Tamagawa Products for Elliptic Curves Over Number Fields

Abstract: In recent work, Griffin, Ono, and Tsai constructs an

L -

series to prove that the proportion of short Weierstrass elliptic curves over

m a t h b b Q

with trivial Tamagawa product is

0.5054 d o t s

and that the average Tamagawa product is

1.8183 d o t s

. Following their work, we generalize their

L -

series over arbitrary number fields

K

to be [L_{mathrm{Tam}}(K; s):=sum_{m=1}^{infty}frac{P_{mathrm{Tam}}(K; m)}{m^s},] where

P_{m a t h r m T a m} (K; m)

is the proportion of short Weierstrass elliptic curves over

K

with Tamagawa product

m

. We then construct Markov chains to compute the exact values of

P_{m a t h r m T a m} (K; m)

for all number fields

K

and positive integers

m

. As a corollary, we also compute the average Tamagawa product

L_{m a t h r m T a m} (K; - 1)

. We then use these results to uniformly bound

P_{m a t h r m T a m} (K; 1)

and

L_{m a t h r m T a m} (K, - 1)

in terms of the degree of

K

. Finally, we show that there exist sequences of

K

for which

P_{m a t h r m T a m} (K; 1)

tends to

0

and

L_{m a t h r m T a m} (K; - 1)

to

i n f t y

, as well as sequences of

K

for which

P_{m a t h r m T a m} (K; 1)

and

L_{m a t h r m T a m} (K; - 1)

tend to

1

.

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