By Terence Tao
Conventional Fourier research, which has been remarkably powerful in lots of contexts, makes use of linear part services to review capabilities. a few questions, reminiscent of difficulties related to mathematics progressions, evidently result in using quadratic or larger order stages. better order Fourier research is a topic that has develop into very lively only in the near past. Gowers, in groundbreaking paintings, constructed a few of the easy thoughts of this thought in an effort to provide a brand new, quantitative evidence of Szemerédi's theorem on mathematics progressions. even though, there also are precursors to this conception in Weyl's classical concept of equidistribution, in addition to in Furstenberg's structural conception of dynamical structures. This publication, that's the 1st monograph during this zone, goals to hide all of those subject matters in a unified demeanour, in addition to to survey the most contemporary advancements, corresponding to the appliance of the speculation to count number linear styles in primes. The publication serves as an creation to the sphere, giving the start graduate scholar within the topic a high-level evaluation of the sphere. The textual content specializes in the easiest illustrative examples of key effects, serving as a better half to the prevailing literature at the topic. there are many workouts with which to check one's wisdom.
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Extra info for Higher Order Fourier Analysis (Graduate Studies in Mathematics)
In the last part of this section, we obtain some preliminary results and make some conjectures as to what should be the case. 1 Definition To define the U operator, we start with the Frobenius endomorphism Frob : V ~ V, which, as was shown in the last section, is locally free of rank p. 3. The U Operator 43 a trace h o m o m o r p h i s m TrFrob : V ~V, defined by (TrFrobf)(E/A, T, z) : ~ f(E1, ~1, zl), where the sum is taken over the triples (E1,~'1,*1) which m a p (by quotient by the f u n d a m e n t a l subgroup) to the given triple (E,T,z).
Note, however, that the difficulty disappears if we consider all weights together, since it is easy to see that any m o d u l a r form of weight 1 over k will always have the same q-expansion as the reduction of some m o d u l a r form of weight p = 1 + (p - 1) (just multiply by Ep-1 and note that the reduction m a p for m o d u l a r forms of weight greater t h a n one is onto). C h a p t e r II T h e H e c k e and U O p e r a t o r s In this chapter we define p-adic versions of the classical Hecke operators.
I=2 with ord(a) > 0 (because E is supersingular), ord(ci) _> 1 for i ~ l(modp), and ord(cv) = 0 (because the formal group is of height 2). Note that since a (rood p) for any nonvanishing differential w on E, we have that, if ord(a) < 1, then ord(a) = ord(Ev_l(E,w)). We want to determine the curves (if any) that are mapped to E by quotient by their fundamental group. T h e o r e m I I . 3 . 5 Let 0 < ord(a) < p/(1 + p), so that the canonical subgroup H0 C E is defined, and let H1, H2, . . , Hp be the other finite fiat subgroup schemes of rank p orE.
Higher Order Fourier Analysis (Graduate Studies in Mathematics) by Terence Tao