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= Sage Days 42: Women in Sage (3) = | == Jen's project == |
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== Jen's project == | The paramodular groups are subgroups of GSp(4) which are good analogues of the congruence subgroups underlying the new and old-form theory for GL(2). They have been studied for a long time for the connection to abelian surfaces with polarizations of type (1,N), but are not as prevalent in the literature on Siegel modular forms as the Siegel congruence subgroup. Interest in these subgroups has increased in the past years in part due to a precise analog of the Taniyama-Shimura conjecture for abelian surfaces by Brumer and Kramer known as the paramodular conjecture. |
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== Kristin's project == | In this project, we will build onto the SiegelModularForms_class which is described in the paper by Raum, Ryan, Skoruppa and Tornari. One project is to implement a method to compute the two Hecke operators for the paramodular group at any level on the Fourier expansion of a paramodular form. There is an explicit description of these Hecke operators in the book by Roberts and Schmidt. Another project would be to compute a set of examples of non-prime level of paramodular forms from the table of Hilbert modular forms over Q(\sqrt(5)) and their associated abelian surfaces by computing the Weil restriction of the elliptic curves and the automorphic induction of the modular forms as described in J-L. and Roberts. == Kristin's project == |
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Recall that the Hilbert class polynomial has as its zeros j-invariants of elliptic curves with complex multiplication (CM). In the case of genus 2, Igusa class polynomials play the analogous role: their zeros are Igusa invariants of genus 2 curves whose Jacobians have CM by a quartic CM field K. These Igusa invariants, in turn, require knowledge of the Siegel modular forms \chi_10 and \chi_12. | Recall that the Hilbert class polynomial has as its zeros j-invariants of elliptic curves with complex multiplication (CM). In the case of genus 2, Igusa class polynomials play the analogous role: their zeros are Igusa invariants of genus 2 curves whose Jacobians have CM by a quartic CM field K. These Igusa invariants, in turn, require knowledge of the Siegel modular forms $\chi_{10}$ and $\chi_{12}$. |
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In genus 3, the analogous computation requires us to properly understand \chi_18. This project will be about various things concerning \chi_18, with the eventual goal of getting some computational data for \chi_18 evaluated at CM points which we could study arithmetically and use to prove bounds on the primes which appear. | In genus 3, the analogous computation requires us to properly understand $\chi_{18}$. This project will be about various things concerning $\chi_{18}$, with the eventual goal of getting some computational data for $\chi_{18}$ evaluated at CM points which we could study arithmetically and use to prove bounds on the primes which appear. |
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0) Understand the definition of \chi_18, as in pp. 850 - 854 of [4]. (Lemma 10 defines \chi_18 as the product of the 36 even theta characteristics and Lemma 11 gives a geometric interpretation.) |
0) Understand the definition of $\chi_{18}$, as in pp. 850 - 854 of [4]. (Lemma 10 defines $\chi_{18}$ as the product of the 36 even theta characteristics and Lemma 11 gives a geometric interpretation.) |
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1) Has \chi_18 been expressed in terms of Eisenstein series, as has been done for \chi_10 and \chi_12? See pp. 189-195 in [5]. In [1], evaluating Igusa functions via the Eisenstein series expansion is investigated, and explicit bounds are proved on the tail of the expansion. Analogous bounds in the genus 3 case could also be helpful. | 1) Has $\chi_{18}$ been expressed in terms of Eisenstein series, as has been done for $\chi_{10}$ and $\chi_{12}$? See pp. 189-195 in [5]. In [1], evaluating Igusa functions via the Eisenstein series expansion is investigated, and explicit bounds are proved on the tail of the expansion. Analogous bounds in the genus 3 case could also be helpful. |
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2) Ritzenthaler used the 36 even theta characteristics definition to do some explicit computations of \chi_18 when the abelian variety is a power of an elliptic curve E with complex multiplication. Some details are in Section 4 of [7]. Compare this computation (timing-wise) to one that uses the Fourier expansion definition. Which one converges faster/is more efficient? Which one allows us to better control precision? | 2) Ritzenthaler used the 36 even theta characteristics definition to do some explicit computations of $\chi_{18}$ when the abelian variety is a power of an elliptic curve E with complex multiplication. Some details are in Section 4 of [7]. Compare this computation (timing-wise) to one that uses the Fourier expansion definition. Which one converges faster/is more efficient? Which one allows us to better control precision? |
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== Michelle's project == | == Michelle's project == [[attachment:ManesProject.pdf]] |
Jen's project
The paramodular groups are subgroups of GSp(4) which are good analogues of the congruence subgroups underlying the new and old-form theory for GL(2). They have been studied for a long time for the connection to abelian surfaces with polarizations of type (1,N), but are not as prevalent in the literature on Siegel modular forms as the Siegel congruence subgroup. Interest in these subgroups has increased in the past years in part due to a precise analog of the Taniyama-Shimura conjecture for abelian surfaces by Brumer and Kramer known as the paramodular conjecture.
In this project, we will build onto the SiegelModularForms_class which is described in the paper by Raum, Ryan, Skoruppa and Tornari. One project is to implement a method to compute the two Hecke operators for the paramodular group at any level on the Fourier expansion of a paramodular form. There is an explicit description of these Hecke operators in the book by Roberts and Schmidt.
Another project would be to compute a set of examples of non-prime level of paramodular forms from the table of Hilbert modular forms over Q(\sqrt(5)) and their associated abelian surfaces by computing the Weil restriction of the elliptic curves and the automorphic induction of the modular forms as described in J-L. and Roberts.
Kristin's project
Recall that the Hilbert class polynomial has as its zeros j-invariants of elliptic curves with complex multiplication (CM). In the case of genus 2, Igusa class polynomials play the analogous role: their zeros are Igusa invariants of genus 2 curves whose Jacobians have CM by a quartic CM field K. These Igusa invariants, in turn, require knowledge of the Siegel modular forms \chi_{10} and \chi_{12}.
One computational difference between genus 1 and 2 is that the Hilbert class polynomial has coefficients in Z, whereas the Igusa class polynomials have coefficients in Q. The difficulty, then, in genus 2, is understanding how bad these denominators are, because recovering the coefficients from approximations requires a bound on the denominators. (See, e.g., [2]. For some other details on the circle of ideas involved, see [3].)
In genus 3, the analogous computation requires us to properly understand \chi_{18}. This project will be about various things concerning \chi_{18}, with the eventual goal of getting some computational data for \chi_{18} evaluated at CM points which we could study arithmetically and use to prove bounds on the primes which appear.
Here are some starting points:
0) Understand the definition of \chi_{18}, as in pp. 850 - 854 of [4]. (Lemma 10 defines \chi_{18} as the product of the 36 even theta characteristics and Lemma 11 gives a geometric interpretation.)
1) Has \chi_{18} been expressed in terms of Eisenstein series, as has been done for \chi_{10} and \chi_{12}? See pp. 189-195 in [5]. In [1], evaluating Igusa functions via the Eisenstein series expansion is investigated, and explicit bounds are proved on the tail of the expansion. Analogous bounds in the genus 3 case could also be helpful.
2) Ritzenthaler used the 36 even theta characteristics definition to do some explicit computations of \chi_{18} when the abelian variety is a power of an elliptic curve E with complex multiplication. Some details are in Section 4 of [7]. Compare this computation (timing-wise) to one that uses the Fourier expansion definition. Which one converges faster/is more efficient? Which one allows us to better control precision?
References:
[1] R. Broker and K. Lauter, Evaluating Igusa functions, preprint. (http://arxiv.org/pdf/1005.1234v2.pdf)
[2] E. Goren and K. Lauter, Class invariants for quartic CM fields. Annales Inst. Fourier 57, 2 (2007), p.457-480.
[3] H. Grundman, J. Johnson-Leung, K. Lauter, A. Salerno, B. Viray and E. Wittenborn, Igusa class polynomials, embeddings of quartic CM fields, and arithmetic intersection theory, WIN— Women in Numbers, Fields Inst. Comm., vol. 60, AMS, Providence, RI, 2011, pp. 35—60.
[4] J.-I. Igusa, Modular forms and projective invariants, Amer. J. Math. 89 (1967), 817-855.
[5] J.-I. Igusa, On Siegel modular forms of genus two, Amer. J. Math. 84 (1962), 175-200.
[6] J.-I. Igusa, On Siegel modular forms of genus two, II, Amer. J. Math. 86 (1964), 392-412.
[7] C. Ritzenthaler, Explicit computations of Serre's obstruction for genus-3 curves and application to optimal curves. LMS J. Comput. Math.13 (2010), 192-207 (http://arxiv.org/pdf/0901.2920v2).