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Search: MSC category 11F46 ( Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms )

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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, filtration
Categories:11F33, 11F46

2. CJM Online first

Bijakowski, Stephane
Partial Hasse invariants, partial degrees, and the canonical subgroup
If the Hasse invariant of a $p$-divisible group is small enough, then one can construct a canonical subgroup inside its $p$-torsion. We prove that, assuming the existence of a subgroup of adequate height in the $p$-torsion with high degree, the expected properties of the canonical subgroup can be easily proved, especially the relation between its degree and the Hasse invariant. When one considers a $p$-divisible group with an action of the ring of integers of a (possibly ramified) finite extension of $\mathbb{Q}_p$, then much more can be said. We define partial Hasse invariants (they are natural in the unramified case, and generalize a construction of Reduzzi and Xiao in the general case), as well as partial degrees. After studying these functions, we compute the partial degrees of the canonical subgroup.

Keywords:canonical subgroup, Hasse invariant, $p$-divisible group
Categories:11F85, 11F46, 11S15

3. CJM 2014 (vol 67 pp. 893)

Mok, Chung Pang; Tan, Fucheng
Overconvergent Families of Siegel-Hilbert Modular Forms
We construct one-parameter families of overconvergent Siegel-Hilbert modular forms. This result has applications to construction of Galois representations for automorphic forms of non-cohomological weights.

Keywords:p-adic automorphic form, rigid analytic geometry
Categories:11F46, 14G22

4. CJM 2010 (vol 62 pp. 1060)

Darmon, Henri; Tian, Ye
Heegner Points over Towers of Kummer Extensions
Let $E$ be an elliptic curve, and let $L_n$ be the Kummer extension generated by a primitive $p^n$-th root of unity and a $p^n$-th root of $a$ for a fixed $a\in \mathbb{Q}^\times-\{\pm 1\}$. A detailed case study by Coates, Fukaya, Kato and Sujatha and V. Dokchitser has led these authors to predict unbounded and strikingly regular growth for the rank of $E$ over $L_n$ in certain cases. The aim of this note is to explain how some of these predictions might be accounted for by Heegner points arising from a varying collection of Shimura curve parametrisations.

Categories:11G05, 11R23, 11F46

5. CJM 2009 (vol 61 pp. 395)

Moriyama, Tomonori
$L$-Functions for $\GSp(2)\times \GL(2)$: Archimedean Theory and Applications
Let $\Pi$ be a generic cuspidal automorphic representation of $\GSp(2)$ defined over a totally real algebraic number field $\gfk$ whose archimedean type is either a (limit of) large discrete series representation or a certain principal series representation. Through explicit computation of archimedean local zeta integrals, we prove the functional equation of tensor product $L$-functions $L(s,\Pi \times \sigma)$ for an arbitrary cuspidal automorphic representation $\sigma$ of $\GL(2)$. We also give an application to the spinor $L$-function of $\Pi$.

Categories:11F70, 11F41, 11F46

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