show/hide this revision's text 3 fix typo

In the case when the Hida algebra is simply $\Lambda$, one can use families of overconvergent modular symbols to compute the $q$-expansion of the Hida family where the coefficients are functions of the weight.

The idea is the following: form a "random" family of overconvergent modular symbols -- that is modular symbols with values in a distribution module tensor power series in the weight. Then iterate $U_p$ on this family until you are in the ordinary subspace (modulo the accuracy at which you are working). Since the Hida algebra is just $\Lambda$, the result will be an eigensymbol. It's Its Hecke-eigenvalues will then be functions of the weight and these are the coefficients of the formal $q$-expansion I mentioned above.

With this $q$-expansion in hand, just plug in your favorite weight and you'll get the form in the Hida family of that weight (of course, computed modulo your fixed accuracy).

This approach actually came out of an Arizona Winter School 2011 student project. Currently three graduate students at University of Madison (Lalit Jain, Marton Hablicsek, and Daniel Ross), together with Rob Harron and myself, are writing up a paper on this.

Our code is still very much in beta form, but hopefully this example works. Take $p=3$ and $N=5$. Then the first few Hecke-eigenvalues (mod $3^3$) of the associated Hida family are:

$$ a_2(w) = 12w^2 + 7w - 1, $$ $$ a_3(w) = 21w^2 + 11w - 1, $$ and $$ a_5(w) = 16w^2 + 17w + 1. $$ Here $w$ is not exactly the weight variable. Instead, to specialize these eigenvalues to weight $k$, one simply sets $w = 4^{k-2} - 1$. (Here is 4 is a topological generator of $1+3{\mathbb Z}_3$.)

Here's the example with $N=1$ and $p=11$ -- below I'm working modulo $11^8$:

$$ a_2(w) = 37w^7 + 880w^6 + 12388w^5 + 151975w^4 + 385840w^3 + 10344442w^2 + 40094463w + 857435522, $$ $$ a_3(w) = 71w^7 + 681w^6 + 8325w^5 + 94314w^4 + 220797w^3 + 2758794w^2 + 16210985w + 428717761, $$ and $$ a_5(w) = 12w^7 + 472w^6 + 2245w^5 + 44445w^4 + 30443w^3 + 14355835w^2 + 180294533w + 1. $$ As before, setting $w = (1+11)^{k-2}-1$ specializes to weight $k$. For instance, taking $k=12$ one should recover the first few Fourier coefficients of $\Delta$ modulo $11^8$.
show/hide this revision's text 2 added N=1 p=11 example and corrected typo on the weight specialization

In the case when the Hida algebra is simply $\Lambda$, one can use families of overconvergent modular symbols to compute the $q$-expansion of the Hida family where the coefficients are functions of the weight.

The idea is the following: form a "random" family of overconvergent modular symbols -- that is modular symbols with values in a distribution module tensor power series in the weight. Then iterate $U_p$ on this family until you are in the ordinary subspace (modulo the accuracy at which you are working). Since the Hida algebra is just $\Lambda$, the result will be an eigensymbol. It's Hecke-eigenvalues will then be functions of the weight and these are the coefficients of the formal $q$-expansion I mentioned above.

With this $q$-expansion in hand, just plug in your favorite weight and you'll get the form in the Hida family of that weight (of course, computed modulo your fixed accuracy).

This approach actually came out of an Arizona Winter School 2011 student project. Currently three graduate students at University of Madison (Lalit Jain, Marton Hablicsek, and Daniel Ross), together with Rob Harron and myself, are writing up a paper on this.

Our code is still very much in beta form, but hopefully this example works. Take $p=3$ and $N=5$. Then the first few Hecke-eigenvalues (mod $3^3$) of the associated Hida family are:

$$ a_2(w) = 12w^2 + 7w - 1, $$ $$ a_3(w) = 21w^2 + 11w - 1, $$ and $$ a_5(w) = 16w^2 + 17w + 1. $$ Here $w$ is not exactly the weight variable. Instead, to specialize these eigenvalues to weight $k$, one simply sets $w = 4^k 4^{k-2} - 1$. (Here is 4 is a topological generator of $1+3{\mathbb Z}_3$.)

Here's the example with $N=1$ and $p=11$ -- below I'm working modulo $11^8$:

$$ a_2(w) = 37w^7 + 880w^6 + 12388w^5 + 151975w^4 + 385840w^3 + 10344442w^2 + 40094463w + 857435522, $$ $$ a_3(w) = 71w^7 + 681w^6 + 8325w^5 + 94314w^4 + 220797w^3 + 2758794w^2 + 16210985w + 428717761, $$ and $$ a_5(w) = 12w^7 + 472w^6 + 2245w^5 + 44445w^4 + 30443w^3 + 14355835w^2 + 180294533w + 1. $$ As before, setting $w = (1+11)^{k-2}-1$ specializes to weight $k$. For instance, taking $k=12$ one should recover the first few Fourier coefficients of $\Delta$ modulo $11^8$.
show/hide this revision's text 1

In the case when the Hida algebra is simply $\Lambda$, one can use families of overconvergent modular symbols to compute the $q$-expansion of the Hida family where the coefficients are functions of the weight.

The idea is the following: form a "random" family of overconvergent modular symbols -- that is modular symbols with values in a distribution module tensor power series in the weight. Then iterate $U_p$ on this family until you are in the ordinary subspace (modulo the accuracy at which you are working). Since the Hida algebra is just $\Lambda$, the result will be an eigensymbol. It's Hecke-eigenvalues will then be functions of the weight and these are the coefficients of the formal $q$-expansion I mentioned above.

With this $q$-expansion in hand, just plug in your favorite weight and you'll get the form in the Hida family of that weight (of course, computed modulo your fixed accuracy).

This approach actually came out of an Arizona Winter School 2011 student project. Currently three graduate students at University of Madison (Lalit Jain, Marton Hablicsek, and Daniel Ross), together with Rob Harron and myself, are writing up a paper on this.

Our code is still very much in beta form, but hopefully this example works. Take $p=3$ and $N=5$. Then the first few Hecke-eigenvalues (mod $3^3$) of the associated Hida family are:

$$ a_2(w) = 12w^2 + 7w - 1, $$ $$ a_3(w) = 21w^2 + 11w - 1, $$ and $$ a_5(w) = 16w^2 + 17w + 1. $$ Here $w$ is not exactly the weight variable. Instead, to specialize these eigenvalues to weight $k$, one simply sets $w = 4^k - 1$. (Here is 4 is a topological generator of $1+3{\mathbb Z}_3$.)