• 충청수학회지
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• 제29권3호
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• pp.509-514
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• 2016

By constructing a canonical basis for the space $M_k^{\sharp}({\Gamma}_0(N))$ explicitly, we find a basis of the space of cusp forms for ${\Gamma}_0(N)$ consisting of $Poincar{\acute{e}}$ series.

• 충청수학회지
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• 제31권3호
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• pp.281-285
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• 2018

Let $S_k(N)$ be the space of cusp forms of weight k for ${\Gamma}_0(N)$. We show that $S_k(N)$ is the direct sum of subspaces $S_k^+(N)$ and $S_k^-(N)$. Where $S_k^+(N)$ is the vector space of cusp forms of weight k for the group ${\Gamma}_0^+(N)$ generated by ${\Gamma}_0(N)$ and $W_N$ and $S_k^-(N)$ is the subspace consisting of elements f in $S_k(N)$ satisfying $f{\mid}_kW_N=-f$. We find generators spanning the space $S_k^-(N)$ from $Poincar{\acute{e}}$ series and give all linear relations among such generators.

• 충청수학회지
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• 제30권1호
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• pp.77-102
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• 2017

As an analogy of $Poincar{\acute{e}}$ series in the space of modular forms, T. Ono associated a module $M_c/P_c$ for ${\gamma}=[c]{\in}H^1(G,R^{\times})$ where finite group G is acting on a ring R. $M_c/P_c$ is regarded as the 0-dimensional twisted Tate cohomology ${\hat{H}}^0(G,R^+)_{\gamma}$. In the case that G is the Galois group of a Galois extension K of a number field k and R is the ring of integers of K, the vanishing properties of $M_c/P_c$ are related to the ramification of K/k. We generalize this to arbitrary n-dimensional twisted cohomology of the ring of integers and obtain the extended version of theorems. Moreover, some explicit examples on quadratic and biquadratic number fields are given.

• 대한수학회보
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• 제57권1호
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• pp.207-218
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• 2020

Some modular representations of reflection groups related to Weyl groups are considered. The rational cohomology of the classifying space of a compact connected Lie group G with a maximal torus T is expressed as the ring of invariants, H*(BG; ℚ) ≅ H*(BT; ℚ)^W(G), which is a polynomial ring. If such Lie groups are locally isomorphic, the rational representations of their Weyl groups are equivalent. However, the integral representations need not be equivalent. Under the mod p reductions, we consider the structure of the rings, particularly for the Weyl group of symplectic groups Sp(n) and for the alternating groups A_n as the subgroup of W(SU(n)). We will ask if such rings of invariants are polynomial rings, and if each of them can be realized as the mod p cohomology of a space. For n = 3, 4, the rings under a conjugate of W(Sp(n)) are shown to be polynomial, and for n = 6, 8, they are non-polynomial. The structures of H*(BT^n-1; 𝔽_p)^A_n will be also discussed for n = 3, 4.