William Yun Chen William Yun Chen chen_w@math.psu.edu Pennsylvania - - PowerPoint PPT Presentation

William Yun Chen

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

William Yun Chen

Institution

• Pennsylvania State University

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

William Yun Chen

Institution

• Pennsylvania State University

Advisor

• Wen-Ching Winnie Li

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

William Yun Chen

Institution

• Pennsylvania State University

Advisor

• Wen-Ching Winnie Li

Status

• Looking for jobs!

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

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.

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

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:

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

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 ≤ SL2(Z) have moduli interpretations. Do noncongruence modular curves also have moduli interpretations?

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

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 ≤ SL2(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.

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

The idea is this:

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

The idea is this: {Γ0(N)-structures on E} ∼ {Cyclic subgroups of E of order N}

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

The idea is this: {Γ0(N)-structures on E} ∼ {Cyclic subgroups of E of order N} · · · ∼ {Cyclic N-isogenies E ′ → E}

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

The idea is this: {Γ0(N)-structures on E} ∼ {Cyclic subgroups of E of order N} · · · ∼ {Cyclic N-isogenies E ′ → E} ∼ {Z/NZ-galois covers of E}

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

The idea is this: {Γ0(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.

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

The idea is this: {Γ0(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.

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

The idea is this: {Γ0(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.

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

The idea is this: {Γ0(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 Sn (n ≥ 4), An (n ≥ 5), PSL2(Fp), any minimal finite simple group, and conjecturally any finite simple group have noncongruence moduli spaces.

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

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.

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

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.

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

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.

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk

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.

William Yun Chen chen_w@math.psu.edu Pennsylvania State University ICERM 5-minute intro talk