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Yimin Ge's Maths Blog | yiminge.wordpress.com Reviews
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March | 2009 | Yimin Ge's Maths Blog
https://yiminge.wordpress.com/2009/03
Yimin Ge’s Maths Blog. Archive for March, 2009. Some More Thoughts on r-Ary Functions over Finite Fields. March 20, 2009. Note, I proved that every. Ary function on a finite field. Elements can be represented by a unique polynomial. A more direct way to see this is, given a function. To consider the polynomial. The polynomial defined above. We know that. But can we find the exact value? For this purpose, consider the following very useful lemma. Be an integral domain and. Then for every integer. Being od...
All Groups of Order n are cyclic iff… | Yimin Ge's Maths Blog
https://yiminge.wordpress.com/2009/01/22/all-groups-of-order-n-are-cyclic-iff
Yimin Ge’s Maths Blog. Laquo; Bezout’s Lemma in Endomorphism Rings of Vector Spaces. Some Remarks on K r-Annihilating Polynomials over Finite Fields. All Groups of Order n are cyclic iff…. Many months ago, I found a very interesting group theory problem (the. Part of the Proposition below) in Lee Zhuo Zhao. I will prove the following result:. Be a positive integer. Then every finite group of order. Is cyclic if and only if. Denotes the Euler’s Totient Function. I will use the following notation: If.
Some More Thoughts on r-Ary Functions over Finite Fields | Yimin Ge's Maths Blog
https://yiminge.wordpress.com/2009/03/20/some-more-thoughts-on-r-ary-functions-over-finite-fields
Yimin Ge’s Maths Blog. Laquo; Hamiltonian Path Exchanges. The Probability of Coprimality. Some More Thoughts on r-Ary Functions over Finite Fields. Note, I proved that every. Ary function on a finite field. Elements can be represented by a unique polynomial. A more direct way to see this is, given a function. To consider the polynomial. The polynomial defined above. We know that. But can we find the exact value? For this purpose, consider the following very useful lemma. Be an integral domain and. You ar...
Algebraic Closures | Yimin Ge's Maths Blog
https://yiminge.wordpress.com/2009/01/18/algebraic-closures
Yimin Ge’s Maths Blog. Laquo; Yimin Ge’s Blog Goes Online. Bezout’s Lemma in Endomorphism Rings of Vector Spaces. This entry contains some serious errors which I was unaware of when I wrote it in the first place. Until I find out how to resolve them, I leave it to the eager reader to find them. Be a field. An. Is an algebraic field extension and. Is algebraically closed, i.e. every polynomial with coefficients in. Splits into linear factors in. Has an algebraic closure. Be the set of all fields. Is algeb...
Hamiltonian Path Exchanges | Yimin Ge's Maths Blog
https://yiminge.wordpress.com/2009/03/07/hamiltonian-path-exchanges
Yimin Ge’s Maths Blog. Laquo; Some Remarks on K r-Annihilating Polynomials over Finite Fields. Some More Thoughts on r-Ary Functions over Finite Fields. The following nice little trick was shown to me by Stephan Wagner. Be a graph and let. Be the set of all vertices of even degree. Then for each. The number of Hamiltonian paths from. To some vertex in. For a Hamiltonian path. Be the Hamiltonian path. We call the transformation. Be the set of all Hamiltonian paths starting in. Where two Hamiltonian paths.
TOTAL PAGES IN THIS WEBSITE
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Modular forms and p-adic L-functions | Tom Lovering's Blog
https://tlovering.wordpress.com/2013/07/05/modular-forms-and-p-adic-l-functions
Tom Lovering's Blog. Modular forms and p-adic L-functions. July 5, 2013 in Mathematics. In this post I want to advertise Serre’s lovely note. Following the opening section which gives a simple definition of p-adic modular forms (via q-expansions) and uses this and some elementary congruences to construct the classical p-adic L-functions. Serre’s writing is excellent so I give only a brief account and let the interested reader consult the master. Which we will take to be odd, but the case. What about p-ad...
February | 2015 | Tom Lovering's Blog
https://tlovering.wordpress.com/2015/02
Tom Lovering's Blog. You are currently browsing the monthly archive for February 2015. Cyclotomic Fields and Fermat’s Last Theorem Tutorial Notes and Problems. February 8, 2015 in Mathematics. During Spring Semester 2015 I am teaching an undergraduate course on “Cyclotomic Fields and Fermat’s Last Theorem” at Harvard. Meeting times and places are currently:. Mondays 4:15-5:45 in SC530. Wednesdays 4:30-6 in SC530. The latest edition of the notes is here: Cyclotomic Fields and FLT. Varieties whose conjugat...
Good reduction of Shimura varieties I: Introduction and motivation | Tom Lovering's Blog
https://tlovering.wordpress.com/2013/01/04/good-reduction-of-shimura-varieties-i-introduction-and-motivation
Tom Lovering's Blog. Good reduction of Shimura varieties I: Introduction and motivation. January 4, 2013 in Mathematics. In what I hope will become a series of posts, I want to think about the following question (to which, at the time of writing of this post, I have no idea of the complete answer). Given a Shimura datum. Giving rise to a Shimura variety. Defined over the number field. Have good reduction at. In this post we sketch the significance of this question in the theory of (nice) automorphic forms.
July | 2014 | Tom Lovering's Blog
https://tlovering.wordpress.com/2014/07
Tom Lovering's Blog. You are currently browsing the monthly archive for July 2014. Mumford’s Hodge type Shimura varieties (not of PEL type). July 9, 2014 in Mathematics. In his note A Note of Shimura’s Paper Discontinuous Groups and Abelian Varieties. More specifically, let. Be a cubic totally real field. We wish to carefully choose a quaternion algebra. Admits a symplectic representation. The trick is to note that corestriction on Brauer groups associates to. Equipped with a natural map. And visibly pre...
Varieties whose conjugates have different fundamental groups | Tom Lovering's Blog
https://tlovering.wordpress.com/2013/01/26/varieties-whose-conjugates-have-different-fundamental-groups
Tom Lovering's Blog. Varieties whose conjugates have different fundamental groups. January 26, 2013 in Mathematics. In this post I want to briefly summarize the following interesting paper of Milne and Suh http:/ www.jmilne.org/math/articles/2009c.pdf in which they give a general method for constructing connected Shimura varieties with the property that one can apply an automorphism of. To their defining equations and obtain a variety with a different fundamental group. Is a symmetric hermitian domain and.
Predicting the dimension of a Galois representation | Tom Lovering's Blog
https://tlovering.wordpress.com/2013/05/07/predicting-the-dimension-of-a-galois-representation
Tom Lovering's Blog. Predicting the dimension of a Galois representation. May 7, 2013 in Mathematics. In this post I want to briefly record the following nice recipe for predicting the dimension of a Galois representation you expect to extract from a Shimura variety (and the local factors you expect to see). More precisely, let’s suppose I have a cohomological cuspidal representation. Of a reductive group. Then Langlands would conjecture the existence of a map. But a cocharacter of. Composing the map pre...
Shimura Varieties for GSp which happen to be Moduli of Abelian Varieties | Tom Lovering's Blog
https://tlovering.wordpress.com/2012/11/15/shimura-varieties-for-gsp-which-happen-to-be-moduli-of-abelian-varieties
Tom Lovering's Blog. Shimura Varieties for GSp which happen to be Moduli of Abelian Varieties. November 15, 2012 in Mathematics. In this reasonably sketchy post [and I’ll include the warning that I’m fairly new to lots of the content, so there may be errors] I want to outline the main example of Deligne’s `Travaux de Shimura’ paper, namely his consideration of the reductive group. Together with a nice class of Hodge structures, and the interpretation of the resulting Shimura variety as parameterising.
Differential equations with regular singularities | Tom Lovering's Blog
https://tlovering.wordpress.com/2014/07/08/differential-equations-with-regular-singularities
Tom Lovering's Blog. Differential equations with regular singularities. July 8, 2014 in Mathematics. In this short post I want to note Deligne’s classic result on vector bundles with flat connections (let’s call them “differential equations”: terminology justified by the `cyclic vector theorem”) and give a stupid example illustrating the basic shape of things. I should thank Ananth Shankar for helping unconfuse me faster than I could unconfuse myself. Be a smooth algebraic variety over. A vector bundle on.
September | 2014 | Tom Lovering's Blog
https://tlovering.wordpress.com/2014/09
Tom Lovering's Blog. You are currently browsing the monthly archive for September 2014. Galois descent for transcendental extensions. September 3, 2014 in Mathematics. In this note, I want to discuss the work of Weil as presented in Milne’s papers (in particuar this one. Given a finite Galois extension. Of fields, many readers will be familiar with the usual “Galois descent” procedure giving an equivalence between (for example) affine varieties over. And affine varieties over. And noting that only the.
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Yimin Ge's Maths Blog
Yimin Ge’s Maths Blog. Automorphism Groups of Simple Graphs. September 26, 2009. Every finite group is the automorphism group of a simple graph. We first show that it is sufficient to construct a weighted directed graph having the given group as automorphism group. A weighted directed graph is a triple. Is a finite set of vertices,. Is a finite set of weights and. Is a partial map. An automorphism of. Such that for all. Is defined if and only if. Is defined in which case. The group of all automorphisms of.
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