We address the modularity questions of Calabi-Yau varieties of dimension ≤ 3 defined over Q. The ... more We address the modularity questions of Calabi-Yau varieties of dimension ≤ 3 defined over Q. The up-to-date reference on the modularity of Calabi-Yau varieties is Yui [Yu03].
The structure of the p-divisible groups arising from Fermat curves over finite fields of characte... more The structure of the p-divisible groups arising from Fermat curves over finite fields of characteristic p > 0 is completely determined, up to isogeny, by purely arithmetic means. In certain cases, the "global" structure of the Jacobian varieties of Fermat curves, up to isogeny, is also determined.
Let X= C, x Cz be the product of two nonsingular projective curves defined over a tinite field k ... more Let X= C, x Cz be the product of two nonsingular projective curves defined over a tinite field k = IF,. We compute the order of the Brauer group Br(X) of X. In the generic case, it is given by the resultant of the characteristic polynomials of the Frobenius endomorphism of the Jacobian variety J(C,) (I = 1, 2). divided by a certain power of 4. In particular, if at least one component C, has ordinary Jacobian variety J(C,), then the order of Br(X) is equal to the resultant divided by y"u, where pq is the geometric genus of X. (' 1986 Academic Pres. Inc Let X be a smooth projective surface defined over a finite field k = IF, of
Let N 1 be an integer and let ; 0 (N) = f a b c d 2 ; j c 0 (mod N)g: Then ; 0 (N) is a subgroup ... more Let N 1 be an integer and let ; 0 (N) = f a b c d 2 ; j c 0 (mod N)g: Then ; 0 (N) is a subgroup of ; of finite index. The cusps of ; 0 (N) consist of the finite set P 1 (Q)=; 0 (N) and they are all defined over Q. Let X 0 (N) = H =; 0 (N) and X 0 (N)+ = X 0 (N)= < W N > where < W N > denotes the group of Atkin-Lehner involutions on X 0 (N) (of order 2 r if r distinct primes divide N ). Associated to X 0 (N)+ there is a subgroup G of P S L 2 (R), which contains ; 0 (N) as a normal subgroup. We denote such a group by ; 0 (N)+. (If N is prime, then ; 0 (N) + = ; 0 (N) + w N where w N = 0 1 ;N 0 is the Fricke involution, and X 0 (N) is a double covering of X 0 (N)+.) There are only finitely many values of N for which X 0 (N) or X 0 (N)+ has genus zero. Let S denote the set of prime values for N giving rise to genus zero Riemann surfaces X 0 (N)+, then
American Mathematical Society eBooks, Dec 14, 2006
Let X be a general complex algebraic K3 surface, and CH 2 (X, 1) Bloch's higher Chow group ). In ... more Let X be a general complex algebraic K3 surface, and CH 2 (X, 1) Bloch's higher Chow group ). In [C-L2] it is proven that the real regulator r 2,1 : CH The question addressed in this paper is whether a similar story holds for products of K3 surfaces. We prove a general regulator result for a product of two surfaces, and then deduce, under the assumptions of [a variant of] the Bloch-Beilinson conjecture regarding the injectivity of the Abel-Jacobi map for smooth quasiprojective varieties defined over number fields, that the (induced) regulator map r 3,1 : CH ) is trivial for a general pair of K3 surfaces, where H 2 tr (X, R) is the space of transcendental classes in H 2 (X, R).
American Mathematical Society eBooks, Dec 14, 2006
We construct an algebraic variety by resolving singularities of a quintic Calabi-Yau threefold. T... more We construct an algebraic variety by resolving singularities of a quintic Calabi-Yau threefold. The middle cohomology of the threefold is shown to contain a piece coming from a pair of elliptic surfaces. The resulting quotient is a two-dimensional Galois representation. By using the Lefschetz fixed-point theorem in étale cohomology and counting points on the variety over finite fields, this Galois representation is shown to be modular.
American Mathematical Society eBooks, Dec 14, 2006
This note reveals a mysterious link between the partition function of certain dimer models on 2-d... more This note reveals a mysterious link between the partition function of certain dimer models on 2-dimensional tori and the L-function of their spectral curves. It also relates the partition function in certain families of dimer models to Eisenstein series.
Motivated by a constructive realization of dihedral groups of prime degree as Galois group over t... more Motivated by a constructive realization of dihedral groups of prime degree as Galois group over the field of rational numbers, we give an explicit construction of the Hilbert class fields of some imaginary quadratic fields with class numbers 7 and 11. This was done by explicitly evaluating the elliptic modular j -invariant at each representative of the ideal class of an imaginary quadratic field, and then forming the class equation. In an appendix we determine the explicit form of the modular equation of order 5 and 7. All computations were carried out on the computer algebra system MACSYMA. We also point out what computational difficulties were encountered and how we resolved them. In this paper we summarize the progress made so far on using the Computer Algebra System MACSYMA [8] to explicitly calculate the defining equations of the Hilbert class fields of imaginary quadratic fields with prime class number. Our motivation for undertaking this investigation is to construct rational polynomials with a given finite Galois group. The groups we try to realize here are the dihedral groups D p for primes p . These groups are non-abelian groups of order 2p and are generated by two elements σ = (1 2 3 . . . p ) and τ = (1)(2 p )(3 p -1) . . . ( 2 p +1 _ ____ 2 p +3 _ ____ ) _ ______________
We present a modification of the Goldwasser-Kilian-Atkin primality test, which, when given an inp... more We present a modification of the Goldwasser-Kilian-Atkin primality test, which, when given an input n, outputs either prime or composite, along with a certificate of correctness which may be verified in polynomial time. Atkin's method computes the order of an elliptic curve whose endomorphism ring is isomorphic to the ring of integers of a given imaginary quadratic field Q( √ -D). Once an appropriate order is found, the parameters of the curve are computed as a function of a root modulo n of the Hilbert class equation for the Hilbert class field of Q( √ -D). The modification we propose determines instead a root of the Watson class equation for Q( √ -D) and applies a transformation to get a root of the corresponding Hilbert equation. This is a substantial improvement, in that the Watson equations have much smaller coefficients than do the Hilbert equations.
BRUEN, JENSEN. AND YUI with certain Frobenius groups as Galois groups. 111.1. Preliminary results... more BRUEN, JENSEN. AND YUI with certain Frobenius groups as Galois groups. 111.1. Preliminary results. 111.2. Realization of Frobenius groups of prime degree as Galois groups (general existence theorem). 111.3. Construction of polynomials with Galois group F,,, , bl over 0. 111.4. Construction of generic family of polynomials with Galois group F,c p I uz (p = 3 (mod 4)). 111.5. Special examples. 111.6. Remarks and problems. IV. Frobenius fields. IV.1. Frobenius fields over 0. IV.2. Frobenius fields over function fields.
ABSTRACT A useful criterion characterizing a monic irreducible polynomial over with Galois group ... more ABSTRACT A useful criterion characterizing a monic irreducible polynomial over with Galois group Dp (the dihedral group of order 2p, p: prime) is given by making use of the geometry of Dp, i.e., Dp is the symmetry group of the regular p-gon. We derive explicit numerical examples of polynomials with dihedral Galois groups D5 and D7.
The proof of Serre's conjecture on Galois representations over finite fields allows us to show, u... more The proof of Serre's conjecture on Galois representations over finite fields allows us to show, using a trick due to Serre himself, that all rigid Calabi-Yau threefolds defined over Q are modular.
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