William Yun Chen. William Yun Chen Pennsylvania State University ICERM 5-minute intro talk

Transcription

1 William Yun Chen

2 William Yun Chen Institution - Pennsylvania State University

3 William Yun Chen Institution - Pennsylvania State University Advisor - Wen-Ching Winnie Li

4 William Yun Chen Institution - Pennsylvania State University Advisor - Wen-Ching Winnie Li Status - Looking for jobs!

5 William Yun Chen Institution - Pennsylvania State University Advisor - Wen-Ching Winnie Li Status - Looking for jobs! Research Interests - Arithmetic geometry, moduli of elliptic curves, noncongruence modular forms, galois theory, group theory, anabelian geometry.

6 William Yun Chen Institution - Pennsylvania State University Advisor - Wen-Ching Winnie Li Status - Looking for jobs! Research Interests - Arithmetic geometry, moduli of elliptic curves, noncongruence modular forms, galois theory, group theory, anabelian geometry. Questions I ve been thinking about:

7 William Yun Chen Institution - Pennsylvania State University Advisor - Wen-Ching Winnie Li Status - Looking for jobs! Research Interests - Arithmetic geometry, moduli of elliptic curves, noncongruence modular forms, galois theory, group theory, anabelian geometry. Questions I ve been thinking about: It s well known that quotients of H by congruence subgroups Γ SL 2 (Z) have moduli interpretations. Do noncongruence modular curves also have moduli interpretations?

8 William Yun Chen Institution - Pennsylvania State University Advisor - Wen-Ching Winnie Li Status - Looking for jobs! Research Interests - Arithmetic geometry, moduli of elliptic curves, noncongruence modular forms, galois theory, group theory, anabelian geometry. Questions I ve been thinking about: It s well known that quotients of H by congruence subgroups Γ SL 2 (Z) have moduli interpretations. Do noncongruence modular curves also have moduli interpretations? Yes! They are moduli spaces for elliptic curves equipped with nonabelian level structures that I discuss in my thesis.

9 The idea is this:

10 The idea is this: {Γ 0 (N)-structures on E} {Cyclic subgroups of E of order N}

11 The idea is this: {Γ 0 (N)-structures on E} {Cyclic subgroups of E of order N} {Cyclic N-isogenies E E}

12 The idea is this: {Γ 0 (N)-structures on E} {Cyclic subgroups of E of order N} {Cyclic N-isogenies E E} {Z/NZ-galois covers of E}

13 The idea is this: {Γ 0 (N)-structures on E} {Cyclic subgroups of E of order N} {Cyclic N-isogenies E E} {Z/NZ-galois covers of E} By allowing for ramification at, we can generalize these level structures to consider G-galois covers of E, where G is a finite nonabelian group.

14 The idea is this: {Γ 0 (N)-structures on E} {Cyclic subgroups of E of order N} {Cyclic N-isogenies E E} {Z/NZ-galois covers of E} By allowing for ramification at, we can generalize these level structures to consider G-galois covers of E, where G is a finite nonabelian group. From this perspective, all congruence level structures on E can be thought of as abelian covers of E.

15 The idea is this: {Γ 0 (N)-structures on E} {Cyclic subgroups of E of order N} {Cyclic N-isogenies E E} {Z/NZ-galois covers of E} By allowing for ramification at, we can generalize these level structures to consider G-galois covers of E, where G is a finite nonabelian group. From this perspective, all congruence level structures on E can be thought of as abelian covers of E. Result: If G is sufficiently nonabelian, then the corresponding moduli space is a noncongruence modular curve.

16 The idea is this: {Γ 0 (N)-structures on E} {Cyclic subgroups of E of order N} {Cyclic N-isogenies E E} {Z/NZ-galois covers of E} By allowing for ramification at, we can generalize these level structures to consider G-galois covers of E, where G is a finite nonabelian group. From this perspective, all congruence level structures on E can be thought of as abelian covers of E. Result: If G is sufficiently nonabelian, then the corresponding moduli space is a noncongruence modular curve. For example, any extension of S n (n 4), A n (n 5), PSL 2 (F p ), any minimal finite simple group, and conjecturally any finite simple group have noncongruence moduli spaces.

17 Related Problems The Unbounded Denominators Conjecture states that a q-expansion for a modular form holomorphic on H with algebraic fourier coefficients has bounded denominators if and only if f is a modular form for a congruence subgroup.

18 Related Problems The Unbounded Denominators Conjecture states that a q-expansion for a modular form holomorphic on H with algebraic fourier coefficients has bounded denominators if and only if f is a modular form for a congruence subgroup. My moduli interpretations of noncongruence modular curves can be used to translate this conjecture into the language of galois theory and the existence of nonabelian covers of the Tate curve.

19 Related Problems The Unbounded Denominators Conjecture states that a q-expansion for a modular form holomorphic on H with algebraic fourier coefficients has bounded denominators if and only if f is a modular form for a congruence subgroup. My moduli interpretations of noncongruence modular curves can be used to translate this conjecture into the language of galois theory and the existence of nonabelian covers of the Tate curve. The Inverse Galois Problem.

20 Related Problems The Unbounded Denominators Conjecture states that a q-expansion for a modular form holomorphic on H with algebraic fourier coefficients has bounded denominators if and only if f is a modular form for a congruence subgroup. My moduli interpretations of noncongruence modular curves can be used to translate this conjecture into the language of galois theory and the existence of nonabelian covers of the Tate curve. The Inverse Galois Problem. Much of the arithmetic geometry of the moduli spaces is encoded in the structure of the finite group G, which is readily accessible by computer computation. Finding rational points on the moduli spaces may lead to new ways of realizing finite groups as galois groups of Q.