98 research outputs found
Finiteness questions for Galois representations
No full textLet p be a prime number. Due to classical work of Shimura and Deligne, to any "newform" (a modular form that is an eigenfunction for the Hecke operators and assumed of level one in the talk) one attaches a p-adic Galois representation. Since there are infinitely many newforms, there are infinitely many attached p-adic Galois representations. However, if one reduces them modulo p, there are only finitely many (up to isomorphism). It is tempting to ask what happens "in between", i.e. whether there is still finiteness modulo fixed prime powers. In the talk, I will motivate and explain a conjecture made with Ian Kiming and Nadim Rustom and explain partial results, including a relation to a strong question by Kevin Buzzard.
The talk is based on joint work with Ian Kiming and Nadim Rustom
Algebra and Arithmetic of Modular Forms
No full textIn [Rus14b] and [Rus14a], we study graded rings of modular forms over congruence subgroups, with coefficients in subrings A of C, and determine bounds of the weights of modular forms constituting a minimal set of generators, as well as on the degree of the generators of the ideal of relations between them. We give an algorithm that computes the structures of these rings, and formulate conjectures on the minimal generating weight for modular forms with coefficients in Z. We discuss questions of finiteness of systems of Hecke eigenvalues modulo pm, for a prime p and an integer m ≥ 2, in analogy to the classical theory that already exists for m = 1. In joint work with Ian Kiming and Gabor Wiese ([KRW14]), we show that these questions are intimately related to a question of Buzzard regarding the boundedness of the eld of denition of Hecke eigenforms (over Qp), and we formulate precise conjectures. We prove the existence of bounds on the weight ltrations of eigenforms modulo pm, which gives evidence as to the truth of these conjectures. These bounds are made explicit in the case N = 1, p = 2
Dihedral Group, 4-Torsion on an Elliptic Curve, and a Peculiar Eigenform Modulo 4
No full textWe work out a non-trivial example of lifting a so-called weak eigenform to a true, characteristic 0 eigenform. The weak eigenform is closely related to Ramanujan's tau function, whereas the characteristic 0 eigenform is attached to an elliptic curve defined over Q. We produce the lift by showing that the coefficients of the initial, weak eigenform (almost all) occur as traces of Frobenii in the Galois representation on the 4-torsion of the elliptic curve. The example is remarkable as the initial form is known not to be liftable to any characteristic 0 eigenform of level 1. We use this example to illustrate certain questions that have arisen lately in the theory of modular forms modulo prime powers. We give a brief survey of those questions.The authors would like to thank Shaunak Deo and Gabor Wiese for interesting discussions relating to this paper, as well as to other questions concerning modular forms modulo prime powers. We thank Ariel Pacetti for comments on the first draft of the paper. We also thank the anonymous referees for comments and suggestions that helped improve the exposition. The second author was supported by a Postdoctoral Fellowship at the National Center for Theoretical Sciences, Taipei, Taiwan. The first author would like to thank Noriko Yui for good contact, collaboration, and interesting exchange over many years
On certain finiteness questions in the arithmetic of Galois representations
No full textLet p be a fixed prime number. It has been known for a long time that there are only finitely many Galois extensions K/Q with Galois group a finite irreducible subgroup of GL_2(F_p^bar) that are imaginary and unramified outside p. In contrast, there are infinitely many such with Galois group inside GL_2(Z_p^bar), even if one restricts to ones coming from modular forms (this restriction is believed to be local at p). It is tempting to ask what happens "in between" F_p^bar and Z_p^bar, i.e. whether there is still finiteness modulo fixed prime powers.
In the talk, I will motivate and explain a conjecture made with Ian Kiming and Nadim Rustom stating that the set of such Galois extensions `modulo p^m' (a proper definition will be given in the talk) coming from modular forms is finite. I will present partial results and a relation of the finiteness conjecture to a strong question by Kevin Buzzard.
The talk is based on joint work with Ian Kiming and Nadim Rustom
Eisenstein series, p-adic modular functions, and overconvergence, II
No full textLet p be a prime number. Continuing and extending our previous paper with the same title, we prove explicit rates of overconvergence for modular functions of the form where is a classical, normalized Eisenstein series on and V the p-adic Frobenius operator. In particular, we extend our previous paper to the primes 2 and 3. For these primes our main theorem improves somewhat upon earlier results by Emerton, Buzzard and Kilford, and Roe. We include a detailed discussion of those earlier results as seen from our perspective. We also give some improvements to our earlier paper for primes . Apart from establishing these improvements, our main purpose here is also to show that all of these results can be obtained by a uniform method, i.e., a method where the main points in the argumentation is the same for all primes. We illustrate the results by some numerical examples
Eisenstein series, p-adic modular functions, and overconvergence
No full textLet p be a prime ≥ 5. We establish explicit rates of overconvergence for some members of the “Eisenstein family”, notably for the p-adic modular function V(E(1,0)∗)/E(1,0)∗ (V the p-adic Frobenius operator) that plays a pivotal role in Coleman’s theory of p-adic families of modular forms. The proof goes via an in-depth analysis of rates of overconvergence of p-adic modular functions of form V(Ek) / Ek where Ek is the classical Eisenstein series of level 1 and weight k divisible by p- 1. Under certain conditions, we extend the latter result to a vast generalization of a theorem of Coleman–Wan regarding the rate of overconvergence of V(Ep-1) / Ep-1. We also comment on previous results in the literature. These include applications of our results for the primes 5 and 7.</p
On certain finiteness questions in the arithmetic of modular forms
No full textpeer reviewedWe investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that, for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on \Gamma_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence
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