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L SERIES and MODULAR FORMS of HALF INTEGRAL WEIGHT I Bnl

L SERIES and MODULAR FORMS of HALF INTEGRAL WEIGHT I Bnl

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uh okay uh well thank you for the introduction and i want to thank the organizers for inviting me to speak here it's a really a great pleasure um so as you'll see in this talk a lot of the uh work in this talk and a lot of my other work has been sort of inspired by work of gordon and his collaborators so it's a special honor to speak here okay uh so the main theories i'm going to talk about uh are jointly spencer leslie so i'm going to start with something very classical the classical theta function so it's got some q expansion theta of z is q to the n squared so 1 plus 2 times the sum of 2 to the n squared for n at least 1 where of course q is e to the 2 pi i z this is a classical modular form of weight one half for the group gamma one four which i'll put in so2 here so this is the congruent subgroup of sl2 of matrices abcd such that this is congruent to 1 star 0 1 mod 4. so what i'm going to do is take this modular form of weight one half and cube it and call this some new modular fo

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