Science Mathematics

Quaternion orders, quadratic forms, and Shimura curves by Montserrat Alsina, Pilar Bayer

By Montserrat Alsina, Pilar Bayer

Shimura curves are a far-reaching generalization of the classical modular curves. They lie on the crossroads of many parts, together with advanced research, hyperbolic geometry, algebraic geometry, algebra, and mathematics. This monograph offers Shimura curves from a theoretical and algorithmic standpoint. the most issues are Shimura curves outlined over the rational quantity box, the development in their basic domain names, and the choice in their advanced multiplication issues. The examine of advanced multiplication issues in Shimura curves results in the learn of households of binary quadratic types with algebraic coefficients and to their class by means of mathematics Fuchsian teams. during this regard, the authors advance a conception filled with new probabilities that parallels Gauss' conception at the class of binary quadratic types with critical coefficients by means of the motion of the modular staff. this can be one of many few to be had books explaining the speculation of Shimura curves on the graduate scholar point. each one subject coated within the ebook starts with a theoretical dialogue via rigorously worked-out examples, getting ready the way in which for additional learn. Titles during this sequence are co-published with the Centre de Recherches Mathématiques.

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2) (-q;q2)ro = T r n n=l 00 •q2""1) = TT L ± ^ n=l 1 + q n 2^ , 2n -n q n=l (1 - q ) ( l + q 2 n ) ~ ( q ; q 2 ) J - q 2 ^ 2 ) ^ 1 which is a famous identity of Euler. 1), we complete the proof of (i). CHAPTER 16 OF RAMANUJAN'S SECOND NOTEBOOK PROOF OF (ii). For f(q,q 3 ) = 1 + I l k=0 |q| < 1, I q 2k(2k-l)/2 k=l k=l k odd 33 q"« + »" + J q 2k(2k + l)/2 k=l + I k=0 k even q' q k(k+D/2 ) which proves the first equality. 2). PROOF OF (iii). 1) of f(a,b). By Entry 19, The first equality follows immediately from the definition f(-q,-q2) = (q;q 3 )Jq 2 ;q 3 )Jq 3 ;q 3 L = ( q ^ L ENTRY 23.

4), we see that it suffices to show that Log G(n) = 4 J ^ W ^ k • 1 )n) k=l (2k - 1)(1 - q 4k ~^) Like the calculation of Log F(n) in the previous proof, the proof of the equality above is quite straightforward. Ramanujan now states two "corollaries". We have not been able to discern why the appellation "corollary" has been given to these two results. Moreover, the "corollaries" are incorrect. We shall give two corrected versions of each corollary. Firstly, we prove versions where the "right sides" are corrected; secondly, we establish reformulations when the "left sides" are corrected.

N > i. n f(bq n+1 ,aq n ) ~ n+1 v x/. 1), and iteration, we deduce that f(b,a) _ -, fTbqTH - bq , m^WZ - bq ! aq + 2 f(bq2>aq) f(bq 3 ,ac 2 ) ^ • ^ - 1+ bq 2 3 N bq bq 1 - aq + 1 - aq + 1 - aq +••• The convergence of this continued fraction follows in the same way as in the proof of Entry 13. This completes the proof. COROLLARY. oo k=Q If |q| < 1, then k k(k+l) (q) kk 2 IM^ k k = 1 aq aq 2 " 1 + 1 + 1 aq3 + T + - k=0 PROOF. Set a = 0 in Entry 15 and then replace b by a. The corollary now readily follows.

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