, generalized Frobenius partition, overpartition, crankCategories:11F33, 11P83
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