32 research outputs found

    On rational Eisenstein primes and the rational cuspidal groups of modular Jacobian varieties

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    Let NN be a non-squarefree positive integer and let ā„“\ell be an odd prime such that ā„“2\ell^2 does not divide NN. Consider the Hecke ring T(N)\mathbb{T}(N) of weight 22 for Ī“0(N)\Gamma_0(N), and its rational Eisenstein primes of T(N)\mathbb{T}(N) containing ā„“\ell, defined in Section 3. If m\mathfrak{m} is such a rational Eisenstein prime, then we prove that m\mathfrak{m} is of the form (ā„“,Ā IM,ND)(\ell, ~\mathcal{I}^D_{M, N}), where the ideal IM,ND\mathcal{I}^D_{M, N} of T(N)\mathbb{T}(N) is also defined in Section 3. Furthermore, we prove that C(N)[m]ā‰ 0\mathcal{C}(N)[\mathfrak{m}] \neq 0, where C(N)\mathcal{C}(N) is the rational cuspidal group of J0(N)J_0(N). To do this, we compute the precise order of the cuspidal divisor CM,ND\mathcal{C}^D_{M, N}, defined in Section 4, and the index of IM,ND\mathcal{I}^D_{M, N} in T(N)āŠ—Zā„“\mathbb{T}(N)\otimes \mathbb{Z}_\ell.Comment: Many arguments are clarified, and many details are filled i

    Rational torsion points on Jacobians of modular curves

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    Let pp be a prime greater than 3. Consider the modular curve X0(3p)X_0(3p) over Q\mathbb{Q} and its Jacobian variety J0(3p)J_0(3p) over Q\mathbb{Q}. Let T(3p)\mathcal{T}(3p) and C(3p)\mathcal{C}(3p) be the group of rational torsion points on J0(3p)J_0(3p) and the cuspidal group of J0(3p)J_0(3p), respectively. We prove that the 33-primary subgroups of T(3p)\mathcal{T}(3p) and C(3p)\mathcal{C}(3p) coincide unless pā‰”1(mod9)p\equiv 1 \pmod 9 and 3pāˆ’13ā‰”1ā€‰ā£(modp)3^{\frac{p-1}{3}} \equiv 1 \!\pmod {p}

    The rational cuspidal divisor class group of X0(N)X_0(N)

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    For any positive integer NN, we completely determine the structure of the rational cuspidal divisor class group of X0(N)X_0(N), which is conjecturally equal to the rational torsion subgroup of J0(N)J_0(N). More specifically, for a given prime ā„“\ell, we construct a rational cuspidal divisor Zā„“(d)Z_\ell(d) for any non-trivial divisor dd of NN. Also, we compute the order of the linear equivalence class of the divisor Zā„“(d)Z_\ell(d) and show that the ā„“\ell-primary subgroup of the rational cuspidal divisor class group of X0(N)X_0(N) is isomorphic to the direct sum of the cyclic subgroups generated by the linear equivalence classes of the divisors Zā„“(d)Z_\ell(d).Comment: Comments are welcom

    Abelian arithmetic Chern-Simons theory and arithmetic linking numbers

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    Following the method of Seifert surfaces in knot theory, we define arithmetic linking numbers and height pairings of ideals using arithmetic duality theorems, and compute them in terms of n-th power residue symbols. This formalism leads to a precise arithmetic analogue of a 'path-integral formula' for linking numbers

    Arithmetic Chern-Simons theory II

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    In this paper, we apply ideas of Dijkgraaf and Witten [6, 32] on 3 dimensional topological quantum field theory to arithmetic curves, that is, the spectra of rings of integers in algebraic number fields. In the first three sections, we define classical Chernā€“Simons actions on spaces of Galois representations. In the subsequent sections, we give formulas for computation in a small class of cases and point towards some arithmetic applications

    NON-OPTIMAL LEVELS OF A REDUCIBLE MOD l MODULAR REPRESENTATION

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    Let l >= 5 be a prime and let N be a square-free integer prime to l. For each prime p dividing N, let ap be either 1 or -1. We give sufficient criteria for the existence of a newform f of weight 2 for G0( N) such that the mod l Galois representation attached to f is reducible and Upf = apf for primes p dividing N. The main techniques used are level raising methods based on an exact sequence due to Ribet. c.2018 American Mathematical Societ
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