{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Dr. Katz DEq Homework Solutions 86

# Dr. Katz DEq Homework Solutions 86 - 86 N.M Katz P roof...

This preview shows page 1. Sign up to view the full content.

86 N.M. Katz: Proof By (5.4.1), (6.2.1).r and by (6.0.4.3), (6.2.2)r Be- cause the hypergeometric equation has regular singular points, (6.2.3)~ (6.2.4). By (5.4.4), we have (6.2.2)~ (6.2.5). By ([24], Theorem 1.3.0), (6.2.5) implies that E(a, b, c) has regular singular points on Sc, and that its local monodromy around each singular point 0, 1, oo is of finite order. In particular, (6.2.5) implies that E(a, b, c) on Sc has rational exponents at each of 0, 1, ~, or, what is the same, that the hypergeometric equa- tion (6.1.2) with parameters a, b, c has rational exponents at 0, 1, ~. As the exponents are 0 and 1 -c at 0, 0 and c- a- b at 1, and a and b at ~, (6.2.5) implies that a, b, c~Q. The rest of the implication (6.2.5)~ (6.2.6) results from the fact that E(a, b, c) "comes from" T-an open subset of Spec(Z), so that we may test for "p-curvature zero for almost all p" by seeing if the reduction modp on E(a, b, c) on Svp has p-curvature zero for almost all p (and perform this latter test prime by prime availing ourselves of (6.0.6.7) and (6.0.7)).
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern