Papers by Daniel Fretwell
Journal of Number Theory, 2014
This is a repository copy of Ramanujan-style congruences of local origin.
We investigate the interplay between Galois and automorphic representations coming from generic H... more We investigate the interplay between Galois and automorphic representations coming from generic Harder type congruences for degree 2 Siegel modular forms. Using Local Langlands results for GSp 4 we see conditions that guarantee the existence of a level p paramodular form satisying the congruence. These conditions provide theoretical justification for numerical evidence found in the author's previous paper.
We investigate certain congruences, as predicted by Harder, between Hecke eigen-values of Siegel ... more We investigate certain congruences, as predicted by Harder, between Hecke eigen-values of Siegel and elliptic modular forms. In particular the Siegel forms will be paramodular of level p. In order to simplify matters on the Siegel side we move into spaces of algebraic modular forms for the group GU 2 (D) for a quaternion algebra D/Q ramified at p, ∞. Under class number assumptions, efficient algorithms for computing with such forms are presented, allowing the congruence to be tested.
We prove that if a prime > 3 divides p k − 1, where p is prime, then there is a congruence modulo... more We prove that if a prime > 3 divides p k − 1, where p is prime, then there is a congruence modulo , like Ramanujan's mod 691 congruence, for the Hecke eigenvalues of some cusp form of weight k and level p. We relate to primes like 691 by viewing it as a divisor of a partial zeta value, and see how a construction of Ribet links the congruence with the Bloch-Kato conjecture (theorem in this case). This viewpoint allows us to give a new proof of a recent theorem of Billerey and Menares. We end with some examples, including where p = 2 and is a Mersenne prime.
The cannonball problem is a very old but interesting problem. It asks when a square-pyramidal sta... more The cannonball problem is a very old but interesting problem. It asks when a square-pyramidal stack of cannonballs can be arranged into a square. The claim is the following:
This was a summer project I undertook after my 3rd undergraduate year, under the supervision of D... more This was a summer project I undertook after my 3rd undergraduate year, under the supervision of Dr. Neil Dummigan.
The first part of my masters dissertation, completed under the supervision of Dr. Neil Dummigan.
... more The first part of my masters dissertation, completed under the supervision of Dr. Neil Dummigan.
This is a quite informal view of global class field theory, viewed from the platform of ideals.
See the second part, "Class Field Theory: Proofs and Applications", for a more detailed view along with proofs, including the introduction of ideles, a bit of cohomology and applications of class field theory to the representation of primes by the quadratic form x^2 + ny^2.
The second part of my masters dissertation, done under the supervision of Dr. Neil Dummigan.
T... more The second part of my masters dissertation, done under the supervision of Dr. Neil Dummigan.
This installment proves everything done informally in the first part. This is quite a difficult and lengthy task and many new devices need to be invented, such as the ideles and the Herbrand quotient.
Finally, we apply the theory to the representation of primes by the quadratic form x^2 + ny^2, giving some examples.
Talks by Daniel Fretwell
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Papers by Daniel Fretwell
This is a quite informal view of global class field theory, viewed from the platform of ideals.
See the second part, "Class Field Theory: Proofs and Applications", for a more detailed view along with proofs, including the introduction of ideles, a bit of cohomology and applications of class field theory to the representation of primes by the quadratic form x^2 + ny^2.
This installment proves everything done informally in the first part. This is quite a difficult and lengthy task and many new devices need to be invented, such as the ideles and the Herbrand quotient.
Finally, we apply the theory to the representation of primes by the quadratic form x^2 + ny^2, giving some examples.
Talks by Daniel Fretwell
This is a quite informal view of global class field theory, viewed from the platform of ideals.
See the second part, "Class Field Theory: Proofs and Applications", for a more detailed view along with proofs, including the introduction of ideles, a bit of cohomology and applications of class field theory to the representation of primes by the quadratic form x^2 + ny^2.
This installment proves everything done informally in the first part. This is quite a difficult and lengthy task and many new devices need to be invented, such as the ideles and the Herbrand quotient.
Finally, we apply the theory to the representation of primes by the quadratic form x^2 + ny^2, giving some examples.
In this talk we study a few basics of algebraic number theory, later specialising to Galois extensions of number fields. It is in this setting that we may study Frobenius elements; specific elements of the Galois group assigned to prime ideals of rings of integers. It is these elements that form the claim of the Cebotarev theorem. Finally we see, as an application, how Dirichlet's theorem is a simple consequence of this theorem.
These notes only deal with the Galois theory of finite extensions.