Computational Algebra Seminar: Fisher and Watkins

This week the computational algebra seminar will feature a double bill.
    

Date : Thursday September 4
Room : Carslaw 535A

2:30 -- 3:30pm : Tom Fisher (Cambridge)

4:00 -- 5:00pm : Mark Watkins (Sydney)



Tom Fisher will speak on

"Minimal models for 6-coverings of elliptic curves"

I will describe a new formula for adding 2-coverings and 3-coverings of
elliptic curves that avoids the need for any field extensions. By
searching for rational points on the 6-coverings obtained, we can then
find generators for the Mordell-Weil group of large height.  However   
before searching for rational points, it helps to make a good choice of   
co-ordinates, by a combination of minimisation and reduction.  I will
discuss the minimisation problem for 6-coverings. In particular it turns
out that adding a minimal 2-covering and a minimal 3-covering (using my     
formula) does not give a minimal 6-covering.



Mark Watkins will speak on

"A database of (degenerated) hypergeometric motives"

A family of hypergeometric motives can be determined by a coprime pair of
products of cyclotomic polynomials, each product having the same degree
$d$. For each $t$ not 0, 1, or infinity, one can then (following Katz)
associate a motive $M_t$ to this data. At each prime $p$, one obtains an
Euler factor by considering the monodromy action (over $Q_p$) on the
solution space of the associated hypergeometric differential equation. For
good primes, this can be explicitly calculated by a trace formula (again
see Katz). There are three kinds of bad primes. The wild primes are those
which divide one of the indices of the cyclotomic polynomials. The tame
primes are with $v_p(t)$ nonzero, and the "multiplicative" primes are
those with $v_p(t-1)$ nonzero. These last are the easiest to understand,
in the complex case they correspond to the fact that there are $(d-1)$ 
independent holomorphic solutions about $t=1$. Indeed, one can consider   
the formal motive $M_1$ at $t=1$ -- as an example, in the case of $\Phi_5$
and $\Phi_1^4$, one obtains the weight 4 modular form of level 25.

We report specifically on a "database" of $t=1$ degenerations, including
all choices (about 1500 total) of cyclotomic data up through degree 6. In
each case, we are able to numerically determine the wild Euler factors and
verify the functional equation of the L-function to (say) 15 digits of
precision. Of additional interest is the fact that approximately half the
relevant (odd motivic weight) examples of even parity appear to have
analytic rank 2.