By C. Herbert Clemens
This high-quality ebook through Herb Clemens quick turned a favourite of many complicated algebraic geometers whilst it used to be first released in 1980. it's been well liked by newbies and specialists ever due to the fact that. it's written as a publication of "impressions" of a trip during the concept of complicated algebraic curves. Many themes of compelling good looks ensue alongside the way in which. A cursory look on the matters visited unearths an it appears eclectic choice, from conics and cubics to theta capabilities, Jacobians, and questions of moduli. through the top of the booklet, the subject matter of theta features turns into transparent, culminating within the Schottky challenge. The author's cause used to be to encourage additional research and to stimulate mathematical job. The attentive reader will study a lot approximately advanced algebraic curves and the instruments used to review them. The publication might be specifically beneficial to a person getting ready a path concerning advanced curves or a person attracted to supplementing his/her examining
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Additional info for A scrapbook of complex curve theory
Since for every n there are maps M ( B / p n B , kn, N;r) -~ M ( B / p " B , k , . N ; 1 ) , taking the inverse limit gives a map M(B, X(i,k), N; r) , M(B, X(,,k), N; 1). It is not clear that this map is an inclusion, because the maps modulo p'~ are not injective. 1 Let the spaces M(B,X(i,~),N;r) and the maps M(B,x(~,k),N;r) , M(B, X(i,k), N; 1) be defined as above. Are the maps c~ inclusions? In other words, can we think of overconvergent forms of weight (i,k) (as defined above) as a certain kind of p-adic modular forms of weight (i, k) ?
P We claim there is an injection D ~ V ( B , N). To see this, let ~" E B be a uniformizer, and let f = ~ f~ E D, where fi E M(K, Np ~',i). T h e n we have f(q) E B[[q]], and, for some n, 7r'~f E ~ M(B,i, Np~), hence ~r"f E V. Then, since (Tr'~f)(q) = 7r'~f(q), f(q) B((q))/V. 1 above), it follows t h a t there exists ] E V such t h a t ](q) = f(q). Hence we m a y define ot D f ,--+ , , V ] . 12) Note t h a t the injectivity follows at once from the equality of the q-expanslons, since B is flat over Zp.
This allows us to define "slope a eigenspaces" for U which generalize (the integral weight case of) Hida's space of "ordinary p-adic modular forms". This will also show that there are few eigenforms for U outside its kernel, in the precise sense that if we fix the weight of f and the valuation of )~, one gets only a finite dimensional space of overconvergent forms of the given weight with eigenvalues of the given valuation. In contrast, it is clear that, even in the overconvergent case, ker(U) is quite large (in fact, infinite-dimensional), because of the Frobenius endomorphism: given any f E M(B,k,N;r), we have f - F r o b V f C ker(U).
A scrapbook of complex curve theory by C. Herbert Clemens