University of MississippieGroveNSF-CBMS Conference: L-functions andMultiplicative Number Theory2019 NSF-CBMS Conference: L-functions andMultiplicative Number TheoryMay 20th, 4:00 PM - 5:00 PMAn effective Chebotarev density theorem forfamilies of fields, with an application to class groupsCaroline Turnage-ButterbaughCarleton College, cturnageb@carleton.eduFollow this and additional works at: https://egrove.olemiss.edu/cbms2019Part of the Number Theory CommonsThis Presentation is brought to you for free and open access by the Mathematics, Department of at eGrove. It has been accepted for inclusion in NSF-CBMS Conference: L-functions and Multiplicative Number Theory by an authorized administrator of eGrove. For more information, please contactegrove@olemiss.edu.Recommended CitationTurnage-Butterbaugh, Caroline, "An effective Chebotarev density theorem for families of fields, with an application to class groups"(2019). NSF-CBMS Conference: L-functions and Multiplicative Number Theory. 22.https://egrove.olemiss.edu/cbms2019/2019/schedule/22A CHEBOTAREV DENSITY THEOREM FORFAMILIES OF FIELDS, WITH AN APPLICATIONTO CLASS GROUPSCaroline Turnage-ButterbaughCarleton College(Joint work with Lillian Pierce and Melanie Matchett Wood)NSF-CBMS ConferenceL-functions and Multiplicative Number TheoryUniversity of MississippiMay 20, 2019BINARY QUADRATIC FORMS(a, b, c) := ax2 + bxy + cy2, a, b, c integers.Gauss• classified the binary quadratic forms with a givendiscriminant D := b2 − 4ac;• formed the class group, the group of equivalence classes ofbinary quadratic forms of a given D with group actionGauss composition;• showed that, for any given discriminant D, there exist onlyfinitely many equivalence classes of binary quadratic forms.1BINARY QUADRATIC FORMS(a, b, c) := ax2 + bxy + cy2, a, b, c integers.Gauss• classified the binary quadratic forms with a givendiscriminant D := b2 − 4ac;• formed the class group, the group of equivalence classes ofbinary quadratic forms of a given D with group actionGauss composition;• showed that, for any given discriminant D, there exist onlyfinitely many equivalence classes of binary quadratic forms.1BINARY QUADRATIC FORMS(a, b, c) := ax2 + bxy + cy2, a, b, c integers.Gauss• classified the binary quadratic forms with a givendiscriminant D := b2 − 4ac;• formed the class group, the group of equivalence classes ofbinary quadratic forms of a given D with group actionGauss composition;• showed that, for any given discriminant D, there exist onlyfinitely many equivalence classes of binary quadratic forms.1BINARY QUADRATIC FORMS(a, b, c) := ax2 + bxy + cy2, a, b, c integers.Gauss• classified the binary quadratic forms with a givendiscriminant D := b2 − 4ac;• formed the class group, the group of equivalence classes ofbinary quadratic forms of a given D with group actionGauss composition;• showed that, for any given discriminant D, there exist onlyfinitely many equivalence classes of binary quadratic forms.1QUADRATIC FORMS AND QUADRATIC NUMBER FIELDSLet K = Q(√D) be a quadratic number field. To each form(a, b, c) := ax2 + bxy + cy2with discriminant D = b2 − 4ac, we may associate an ideal I ofOK, whereI =〈a,−b +√D2〉.2Binary quadratic forms ←→ Nonzero ideals of OQ[√D](a, b, c) := ax2 + bxy + cy2 I =〈a, −b+√D2〉equivalent ←→ equivalentbinary quadratic forms idealscomposition of ←→ multiplication ofequivalence classes of forms equivalence classes of idealsClK := the ideal class group of K = Q(√D)h(K) = |ClK| := the class number of K = Q(√D)Note: h(K) is finite via the correspondence.3Binary quadratic forms ←→ Nonzero ideals of OQ[√D](a, b, c) := ax2 + bxy + cy2 I =〈a, −b+√D2〉equivalent ←→ equivalentbinary quadratic forms idealscomposition of ←→ multiplication ofequivalence classes of forms equivalence classes of idealsClK := the ideal class group of K = Q(√D)h(K) = |ClK| := the class number of K = Q(√D)Note: h(K) is finite via the correspondence.3Binary quadratic forms ←→ Nonzero ideals of OQ[√D](a, b, c) := ax2 + bxy + cy2 I =〈a, −b+√D2〉equivalent ←→ equivalentbinary quadratic forms idealscomposition of ←→ multiplication ofequivalence classes of forms equivalence classes of idealsClK := the ideal class group of K = Q(√D)h(K) = |ClK| := the class number of K = Q(√D)Note: h(K) is finite via the correspondence.3CLASS GROUP OF K, [K : Q] ≥ 2The ideal class group of K is defined byClK := JK/PK• JK := the group of fractional ideals of K• PK := the subgroup of principal ideals of K.The class number of K is defined byh(K) = |ClK|.h(K) = 1 ⇐⇒ ClK = {id} ⇐⇒ OK is a PID ⇐⇒ OK is a UFDQuestion: How big is |ClK| in general?4CLASS GROUP OF K, [K : Q] ≥ 2The ideal class group of K is defined byClK := JK/PK• JK := the group of fractional ideals of K• PK := the subgroup of principal ideals of K.The class number of K is defined byh(K) = |ClK|.h(K) = 1 ⇐⇒ ClK = {id}⇐⇒ OK is a PID ⇐⇒ OK is a UFDQuestion: How big is |ClK| in general?4CLASS GROUP OF K, [K : Q] ≥ 2The ideal class group of K is defined byClK := JK/PK• JK := the group of fractional ideals of K• PK := the subgroup of principal ideals of K.The class number of K is defined byh(K) = |ClK|.h(K) = 1 ⇐⇒ ClK = {id} ⇐⇒ OK is a PID⇐⇒ OK is a UFDQuestion: How big is |ClK| in general?4CLASS GROUP OF K, [K : Q] ≥ 2The ideal class group of K is defined byClK := JK/PK• JK := the group of fractional ideals of K• PK := the subgroup of principal ideals of K.The class number of K is defined byh(K) = |ClK|.h(K) = 1 ⇐⇒ ClK = {id} ⇐⇒ OK is a PID ⇐⇒ OK is a UFDQuestion: How big is |ClK| in general?4CLASS GROUP OF K, [K : Q] ≥ 2The ideal class group of K is defined byClK := JK/PK• JK := the group of fractional ideals of K• PK := the subgroup of principal ideals of K.The class number of K is defined byh(K) = |ClK|.h(K) = 1 ⇐⇒ ClK = {id} ⇐⇒ OK is a PID ⇐⇒ OK is a UFDQuestion: How big is |ClK| in general?4Landau observed that if [K : Q] = n, then|ClK| n D1/2+εKWe may conclude that ClK is a finite abelian group.For any integer ` > 1, the `-torsion subgroup of ClK isgiven byClK[`] :={[a] ∈ ClK : [a]` = Id}Natural Question: What is the size of ClK[`] as K varies withina family of fields of fixed degree?5Landau observed that if [K : Q] = n, then|ClK| n D1/2+εKWe may conclude that ClK is a finite abelian group.For any integer ` > 1, the `-torsion subgroup of ClK isgiven byClK[`] :={[a] ∈ ClK : [a]` = Id}Natural Question: What is the size of ClK[`] as K varies withina family of fields of fixed degree?5Landau observed that if [K : Q] = n, then|ClK| n D1/2+εKWe may conclude that ClK is a finite abelian group.For any integer ` > 1, the `-torsion subgroup of ClK isgiven byClK[`] :={[a] ∈ ClK : [a]` = Id}Natural Question: What is the size of ClK[`] as K varies withina family of fields of fixed degree?5HOW BIG IS |ClK[`]|?Trivial Bound – For [K : Q] = n, any integer ` ≥ 1, andε > 0|ClK[`]| ≤ |ClK| n,ε D1/2+εKConjecture – For [K : Q] = n, any integer ` ≥ 1, and ε > 0|ClK[`]| n,`,ε DεK.Recorded by• Brumer-Silverman, ’96• Duke, ’98• Zhang, ’05• Ellenberg-Venkatesh, ’07Implied by• Cohen-Lenstra-Martinetheuristics on thedistribution of class groupsand `-torsion subgroupswithin families6HOW BIG IS |ClK[`]|?Trivial Bound – For [K : Q] = n, any integer ` ≥ 1, andε > 0|ClK[`]| ≤ |ClK| n,ε D1/2+εKConjecture – For [K : Q] = n, any integer ` ≥ 1, and ε > 0|ClK[`]| n,`,ε DεK.Recorded by• Brumer-Silverman, ’96• Duke, ’98• Zhang, ’05• Ellenberg-Venkatesh, ’07Implied by• Cohen-Lenstra-Martinetheuristics on thedistribution of class groupsand `-torsion subgroupswithin families6HOW BIG IS |ClK[`]|?Trivial Bound – For [K : Q] = n, any integer ` ≥ 1, andε > 0|ClK[`]| ≤ |ClK| n,ε D1/2+εKConjecture – For [K : Q] = n, any integer ` ≥ 1, and ε > 0|ClK[`]| n,`,ε DεK.Recorded by• Brumer-Silverman, ’96• Duke, ’98• Zhang, ’05• Ellenberg-Venkatesh, ’07Implied by• Cohen-Lenstra-Martinetheuristics on thedistribution of class groupsand `-torsion subgroupswithin families6HOW BIG IS |ClK[`]|?Trivial Bound – For [K : Q] = n, any integer ` ≥ 1, andε > 0|ClK[`]| ≤ |ClK| n,ε D1/2+εKConjecture – For [K : Q] = n, any integer ` ≥ 1, and ε > 0|ClK[`]| n,`,ε DεK.Recorded by• Brumer-Silverman, ’96• Duke, ’98• Zhang, ’05• Ellenberg-Venkatesh, ’07Implied by• Cohen-Lenstra-Martinetheuristics on thedistribution of class groupsand `-torsion subgroupswithin families6WHAT DO WE KNOW IS TRUE?Conjecture – For [K : Q] = n, any integer ` ≥ 1, and ε > 0|ClK[`]| n,`,ε DεK.Theorem (Gauss)For all quadratic fields K, we have |ClK[2]| ε DεK.• This is the only case (for ` prime) in which the conjecturehas been proved.• Question: Are there cases for which nontrivial boundsknown?7WHAT DO WE KNOW IS TRUE?Conjecture – For [K : Q] = n, any integer ` ≥ 1, and ε > 0|ClK[`]| n,`,ε DεK.Theorem (Gauss)For all quadratic fields K, we have |ClK[2]| ε DεK.• This is the only case (for ` prime) in which the conjecturehas been proved.• Question: Are there cases for which nontrivial boundsknown?7WHAT DO WE KNOW IS TRUE?Conjecture – For [K : Q] = n, any integer ` ≥ 1, and ε > 0|ClK[`]| n,`,ε DεK.Theorem (Gauss)For all quadratic fields K, we have |ClK[2]| ε DεK.• This is the only case (for ` prime) in which the conjecturehas been proved.• Question: Are there cases for which nontrivial boundsknown?7WHAT DO WE KNOW IS TRUE?Conjecture – For [K : Q] = n, any integer ` ≥ 1, and ε > 0|ClK[`]| n,`,ε DεK.Theorem (Gauss)For all quadratic fields K, we have |ClK[2]| ε DεK.• This is the only case (for ` prime) in which the conjecturehas been proved.• Question: Are there cases for which nontrivial boundsknown?7NONTRIVIAL BOUNDS ON |ClK[`]|Theorem (Ellenberg & Venkatesh, 2007)Let K/Q be a number field of degree 2 or 3. We have|ClK[3]| n,ε D13+εK .Let K/Q be a non-D4 number field of degree 4. We have|ClK[3]| ε D12− 1168+εK .Theorem (Bhargava, Shankar, Taniguchi, Thorne, Tsimerman &Zhao, 2017)Let K/Q be a number field of degree n > 2. For some δn > 0 we have|ClK[2]| n,ε D12−δn+εK .8NONTRIVIAL BOUNDS ON |ClK[`]|Theorem (Ellenberg & Venkatesh, 2007)Let K/Q be a number field of degree 2 or 3. We have|ClK[3]| n,ε D13+εK .Let K/Q be a non-D4 number field of degree 4. We have|ClK[3]| ε D12− 1168+εK .Theorem (Bhargava, Shankar, Taniguchi, Thorne, Tsimerman &Zhao, 2017)Let K/Q be a number field of degree n > 2. For some δn > 0 we have|ClK[2]| n,ε D12−δn+εK .8NONTRIVIAL BOUNDS ON |ClK[`]|Theorem (Ellenberg & Venkatesh, 2007)Let K/Q be a number field of degree 2 or 3. We have|ClK[3]| n,ε D13+εK .Let K/Q be a non-D4 number field of degree 4. We have|ClK[3]| ε D12− 1168+εK .Theorem (Bhargava, Shankar, Taniguchi, Thorne, Tsimerman &Zhao, 2017)Let K/Q be a number field of degree n > 2. For some δn > 0 we have|ClK[2]| n,ε D12−δn+εK .8NONTRIVIAL BOUNDS ON |ClK[`]| . . . UNDER GRHTheorem (Ellenberg & Venkatesh, 2007)Let K/Q be a number field of degree n and ` a positive integer.Assuming GRH, we have|ClK[`]| n,`,ε D12− 12`(n−1)+εK .• Question: What can we say unconditionally for all but apossible exceptional set of fields K within a family?9NONTRIVIAL BOUNDS ON |ClK[`]| . . . UNDER GRHTheorem (Ellenberg & Venkatesh, 2007)Let K/Q be a number field of degree n and ` a positive integer.Assuming GRH, we have|ClK[`]| n,`,ε D12− 12`(n−1)+εK .• Question: What can we say unconditionally for all but apossible exceptional set of fields K within a family?9NONTRIVIAL BOUNDS ON |ClK[`]| . . . IN FAMILIESTheorem (Soundararajan, 2000)Let ` be prime. For all but a possible zero-density exceptional familyof imaginary quadratic fields K/Q, we have|ClK[`]| `,ε D12− 12`+εK .Theorem (Heath-Brown & Pierce, 2014)Let ` ≥ 5 be prime. For all but a possible zero-density exceptionalfamily of imaginary quadratic fields K/Q, we have|ClK[`]| `,ε D12− 32`+2+εK .10NONTRIVIAL BOUNDS ON |ClK[`]| . . . IN FAMILIESTheorem (Soundararajan, 2000)Let ` be prime. For all but a possible zero-density exceptional familyof imaginary quadratic fields K/Q, we have|ClK[`]| `,ε D12− 12`+εK .Theorem (Heath-Brown & Pierce, 2014)Let ` ≥ 5 be prime. For all but a possible zero-density exceptionalfamily of imaginary quadratic fields K/Q, we have|ClK[`]| `,ε D12− 32`+2+εK .10NONTRIVIAL BOUNDS ON |ClK[`]| . . . IN FAMILIESTheorem (Ellenberg, Pierce, & Wood, 2016)Let ` ≥ 1, and let [K : Q] = 2, 3 or 5. For all but a possiblezero-density exceptional family of fields K/Q, we have|ClK[`]| n,`,ε D12− 12`(n−1)+εK .If [K : Q] = 4, then the same bound applies for K non-D4.• Note that the bound is as strong as on GRH.Pierce, T., and Wood, (2017 preprint)Under certain conditions (but never under GRH), we ex-tend this result to different families in which [K : Q] ≥ 2.11NONTRIVIAL BOUNDS ON |ClK[`]| . . . IN FAMILIESTheorem (Ellenberg, Pierce, & Wood, 2016)Let ` ≥ 1, and let [K : Q] = 2, 3 or 5. For all but a possiblezero-density exceptional family of fields K/Q, we have|ClK[`]| n,`,ε D12− 12`(n−1)+εK .If [K : Q] = 4, then the same bound applies for K non-D4.• Note that the bound is as strong as on GRH.Pierce, T., and Wood, (2017 preprint)Under certain conditions (but never under GRH), we ex-tend this result to different families in which [K : Q] ≥ 2.11NONTRIVIAL BOUNDS ON |ClK[`]| . . . IN FAMILIESTheorem (Ellenberg, Pierce, & Wood, 2016)Let ` ≥ 1, and let [K : Q] = 2, 3 or 5. For all but a possiblezero-density exceptional family of fields K/Q, we have|ClK[`]| n,`,ε D12− 12`(n−1)+εK .If [K : Q] = 4, then the same bound applies for K non-D4.• Note that the bound is as strong as on GRH.Pierce, T., and Wood, (2017 preprint)Under certain conditions (but never under GRH), we ex-tend this result to different families in which [K : Q] ≥ 2.11STARTING POINTTheorem (Ellenberg & Venkatesh, 2007)Suppose that there are M rational primesp1, p2, . . . , pMthat split completely in K, where pj ≤ DδK and δ < 12`(n−1) . Then forany ε > 0,|ClK[`]| n,`,ε D12+εK M−1.Question: How might one go about finding small primes thatsplit completely in K?Answer: via a Chebotarev Density Theorem12STARTING POINTTheorem (Ellenberg & Venkatesh, 2007)Suppose that there are M rational primesp1, p2, . . . , pMthat split completely in K, where pj ≤ DδK and δ < 12`(n−1) . Then forany ε > 0,|ClK[`]| n,`,ε D12+εK M−1.Question: How might one go about finding small primes thatsplit completely in K?Answer: via a Chebotarev Density Theorem13STARTING POINTTheorem (Ellenberg & Venkatesh, 2007)Suppose that there are M rational primesp1, p2, . . . , pMthat split completely in K, where pj ≤ DδK and δ < 12`(n−1) . Then forany ε > 0,|ClK[`]| n,`,ε D12+εK M−1.Question: How might one go about finding small primes thatsplit completely in K?Answer: via a Chebotarev Density Theorem13STARTING POINTTheorem (Ellenberg & Venkatesh, 2007)Suppose that there are M rational primesp1, p2, . . . , pMthat split completely in K, where pj ≤ DδK and δ < 12`(n−1) . Then forany ε > 0,|ClK[`]| n,`,ε D12+εK M−1.Question: How might one go about finding small primes thatsplit completely in K?Answer: via a Chebotarev Density Theorem13AN EFFECTIVE CHEBOTAREV DENSITY THEOREMKQGal(K/Q) ∼= GnTheorem (Lagarias-Odlyzko*, 1975)If GRH holds for ζK(s), then∣∣∣∣#{p ≤ x that split completely in K} − Li(x)|G|∣∣∣∣≤ C0|G|x1/2 log(DKxnK)for every x ≥ 2 and C0 is effectively computable.*This is a special case of their theorem.14AN EFFECTIVE CHEBOTAREV DENSITY THEOREMKQGal(K/Q) ∼= GnTheorem (Lagarias-Odlyzko*, 1975)If GRH holds for ζK(s), then∣∣∣∣#{p ≤ x that split completely in K} − Li(x)|G|∣∣∣∣≤ C0|G|x1/2 log(DKxnK)for every x ≥ 2 and C0 is effectively computable.*This is a special case of their theorem.14AN EFFECTIVE CHEBOTAREV DENSITY THEOREMKQGal(K/Q) ∼= GnTheorem (Lagarias-Odlyzko*, 1975)If GRH holds for ζK(s), then∣∣∣∣#{p ≤ x that split completely in K} − Li(x)|G|∣∣∣∣≤ C0|G|x1/2 log(DKxnK)for every x ≥ 2 and C0 is effectively computable.*This is a special case of their theorem.• We may take x = Dδ−0K , with δ =12`(n− 1) .• Obtain at least M D1/(2`(n−1))−ε0K sufficiently smallprimes that split completely in K.15AN EFFECTIVE CHEBOTAREV DENSITY THEOREMKQGal(K/Q) ∼= GnTheorem (Lagarias-Odlyzko*, 1975)If GRH holds for ζK(s), then∣∣∣∣#{p ≤ x that split completely in K} − Li(x)|G|∣∣∣∣≤ C0|G|x1/2 log(DKxnK)for every x ≥ 2 and C0 is effectively computable.*This is a special case of their theorem.• We may take x = Dδ−0K , with δ =12`(n− 1) .• Obtain at least M D1/(2`(n−1))−ε0K sufficiently smallprimes that split completely in K.15AN EFFECTIVE CHEBOTAREV DENSITY THEOREMKQGal(K/Q) ∼= GnTheorem (Lagarias-Odlyzko*, 1975)If GRH holds for ζK(s), then∣∣∣∣#{p ≤ x that split completely in K} − Li(x)|G|∣∣∣∣≤ C0|G|x1/2 log(DKxnK)for every x ≥ 2 and C0 is effectively computable.*This is a special case of their theorem.• We may take x = Dδ−0K , with δ =12`(n− 1) .• Obtain at least M D1/(2`(n−1))−ε0K sufficiently smallprimes that split completely in K.15BOUNDING `-TORSION ASSUMING GRHEllenberg-Venkatesh (2007)|ClK[`]| `,n,ε D12+εK M−1Lagarias-Odlyzko (1975)Conditional Effective Chebotarev Density TheoremM D1/(2`(n−1))−ε0KEllenberg-Venkatesh (2007)Assuming GRH, we have |ClK[`]| `,n,ε D12− 12`(n−1)+εKGoal: Remove GRH and obtain the same `-torsion bound.– We can do this at the cost of proving the result for all but apossible zero-density family of fields.16BOUNDING `-TORSION ASSUMING GRHEllenberg-Venkatesh (2007)|ClK[`]| `,n,ε D12+εK M−1Lagarias-Odlyzko (1975)Conditional Effective Chebotarev Density TheoremM D1/(2`(n−1))−ε0KEllenberg-Venkatesh (2007)Assuming GRH, we have |ClK[`]| `,n,ε D12− 12`(n−1)+εKGoal: Remove GRH and obtain the same `-torsion bound.– We can do this at the cost of proving the result for all but apossible zero-density family of fields.16BOUNDING `-TORSION ASSUMING GRHEllenberg-Venkatesh (2007)|ClK[`]| `,n,ε D12+εK M−1Lagarias-Odlyzko (1975)Conditional Effective Chebotarev Density TheoremM D1/(2`(n−1))−ε0KEllenberg-Venkatesh (2007)Assuming GRH, we have |ClK[`]| `,n,ε D12− 12`(n−1)+εKGoal: Remove GRH and obtain the same `-torsion bound.– We can do this at the cost of proving the result for all but apossible zero-density family of fields.16BOUNDING `-TORSION ASSUMING GRHEllenberg-Venkatesh (2007)|ClK[`]| `,n,ε D12+εK M−1Lagarias-Odlyzko (1975)Conditional Effective Chebotarev Density TheoremM D1/(2`(n−1))−ε0KEllenberg-Venkatesh (2007)Assuming GRH, we have |ClK[`]| `,n,ε D12− 12`(n−1)+εKGoal: Remove GRH and obtain the same `-torsion bound.– We can do this at the cost of proving the result for all but apossible zero-density family of fields.16BOUNDING `-TORSION ASSUMING GRHEllenberg-Venkatesh (2007)|ClK[`]| `,n,ε D12+εK M−1Lagarias-Odlyzko (1975)Conditional Effective Chebotarev Density TheoremM D1/(2`(n−1))−ε0KEllenberg-Venkatesh (2007)Assuming GRH, we have |ClK[`]| `,n,ε D12− 12`(n−1)+εKGoal: Remove GRH and obtain the same `-torsion bound.– We can do this at the cost of proving the result for all but apossible zero-density family of fields.16SAME STARTING POINT AS BEFORETheorem (Ellenberg & Venkatesh, 2007)Suppose that there are M rational primesp1, p2, . . . , pMthat split completely in K, where pj ≤ DδK and δ < 12`(n−1) . Then forany ε > 0,|ClK[`]| n,`,ε D12+εK M−1.We need an effective Chebotarev density theorem for a familyof fields K• that does not assume GRH, and• has a low threshold on x.Let us first recall how to count primes.17SAME STARTING POINT AS BEFORETheorem (Ellenberg & Venkatesh, 2007)Suppose that there are M rational primesp1, p2, . . . , pMthat split completely in K, where pj ≤ DδK and δ < 12`(n−1) . Then forany ε > 0,|ClK[`]| n,`,ε D12+εK M−1.We need an effective Chebotarev density theorem for a familyof fields K• that does not assume GRH, and• has a low threshold on x.Let us first recall how to count primes.17SAME STARTING POINT AS BEFORETheorem (Ellenberg & Venkatesh, 2007)Suppose that there are M rational primesp1, p2, . . . , pMthat split completely in K, where pj ≤ DδK and δ < 12`(n−1) . Then forany ε > 0,|ClK[`]| n,`,ε D12+εK M−1.We need an effective Chebotarev density theorem for a familyof fields K• that does not assume GRH, and• has a low threshold on x.Let us first recall how to count primes.17SAME STARTING POINT AS BEFORETheorem (Ellenberg & Venkatesh, 2007)Suppose that there are M rational primesp1, p2, . . . , pMthat split completely in K, where pj ≤ DδK and δ < 12`(n−1) . Then forany ε > 0,|ClK[`]| n,`,ε D12+εK M−1.We need an effective Chebotarev density theorem for a familyof fields K• that does not assume GRH, and• has a low threshold on x.Let us first recall how to count primes.17COUNTING PRIMESMotivating QuestionGiven a large number x, how many primes are there lessthan or equal to x?That is, if we letpi(x) :=∑p≤ x1,how does pi(x) behave as x→∞ ?18COUNTING PRIMESMotivating QuestionGiven a large number x, how many primes are there lessthan or equal to x?That is, if we letpi(x) :=∑p≤ x1,how does pi(x) behave as x→∞ ?18COUNTING PRIMESMotivating QuestionGiven a large number x, how many primes are there lessthan or equal to x?That is, if we letpi(x) :=∑p≤ x1,how does pi(x) behave as x→∞ ?18Prime Number Theorem (Hadamard, de la Valle´e Poussin 1896)pi(x) ∼ Li(x), x→∞Count primes with a weight:ψ(x) :=∑n≤xΛ(n), Λ(n) ={log p, if n = p k, k ≥ 1,0, else.Heuristic:ψ(x) =∑n≤xΛ(n) ≈∑p≤xlog p ≈ pi(x) log xψ(x) ∼ x ⇐⇒ pi(x) ∼ xlog x19Prime Number Theorem (Hadamard, de la Valle´e Poussin 1896)pi(x) ∼ Li(x), x→∞Count primes with a weight:ψ(x) :=∑n≤xΛ(n),Λ(n) ={log p, if n = p k, k ≥ 1,0, else.Heuristic:ψ(x) =∑n≤xΛ(n) ≈∑p≤xlog p ≈ pi(x) log xψ(x) ∼ x ⇐⇒ pi(x) ∼ xlog x19Prime Number Theorem (Hadamard, de la Valle´e Poussin 1896)pi(x) ∼ Li(x), x→∞Count primes with a weight:ψ(x) :=∑n≤xΛ(n), Λ(n) ={log p, if n = p k, k ≥ 1,0, else.Heuristic:ψ(x) =∑n≤xΛ(n) ≈∑p≤xlog p ≈ pi(x) log xψ(x) ∼ x ⇐⇒ pi(x) ∼ xlog x19Prime Number Theorem (Hadamard, de la Valle´e Poussin 1896)pi(x) ∼ Li(x), x→∞Count primes with a weight:ψ(x) :=∑n≤xΛ(n), Λ(n) ={log p, if n = p k, k ≥ 1,0, else.Heuristic:ψ(x) =∑n≤xΛ(n)≈∑p≤xlog p ≈ pi(x) log xψ(x) ∼ x ⇐⇒ pi(x) ∼ xlog x19Prime Number Theorem (Hadamard, de la Valle´e Poussin 1896)pi(x) ∼ Li(x), x→∞Count primes with a weight:ψ(x) :=∑n≤xΛ(n), Λ(n) ={log p, if n = p k, k ≥ 1,0, else.Heuristic:ψ(x) =∑n≤xΛ(n) ≈∑p≤xlog p≈ pi(x) log xψ(x) ∼ x ⇐⇒ pi(x) ∼ xlog x19Prime Number Theorem (Hadamard, de la Valle´e Poussin 1896)pi(x) ∼ Li(x), x→∞Count primes with a weight:ψ(x) :=∑n≤xΛ(n), Λ(n) ={log p, if n = p k, k ≥ 1,0, else.Heuristic:ψ(x) =∑n≤xΛ(n) ≈∑p≤xlog p ≈ pi(x) log xψ(x) ∼ x ⇐⇒ pi(x) ∼ xlog x19Prime Number Theorem (Hadamard, de la Valle´e Poussin 1896)pi(x) ∼ Li(x), x→∞Count primes with a weight:ψ(x) :=∑n≤xΛ(n), Λ(n) ={log p, if n = p k, k ≥ 1,0, else.Heuristic:ψ(x) =∑n≤xΛ(n) ≈∑p≤xlog p ≈ pi(x) log xψ(x) ∼ x ⇐⇒ pi(x) ∼ xlog x19PROVING ψ(x) ∼ xExplicit Formula (truncated version)We haveψ(x) = x−∑|γ| ≤ xxρρ+ O(log2 x)where the sum is over the nontrivial zeros of ζ(s).ρ = β + iγ is a nontrivial zero of ζ(s):ζ(s) =∞∑n=11ns=∏p prime(1− 1ps)−1, <(s) > 1• Since |xρ| = xβ , if β < 1, then the contribution from thenontrivial zeros is not too big.• Key to proof of the Prime Number Theorem:ζ(s) 6= 0 for <(s) = 120PROVING ψ(x) ∼ xExplicit Formula (truncated version)We haveψ(x) = x−∑|γ| ≤ xxρρ+ O(log2 x)where the sum is over the nontrivial zeros of ζ(s).ρ = β + iγ is a nontrivial zero of ζ(s):ζ(s) =∞∑n=11ns=∏p prime(1− 1ps)−1, <(s) > 1• Since |xρ| = xβ , if β < 1, then the contribution from thenontrivial zeros is not too big.• Key to proof of the Prime Number Theorem:ζ(s) 6= 0 for <(s) = 120PROVING ψ(x) ∼ xExplicit Formula (truncated version)We haveψ(x) = x−∑|γ| ≤ xxρρ+ O(log2 x)where the sum is over the nontrivial zeros of ζ(s).ρ = β + iγ is a nontrivial zero of ζ(s):ζ(s) =∞∑n=11ns=∏p prime(1− 1ps)−1, <(s) > 1• Since |xρ| = xβ , if β < 1, then the contribution from thenontrivial zeros is not too big.• Key to proof of the Prime Number Theorem:ζ(s) 6= 0 for <(s) = 120PROVING ψ(x) ∼ xExplicit Formula (truncated version)We haveψ(x) = x−∑|γ| ≤ xxρρ+ O(log2 x)where the sum is over the nontrivial zeros of ζ(s).ρ = β + iγ is a nontrivial zero of ζ(s):ζ(s) =∞∑n=11ns=∏p prime(1− 1ps)−1, <(s) > 1• Since |xρ| = xβ , if β < 1, then the contribution from thenontrivial zeros is not too big.• Key to proof of the Prime Number Theorem:ζ(s) 6= 0 for <(s) = 120PROVING ψ(x) ∼ xExplicit Formula (truncated version)We haveψ(x) = x−∑|γ| ≤ xxρρ+ O(log2 x)where the sum is over the nontrivial zeros of ζ(s).ρ = β + iγ is a nontrivial zero of ζ(s):ζ(s) =∞∑n=11ns=∏p prime(1− 1ps)−1, <(s) > 1• Since |xρ| = xβ , if β < 1, then the contribution from thenontrivial zeros is not too big.• Key to proof of the Prime Number Theorem:ζ(s) 6= 0 for <(s) = 120COUNTING PRIMES IN ARITHMETIC PROGRESSIONSSiegel-Walfisz Theorem (1935)If n ≥ 2 and a is coprime to q then as x→∞,pi(x; a, q) :=∑p≤xp≡ a (mod q)1 =1ϕ(q)Li(x) + ”error term”.• The error term depends on the zero-free region of theDirichlet L-function:L(s, χq) :=∞∑n=1χq(n)ns, <(s) > 121COUNTING PRIMES IN ARITHMETIC PROGRESSIONSSiegel-Walfisz Theorem (1935)If n ≥ 2 and a is coprime to q then as x→∞,pi(x; a, q) :=∑p≤xp≡ a (mod q)1 =1ϕ(q)Li(x) + ”error term”.• The error term depends on the zero-free region of theDirichlet L-function:L(s, χq) :=∞∑n=1χq(n)ns, <(s) > 121COUNTING PRIME IDEALS IN NUMBER FIELDSkOkQZPrime Ideal Theorem (Landau 1918)As x→∞,pi(x; k) :=∑p⊂OkNmk/Qp≤ x1 = Li(x) + ”error term”.The error term depends on the zero-free region of the Dedekindzeta-function of k:ζk(s) :=∑I⊂Ok1(Nmk/QI)s=∏p⊂Ok(1− 1(Nmk/Qp)s)−1, <(s) > 1Example 1: When k = Q, we have ζk(s) = ζ(s).Example 2: When k = Q(√q), one can show ζk(s) = ζ(s)L(s, χq).22COUNTING PRIME IDEALS IN NUMBER FIELDSkOkQZPrime Ideal Theorem (Landau 1918)As x→∞,pi(x; k) :=∑p⊂OkNmk/Qp≤ x1 = Li(x) + ”error term”.The error term depends on the zero-free region of the Dedekindzeta-function of k:ζk(s) :=∑I⊂Ok1(Nmk/QI)s=∏p⊂Ok(1− 1(Nmk/Qp)s)−1, <(s) > 1Example 1: When k = Q, we have ζk(s) = ζ(s).Example 2: When k = Q(√q), one can show ζk(s) = ζ(s)L(s, χq).22COUNTING PRIME IDEALS IN NUMBER FIELDSkOkQZPrime Ideal Theorem (Landau 1918)As x→∞,pi(x; k) :=∑p⊂OkNmk/Qp≤ x1 = Li(x) + ”error term”.The error term depends on the zero-free region of the Dedekindzeta-function of k:ζk(s) :=∑I⊂Ok1(Nmk/QI)s=∏p⊂Ok(1− 1(Nmk/Qp)s)−1, <(s) > 1Example 1: When k = Q, we have ζk(s) = ζ(s).Example 2: When k = Q(√q), one can show ζk(s) = ζ(s)L(s, χq).22COUNTING PRIME IDEALS IN NUMBER FIELDSkOkQZPrime Ideal Theorem (Landau 1918)As x→∞,pi(x; k) :=∑p⊂OkNmk/Qp≤ x1 = Li(x) + ”error term”.The error term depends on the zero-free region of the Dedekindzeta-function of k:ζk(s) :=∑I⊂Ok1(Nmk/QI)s=∏p⊂Ok(1− 1(Nmk/Qp)s)−1, <(s) > 1Example 1: When k = Q, we have ζk(s) = ζ(s).Example 2: When k = Q(√q), one can show ζk(s) = ζ(s)L(s, χq).22COUNTING PRIME IDEALS IN NUMBER FIELDSkOkQZPrime Ideal Theorem (Landau 1918)As x→∞,pi(x; k) :=∑p⊂OkNmk/Qp≤ x1 = Li(x) + ”error term”.The error term depends on the zero-free region of the Dedekindzeta-function of k:ζk(s) :=∑I⊂Ok1(Nmk/QI)s=∏p⊂Ok(1− 1(Nmk/Qp)s)−1, <(s) > 1Example 1: When k = Q, we have ζk(s) = ζ(s).Example 2: When k = Q(√q), one can show ζk(s) = ζ(s)L(s, χq).22COUNTING PRIME IDEALS IN NUMBER FIELDSkOkQZPrime Ideal Theorem (Landau 1918)As x→∞,pi(x; k) :=∑p⊂OkNmk/Qp≤ x1 = Li(x) + ”error term”.The error term depends on the zero-free region of the Dedekindzeta-function of k:ζk(s) :=∑I⊂Ok1(Nmk/QI)s=∏p⊂Ok(1− 1(Nmk/Qp)s)−1, <(s) > 1Generalized Riemann Hypothesis: Nontrivial zeros ofζK(s) have real part equal to 1/2.22COUNTING PRIME IDEALS IN CONJUGACY CLASSESLet L/k be a normal extension with Galois group G = Gal(L/k).piC(x,L/k) := #{p ⊂ Ok : p unramified in L,[L/kp]= C,Nmk/Qp ≤ x}LkQGal(L/k) ∼= G• p is a prime ideal in Ok which isunramified in L.•[L/kp]is the Artin symbol, whichdenotes the fixed, targetedconjugacy class C within G.23COUNTING PRIME IDEALS IN CONJUGACY CLASSESLet L/k be a normal extension with Galois group G = Gal(L/k).piC(x,L/k) := #{p ⊂ Ok : p unramified in L,[L/kp]= C,Nmk/Qp ≤ x}LkQGal(L/k) ∼= G• p is a prime ideal in Ok which isunramified in L.•[L/kp]is the Artin symbol, whichdenotes the fixed, targetedconjugacy class C within G.23COUNTING PRIME IDEALS IN CONJUGACY CLASSESLet L/k be a normal extension with Galois group G = Gal(L/k).piC(x,L/k) := #{p ⊂ Ok : p unramified in L,[L/kp]= C,Nmk/Qp ≤ x}LkQGal(L/k) ∼= G• p is a prime ideal in Ok which isunramified in L.•[L/kp]is the Artin symbol, whichdenotes the fixed, targetedconjugacy class C within G.23COUNTING PRIME IDEALS IN CONJUGACY CLASSESLet L/k be a normal extension with Galois group G = Gal(L/k).piC(x,L/k) := #{p ⊂ Ok : p unramified in L,[L/kp]= C,Nmk/Qp ≤ x}LkQGal(L/k) ∼= G• p is a prime ideal in Ok which isunramified in L.•[L/kp]is the Artin symbol, whichdenotes the fixed, targetedconjugacy class C within G.23COUNTING PRIME IDEALS IN CONJUGACY CLASSESLet L/k be a normal extension with Galois group G = Gal(L/k).piC(x,L/k) := #{p ⊂ Ok : p unramified in L,[L/kp]= C,Nmk/Qp ≤ x}LkQGal(L/k) ∼= G• p is a prime ideal in Ok which isunramified in L.•[L/kp]is the Artin symbol, whichdenotes the fixed, targetedconjugacy class C within G.23COUNTING PRIME IDEALS IN CONJUGACY CLASSESChebotarev Density Theorem(1922)piC(x; L/k) ∼ |C||G|Li(x), x→∞LkQGal(L/k) ∼= G24COUNTING PRIME IDEALS IN CONJUGACY CLASSESEffective Chebotarev Density Theorem(Lagarias & Odlyzko 1975)piC(x; L/k) =|C||G|Li(x) + ”error term”, x→∞The error term depends on the zero-free region of the Dedekindzeta-function of L.ζL(s) := ζk(s)∏ρ∈Gˆρ 6=ρ0L(s, ρ,L/k)dim ρ• Each L(s, ρ,L/k) is an Artin L-function.• The product is over the nontrivialirreducible representations of G.LkQGal(L/k) ∼= G25COUNTING PRIME IDEALS IN CONJUGACY CLASSESEffective Chebotarev Density Theorem(Lagarias & Odlyzko 1975)piC(x; L/k) =|C||G|Li(x) + ”error term”, x→∞The error term depends on the zero-free region of the Dedekindzeta-function of L.ζL(s) := ζk(s)∏ρ∈Gˆρ 6=ρ0L(s, ρ,L/k)dim ρ• Each L(s, ρ,L/k) is an Artin L-function.• The product is over the nontrivialirreducible representations of G.LkQGal(L/k) ∼= G25COUNTING PRIME IDEALS IN CONJUGACY CLASSESEffective Chebotarev Density Theorem(Lagarias & Odlyzko 1975)piC(x; L/k) =|C||G|Li(x) + ”error term”, x→∞The error term depends on the zero-free region of the Dedekindzeta-function of L.ζL(s) := ζk(s)∏ρ∈Gˆρ 6=ρ0L(s, ρ,L/k)dim ρ• Each L(s, ρ,L/k) is an Artin L-function.• The product is over the nontrivialirreducible representations of G.LkQGal(L/k) ∼= G25COUNTING PRIME IDEALS IN CONJUGACY CLASSESEffective Chebotarev Density Theorem(Lagarias & Odlyzko 1975)piC(x; L/k) =|C||G|Li(x) + ”error term”, x→∞The error term depends on the zero-free region of the Dedekindzeta-function of L.ζL(s) := ζk(s)∏ρ∈Gˆρ 6=ρ0L(s, ρ,L/k)dim ρ• Each L(s, ρ,L/k) is an Artin L-function.• The product is over the nontrivialirreducible representations of G.LkQGal(L/k) ∼= G25COUNTING PRIME IDEALS IN CONJUGACY CLASSESEffective Chebotarev Density Theorem(Lagarias & Odlyzko 1975)piC(x; L/k) =|C||G|Li(x) + ”error term”, x→∞The error term depends on the zero-free region of the Dedekindzeta-function of L.ζL(s) := ζk(s)∏ρ∈Gˆρ 6=ρ0L(s, ρ,L/k)dim ρ• Each L(s, ρ,L/k) is an Artin L-function.• The product is over the nontrivialirreducible representations of G.LkQGal(L/k) ∼= G25EXAMPLE OF A DEDEKIND ZETA-FUNCTION ζL(s)Let k = Q and G = Gal(L/Q) ∼= S3.S3 has the following Galois representations:• ρ0 – trivial representation, 1-dimensional• ρ1 – sign representation, 1-dimensional• ρ2 – standard representation, 2-dimensionalζL(s) =ζ(s) L(s, ρ1) L(s, ρ2)226EXAMPLE OF A DEDEKIND ZETA-FUNCTION ζL(s)Let k = Q and G = Gal(L/Q) ∼= S3.S3 has the following Galois representations:• ρ0 – trivial representation, 1-dimensional• ρ1 – sign representation, 1-dimensional• ρ2 – standard representation, 2-dimensionalζL(s) =ζ(s) L(s, ρ1) L(s, ρ2)226EXAMPLE OF A DEDEKIND ZETA-FUNCTION ζL(s)Let k = Q and G = Gal(L/Q) ∼= S3.S3 has the following Galois representations:• ρ0 – trivial representation, 1-dimensional• ρ1 – sign representation, 1-dimensional• ρ2 – standard representation, 2-dimensionalζL(s) = ζ(s)L(s, ρ1) L(s, ρ2)226EXAMPLE OF A DEDEKIND ZETA-FUNCTION ζL(s)Let k = Q and G = Gal(L/Q) ∼= S3.S3 has the following Galois representations:• ρ0 – trivial representation, 1-dimensional• ρ1 – sign representation, 1-dimensional• ρ2 – standard representation, 2-dimensionalζL(s) = ζ(s) L(s, ρ1)L(s, ρ2)226EXAMPLE OF A DEDEKIND ZETA-FUNCTION ζL(s)Let k = Q and G = Gal(L/Q) ∼= S3.S3 has the following Galois representations:• ρ0 – trivial representation, 1-dimensional• ρ1 – sign representation, 1-dimensional• ρ2 – standard representation, 2-dimensionalζL(s) = ζ(s) L(s, ρ1) L(s, ρ2)226AN EFFECTIVE CHEBOTAREV DENSITY THEOREMLet L/k be a normal extension with Galois group G = Gal(L/k),DL = |Disc L/Q|, and nL = [L : Q].Theorem (Lagarias-Odlyzko, 1975)For any fixed conjugacy class C ⊂ G,∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ |C||G|Li(xβ0) + c1x exp(−c2n1/2L (log x)1/2)︸︷︷︸Error term depends on zero-free region of ζL(s).for x ≥ exp(10nL(log DL)2), where• β0 is a real, simple exceptional zero of ζL(s);• c1, c2 are effectively computable constants.26AN EFFECTIVE CHEBOTAREV DENSITY THEOREMLet L/k be a normal extension with Galois group G = Gal(L/k),DL = |Disc L/Q|, and nL = [L : Q].Theorem (Lagarias-Odlyzko, 1975)For any fixed conjugacy class C ⊂ G,∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ |C||G|Li(xβ0) + c1x exp(−c2n1/2L (log x)1/2)︸︷︷︸Error term depends on zero-free region of ζL(s).for x ≥ exp(10nL(log DL)2), where• β0 is a real, simple exceptional zero of ζL(s);• c1, c2 are effectively computable constants.25A Conditional EFFECTIVE CHEBOTAREV DENSITY THEOREMLet L/k be a normal extension with Galois group G = Gal(L/k),DL = |Disc L/Q|, and nL = [L : Q].Theorem (Lagarias-Odlyzko, 1975)If the generalized Riemann hypothesis holds for the Dedekindzeta-function ζL(s), then for any fixed conjugacy class C ⊂ G∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ C0 |C||G|x1/2 log(DLxnL)︸︷︷︸Error term relies on GRH for ζL(s).for every x ≥ 2, where• C0 is an effectively computable constant.25A Conditional EFFECTIVE CHEBOTAREV DENSITY THEOREMLet L/k be a normal extension with Galois group G = Gal(L/k),DL = |Disc L/Q|, and nL = [L : Q].Theorem (Lagarias-Odlyzko, 1975)If the generalized Riemann hypothesis holds for the Dedekindzeta-function ζL(s), then for any fixed conjugacy class C ⊂ G∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ C0 |C||G|x1/2 log(DLxnL)︸︷︷︸Error term relies on GRH for ζL(s).for every x ≥ 2, where• C0 is an effectively computable constant.24COMPARING THE THEOREMS (LAGARIAS-ODLYZKO, 1975)Theorem (Unconditional)For any fixed conjugacy class C ⊂ G,∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ |C||G|Li(xβ0)+c1x exp(−c2n1/2L (log x)1/2)for x ≥ exp(10nL(log DL)2.Theorem (Conditional)If GRH holds for ζL(s), then for any fixed conjugacy class C ⊂ G∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ C0 |C||G|x1/2 log(DLxnL).for every x ≥ 2.Question: What do a lower threshold and no β0 term get you?25COMPARING THE THEOREMS (LAGARIAS-ODLYZKO, 1975)Theorem (Unconditional)For any fixed conjugacy class C ⊂ G,∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ |C||G|Li(xβ0)+c1x exp(−c2n1/2L (log x)1/2)for x ≥ exp(10nL(log DL)2.Theorem (Conditional)If GRH holds for ζL(s), then for any fixed conjugacy class C ⊂ G∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ C0 |C||G|x1/2 log(DLxnL).for every x ≥ 2.Question: What do a lower threshold and no β0 term get you?25COMPARING THE THEOREMS (LAGARIAS-ODLYZKO, 1975)Theorem (Unconditional)For any fixed conjugacy class C ⊂ G,∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ |C||G|Li(xβ0)+c1x exp(−c2n1/2L (log x)1/2)for x ≥ exp(10nL(log DL)2.Theorem (Conditional)If GRH holds for ζL(s), then for any fixed conjugacy class C ⊂ G∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ C0 |C||G|x1/2 log(DLxnL).for every x ≥ 2.Question: What do a lower threshold and no β0 term get you?26COMPARING THE THEOREMS (LAGARIAS-ODLYZKO, 1975)Theorem (Unconditional)For any fixed conjugacy class C ⊂ G,∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ |C||G|Li(xβ0)+c1x exp(−c2n1/2L (log x)1/2)for x ≥ exp(10nL(log DL)2.Theorem (Conditional)If GRH holds for ζL(s), then for any fixed conjugacy class C ⊂ G∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ C0 |C||G|x1/2 log(DLxnL).for every x ≥ 2.Question: What do a lower threshold and no β0 term get you?26COMPARING THE THEOREMS (LAGARIAS-ODLYZKO, 1975)Theorem (Unconditional)For any fixed conjugacy class C ⊂ G,∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ |C||G|Li(xβ0)+c1x exp(−c2n1/2L (log x)1/2)for x ≥ exp(10nL(log DL)2 ≥ D10nLL .Theorem (Conditional)If GRH holds for ζL(s), then for any fixed conjugacy class C ⊂ G∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ C0 |C||G|x1/2 log(DLxnL).for every x ≥ 2.Want: An unconditional effective CDT with a low threshold onx, no β0 term, and an acceptable error term.27COMPARING THE THEOREMS (LAGARIAS-ODLYZKO, 1975)Theorem (Unconditional)For any fixed conjugacy class C ⊂ G,∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ |C||G|Li(xβ0)+c1x exp(−c2n1/2L (log x)1/2)for x ≥ exp(10nL(log DL)2 ≥ D10nLL .Theorem (Conditional)If GRH holds for ζL(s), then for any fixed conjugacy class C ⊂ G∣∣∣∣piC(x,L/k)− |C||G|Li(x)∣∣∣∣ ≤ C0 |C||G|x1/2 log(DLxnL).for every x ≥ 2.Want: An unconditional effective CDT with a low threshold onx, no β0 term, and an acceptable error term. 27Skeleton of Theorem (Pierce, T., Wood)Let F(X) be a family of fields, where K ∈ F(X) have• fixed degree n over Q• fixed Galois Group G = Gal(K˜/Q)• DK ≤ X• a possible ramification restriction on tamely ramified primes;Suppose it is known that |F(X)| Xa for some a > 0. Then for atmost O(Xb) exceptions, with b < a, for fixed A ≥ 2, we have∣∣∣∣piC(x, K˜/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)Awhere x ≥ κ1 exp{κ2(log log Dκ3K˜ )2}, and the κi depend onn, |G|,DK˜, a, b, and A.• No β0 term. Can take x = DηK˜ for η small.We prove most Dedekind zeta-functions in the family satisfy acertain zero-free region.28Skeleton of Theorem (Pierce, T., Wood)Let F(X) be a family of fields, where K ∈ F(X) have• fixed degree n over Q• fixed Galois Group G = Gal(K˜/Q)• DK ≤ X• a possible ramification restriction on tamely ramified primes;Suppose it is known that |F(X)| Xa for some a > 0. Then for atmost O(Xb) exceptions, with b < a, for fixed A ≥ 2, we have∣∣∣∣piC(x, K˜/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)Awhere x ≥ κ1 exp{κ2(log log Dκ3K˜ )2}, and the κi depend onn, |G|,DK˜, a, b, and A.• No β0 term. Can take x = DηK˜ for η small.We prove most Dedekind zeta-functions in the family satisfy acertain zero-free region.28Skeleton of Theorem (Pierce, T., Wood)Let F(X) be a family of fields, where K ∈ F(X) have• fixed degree n over Q• fixed Galois Group G = Gal(K˜/Q)• DK ≤ X• a possible ramification restriction on tamely ramified primes;Suppose it is known that |F(X)| Xa for some a > 0. Then for atmost O(Xb) exceptions, with b < a, for fixed A ≥ 2, we have∣∣∣∣piC(x, K˜/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)Awhere x ≥ κ1 exp{κ2(log log Dκ3K˜ )2}, and the κi depend onn, |G|,DK˜, a, b, and A.• No β0 term. Can take x = DηK˜ for η small.We prove most Dedekind zeta-functions in the family satisfy acertain zero-free region.28Skeleton of Theorem (Pierce, T., Wood)Let F(X) be a family of fields, where K ∈ F(X) have• fixed degree n over Q• fixed Galois Group G = Gal(K˜/Q)• DK ≤ X• a possible ramification restriction on tamely ramified primes;Suppose it is known that |F(X)| Xa for some a > 0. Then for atmost O(Xb) exceptions, with b < a, for fixed A ≥ 2, we have∣∣∣∣piC(x, K˜/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)Awhere x ≥ κ1 exp{κ2(log log Dκ3K˜ )2}, and the κi depend onn, |G|,DK˜, a, b, and A.• No β0 term. Can take x = DηK˜ for η small.We prove most Dedekind zeta-functions in the family satisfy acertain zero-free region.28Skeleton of Theorem (Pierce, T., Wood)Let F(X) be a family of fields, where K ∈ F(X) have• fixed degree n over Q• fixed Galois Group G = Gal(K˜/Q)• DK ≤ X• a possible ramification restriction on tamely ramified primes;Suppose it is known that |F(X)| Xa for some a > 0. Then for atmost O(Xb) exceptions, with b < a, for fixed A ≥ 2, we have∣∣∣∣piC(x, K˜/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)Awhere x ≥ κ1 exp{κ2(log log Dκ3K˜ )2}, and the κi depend onn, |G|,DK˜, a, b, and A.• No β0 term. Can take x = DηK˜ for η small.We prove most Dedekind zeta-functions in the family satisfy acertain zero-free region.28Skeleton of Theorem (Pierce, T., Wood)Let F(X) be a family of fields, where K ∈ F(X) have• fixed degree n over Q• fixed Galois Group G = Gal(K˜/Q)• DK ≤ X• a possible ramification restriction on tamely ramified primes;Suppose it is known that |F(X)| Xa for some a > 0.Then for atmost O(Xb) exceptions, with b < a, for fixed A ≥ 2, we have∣∣∣∣piC(x, K˜/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)Awhere x ≥ κ1 exp{κ2(log log Dκ3K˜ )2}, and the κi depend onn, |G|,DK˜, a, b, and A.• No β0 term. Can take x = DηK˜ for η small.We prove most Dedekind zeta-functions in the family satisfy acertain zero-free region.28Skeleton of Theorem (Pierce, T., Wood)Let F(X) be a family of fields, where K ∈ F(X) have• fixed degree n over Q• fixed Galois Group G = Gal(K˜/Q)• DK ≤ X• a possible ramification restriction on tamely ramified primes;Suppose it is known that |F(X)| Xa for some a > 0. Then for atmost O(Xb) exceptions, with b < a, for fixed A ≥ 2, we have∣∣∣∣piC(x, K˜/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)Awhere x ≥ κ1 exp{κ2(log log Dκ3K˜ )2}, and the κi depend onn, |G|,DK˜, a, b, and A.• No β0 term. Can take x = DηK˜ for η small.We prove most Dedekind zeta-functions in the family satisfy acertain zero-free region.28Skeleton of Theorem (Pierce, T., Wood)Let F(X) be a family of fields, where K ∈ F(X) have• fixed degree n over Q• fixed Galois Group G = Gal(K˜/Q)• DK ≤ X• a possible ramification restriction on tamely ramified primes;Suppose it is known that |F(X)| Xa for some a > 0. Then for atmost O(Xb) exceptions, with b < a, for fixed A ≥ 2, we have∣∣∣∣piC(x, K˜/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)Awhere x ≥ κ1 exp{κ2(log log Dκ3K˜ )2}, and the κi depend onn, |G|,DK˜, a, b, and A.• No β0 term. Can take x = DηK˜ for η small.We prove most Dedekind zeta-functions in the family satisfy acertain zero-free region.28Skeleton of Theorem (Pierce, T., Wood)Let F(X) be a family of fields, where K ∈ F(X) have• fixed degree n over Q• fixed Galois Group G = Gal(K˜/Q)• DK ≤ X• a possible ramification restriction on tamely ramified primes;Suppose it is known that |F(X)| Xa for some a > 0. Then for atmost O(Xb) exceptions, with b < a, for fixed A ≥ 2, we have∣∣∣∣piC(x, K˜/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)Awhere x ≥ κ1 exp{κ2(log log Dκ3K˜ )2}, and the κi depend onn, |G|,DK˜, a, b, and A.• No β0 term.Can take x = DηK˜for η small.We prove most Dedekind zeta-functions in the family satisfy acertain zero-free region.28Skeleton of Theorem (Pierce, T., Wood)Let F(X) be a family of fields, where K ∈ F(X) have• fixed degree n over Q• fixed Galois Group G = Gal(K˜/Q)• DK ≤ X• a possible ramification restriction on tamely ramified primes;Suppose it is known that |F(X)| Xa for some a > 0. Then for atmost O(Xb) exceptions, with b < a, for fixed A ≥ 2, we have∣∣∣∣piC(x, K˜/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)Awhere x ≥ κ1 exp{κ2(log log Dκ3K˜ )2}, and the κi depend onn, |G|,DK˜, a, b, and A.• No β0 term. Can take x = DηK˜ for η small.We prove most Dedekind zeta-functions in the family satisfy acertain zero-free region.28Skeleton of Theorem (Pierce, T., Wood)Let F(X) be a family of fields, where K ∈ F(X) have• fixed degree n over Q• fixed Galois Group G = Gal(K˜/Q)• DK ≤ X• a possible ramification restriction on tamely ramified primes;Suppose it is known that |F(X)| Xa for some a > 0. Then for atmost O(Xb) exceptions, with b < a, for fixed A ≥ 2, we have∣∣∣∣piC(x, K˜/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)Awhere x ≥ κ1 exp{κ2(log log Dκ3K˜ )2}, and the κi depend onn, |G|,DK˜, a, b, and A.• No β0 term. Can take x = DηK˜ for η small.We prove most Dedekind zeta-functions in the family satisfy acertain zero-free region.28APPLICATION TO BOUNDING `-TORSIONSkeleton of Corollary (Pierce, T., Wood)Let F(X) be a family of fields for which the previous ChebotarevDensity Theorem holds. For the nonexceptional fields K ∈ F(X), wehave|ClK[`]| n,`,ε D12− 12`(n−1)+εK .Question:To which families does our Chebotarev Density Theoremapply?29APPLICATION TO BOUNDING `-TORSIONSkeleton of Corollary (Pierce, T., Wood)Let F(X) be a family of fields for which the previous ChebotarevDensity Theorem holds. For the nonexceptional fields K ∈ F(X), wehave|ClK[`]| n,`,ε D12− 12`(n−1)+εK .Question:To which families does our Chebotarev Density Theoremapply?29[K : Q] Gal(K˜/Q) restriction on size of size oftamely ramified primes exceptional family total familyn ≥ 2 Cn totally ramified Xε, ε > 0 ∼ cX1/(n−1)3 S3 transposition X1/3 ∼ cXEllenberg-Venkatesh Bhargava4 S4 transposition X1/2+ε, ε > 0 ∼ cXKlu¨ners Bhargava4 A4 K4 subgroup X0.27 X1/2p ≥ 5 Dp reflection X1/(p−1) X2/(p−1)order 2p30[K : Q] Gal(K˜/Q) restriction on size of size oftamely ramified primes exceptional family total familyn ≥ 2 Cn totally ramified Xε, ε > 0 ∼ cX1/(n−1)3 S3 transposition X1/3 ∼ cXEllenberg-Venkatesh Bhargava4 S4 transposition X1/2+ε, ε > 0 ∼ cXKlu¨ners Bhargava4 A4 K4 subgroup X0.27 X1/2p ≥ 5 Dp reflection X1/(p−1) X2/(p−1)order 2p30[K : Q] Gal(K˜/Q) restriction on size of size oftamely ramified primes exceptional family total familyn ≥ 2 Cn totally ramified Xε, ε > 0 ∼ cX1/(n−1)3 S3 transposition X1/3 ∼ cXEllenberg-Venkatesh Bhargava4 S4 transposition X1/2+ε, ε > 0 ∼ cXKlu¨ners Bhargava4 A4 K4 subgroup X0.27 X1/2p ≥ 5 Dp reflection X1/(p−1) X2/(p−1)order 2p30[K : Q] Gal(K˜/Q) restriction on size of size oftamely ramified primes exceptional family total familyn ≥ 2 Cn totally ramified Xε, ε > 0 ∼ cX1/(n−1)3 S3 transposition X1/3 ∼ cXEllenberg-Venkatesh Bhargava4 S4 transposition X1/2+ε, ε > 0 ∼ cXKlu¨ners Bhargava4 A4 K4 subgroup X0.27 X1/2p ≥ 5 Dp reflection X1/(p−1) X2/(p−1)order 2p30[K : Q] Gal(K˜/Q) restriction on size of size oftamely ramified primes exceptional family total familyn ≥ 2 Cn totally ramified Xε, ε > 0 ∼ cX1/(n−1)3 S3 transposition X1/3 ∼ cXEllenberg-Venkatesh Bhargava4 S4 transposition X1/2+ε, ε > 0 ∼ cXKlu¨ners Bhargava4 A4 K4 subgroup X0.27 X1/2p ≥ 5 Dp reflection X1/(p−1) X2/(p−1)order 2p30[K : Q] Gal(K˜/Q) restriction on size of size oftamely ramified primes exceptional family total familyn ≥ 2 Cn totally ramified Xε, ε > 0 ∼ cX1/(n−1)3 S3 transposition X1/3 ∼ cXEllenberg-Venkatesh Bhargava4 S4 transposition X1/2+ε, ε > 0 ∼ cXKlu¨ners Bhargava4 A4 K4 subgroup X0.27 X1/2p ≥ 5 Dp reflection X1/(p−1) X2/(p−1)order 2p30CONDITIONAL ON THE STRONG ARTIN CONJECTURE[K : Q] Gal(K˜/Q) restriction on size of size oftamely ramified primes exceptional family total family5 S5 transposition X199/200 XBhargavan ≥ 6 Sn transposition X∆ if there X1/2+1/nexists D∆ degree Bhargava, Shankurn fields such that WangDK = D.n ≥ 5 An none Xε, ε > 0 Xβn−εβn =1−2/n!4n−431CONDITIONAL ON THE STRONG ARTIN CONJECTURE[K : Q] Gal(K˜/Q) restriction on size of size oftamely ramified primes exceptional family total family5 S5 transposition X199/200 XBhargavan ≥ 6 Sn transposition X∆ if there X1/2+1/nexists D∆ degree Bhargava, Shankurn fields such that WangDK = D.n ≥ 5 An none Xε, ε > 0 Xβn−εβn =1−2/n!4n−431CONDITIONAL ON THE STRONG ARTIN CONJECTURE[K : Q] Gal(K˜/Q) restriction on size of size oftamely ramified primes exceptional family total family5 S5 transposition X199/200 XBhargavan ≥ 6 Sn transposition X∆ if there X1/2+1/nexists D∆ degree Bhargava, Shankurn fields such that WangDK = D.n ≥ 5 An none Xε, ε > 0 Xβn−εβn =1−2/n!4n−431CONDITIONAL ON THE STRONG ARTIN CONJECTURE[K : Q] Gal(K˜/Q) restriction on size of size oftamely ramified primes exceptional family total family5 S5 transposition X199/200 XBhargavan ≥ 6 Sn transposition X∆ if there X1/2+1/nexists D∆ degree Bhargava, Shankurn fields such that WangDK = D.n ≥ 5 An none Xε, ε > 0 Xβn−εβn =1−2/n!4n−431OVERVIEW OF ARGUMENTEllenberg-Venkatesh|ClK[`]| `,n,ε D12+εK M−1Effective Chebotarev Density Theoremassuming non-GRH zero-free regionShow assumed zero-free region is obeyedby ”most” number fields in an appropriate familyControl the propagation of ”bad” fields within the familyWithout assuming GRH, conclude|ClK[`]| `,n,ε D12− 12`(n−1)+εK for non-exceptional K.32OVERVIEW OF ARGUMENTEllenberg-Venkatesh|ClK[`]| `,n,ε D12+εK M−1Effective Chebotarev Density Theoremassuming non-GRH zero-free regionShow assumed zero-free region is obeyedby ”most” number fields in an appropriate familyControl the propagation of ”bad” fields within the familyWithout assuming GRH, conclude|ClK[`]| `,n,ε D12− 12`(n−1)+εK for non-exceptional K.32OVERVIEW OF ARGUMENTEllenberg-Venkatesh|ClK[`]| `,n,ε D12+εK M−1Effective Chebotarev Density Theoremassuming non-GRH zero-free regionShow assumed zero-free region is obeyedby ”most” number fields in an appropriate familyControl the propagation of ”bad” fields within the familyWithout assuming GRH, conclude|ClK[`]| `,n,ε D12− 12`(n−1)+εK for non-exceptional K.32OVERVIEW OF ARGUMENTEllenberg-Venkatesh|ClK[`]| `,n,ε D12+εK M−1Effective Chebotarev Density Theoremassuming non-GRH zero-free regionShow assumed zero-free region is obeyedby ”most” number fields in an appropriate familyControl the propagation of ”bad” fields within the familyWithout assuming GRH, conclude|ClK[`]| `,n,ε D12− 12`(n−1)+εK for non-exceptional K.32OVERVIEW OF ARGUMENTEllenberg-Venkatesh|ClK[`]| `,n,ε D12+εK M−1Effective Chebotarev Density Theoremassuming non-GRH zero-free regionShow assumed zero-free region is obeyedby ”most” number fields in an appropriate familyControl the propagation of ”bad” fields within the familyWithout assuming GRH, conclude|ClK[`]| `,n,ε D12− 12`(n−1)+εK for non-exceptional K.32OVERVIEW OF ARGUMENTEllenberg-Venkatesh|ClK[`]| `,n,ε D12+εK M−1Effective Chebotarev Density Theoremassuming non-GRH zero-free regionShow assumed zero-free region is obeyedby ”most” number fields in an appropriate familyControl the propagation of ”bad” fields within the familyWithout assuming GRH, conclude|ClK[`]| `,n,ε D12− 12`(n−1)+εK for non-exceptional K.32OVERVIEW OF ARGUMENTEllenberg-Venkatesh|ClK[`]| `,n,ε D12+εK M−1Effective Chebotarev Density Theoremassuming non-GRH zero-free regionShow assumed zero-free region is obeyedby ”most” number fields in an appropriate familyControl the propagation of ”bad” fields within the familyWithout assuming GRH, conclude|ClK[`]| `,n,ε D12− 12`(n−1)+εK for non-exceptional K.32THE ZERO-FREE REGIONL = K˜KQGal(K˜/Q) ∼= GnζK˜(s) = ζ(s)∏ρ∈Gˆρ 6=ρ0 irreducibleL(s, ρ, K˜/Q)dim ρKnown zero-free region for ζ(s):σ > 1− clog2/3(|t|+ 2) log log1/3(|t|+ 3) .Assumed zero-free region for ζK˜(s)/ζ(s):[1− δ, 1]× [−(log DK˜)2/δ, (log DK˜)2/δ].33THE ZERO-FREE REGIONL = K˜KQGal(K˜/Q) ∼= GnζK˜(s) = ζ(s)∏ρ∈Gˆρ 6=ρ0 irreducibleL(s, ρ, K˜/Q)dim ρKnown zero-free region for ζ(s):σ > 1− clog2/3(|t|+ 2) log log1/3(|t|+ 3) .Assumed zero-free region for ζK˜(s)/ζ(s):[1− δ, 1]× [−(log DK˜)2/δ, (log DK˜)2/δ].33THE ZERO-FREE REGIONL = K˜KQGal(K˜/Q) ∼= GnζK˜(s) = ζ(s)∏ρ∈Gˆρ 6=ρ0 irreducibleL(s, ρ, K˜/Q)dim ρKnown zero-free region for ζ(s):σ > 1− clog2/3(|t|+ 2) log log1/3(|t|+ 3) .Assumed zero-free region for ζK˜(s)/ζ(s):[1− δ, 1]× [−(log DK˜)2/δ, (log DK˜)2/δ].33ζK˜(s) = ζ(s)∏ρ∈Gˆρ6=ρ0 irreducibleL(s, ρ, K˜/Q)dim ρPROVING THE CHEBOTAREV DENSITY THEOREMIdea of the proof• We return to the method of Lagarias-Odlyzko.• We insert our assumed zero-free region for ζL(s)/ζ(s) at akey point.• We work delicately to provide both an acceptable effectiveerror term, and a sufficiently small threshold for xdepending on DL.34PROVING THE CHEBOTAREV DENSITY THEOREMIdea of the proof• We return to the method of Lagarias-Odlyzko.• We insert our assumed zero-free region for ζL(s)/ζ(s) at akey point.• We work delicately to provide both an acceptable effectiveerror term, and a sufficiently small threshold for xdepending on DL.34PROVING THE CHEBOTAREV DENSITY THEOREMIdea of the proof• We return to the method of Lagarias-Odlyzko.• We insert our assumed zero-free region for ζL(s)/ζ(s) at akey point.• We work delicately to provide both an acceptable effectiveerror term, and a sufficiently small threshold for xdepending on DL.34Theorem (Pierce, T., Wood)Let 0 < δ ≤ 1/4 be a fixed positive constant. For any normalextension of number fields L/Q with [L : Q] = nL such that DL issufficiently large and ζL(s) obeys the assumed zero-free region, wehave that for any A ≥ 2 and any conjugacy class C ⊂ G = Gal(L/Q)∣∣∣∣piC(x,L/Q)− |C||G|Li(x)∣∣∣∣ ≤ |C||G| x(log x)A︸︷︷︸error term dependson assumed zero-free regionfor allx ≥ c1 exp{c2(log log(Dc3L )3/2 log log log(Dc4L ))1/2}︸︷︷︸≥(logDL)small power,where all the constants can be written explicitly.35BOUNDING `-TORSION WITHOUT ASSUMING GRHEllenberg-Venkatesh|ClK[`]| `,n,ε D12+εK M−1Prove an effective Chebotarev Density Theoremassuming non-GRH zero-free regionShow that within an appropriate family offields K, most ζK˜(s) obey the zero-free regionControl the propagation of ”bad” fields within the familyWithout assuming GRH, conclude|ClK[`]| `,n,ε D12− 12`(n−1)+εK for non-exceptional K.35BOUNDING `-TORSION WITHOUT ASSUMING GRHEllenberg-Venkatesh|ClK[`]| `,n,ε D12+εK M−1Prove an effective Chebotarev Density Theoremassuming non-GRH zero-free regionShow that within an appropriate family offields K, most ζK˜(s) obey the zero-free regionControl the propagation of ”bad” fields within the familyWithout assuming GRH, conclude|ClK[`]| `,n,ε D12− 12`(n−1)+εK for non-exceptional K.35KEY TOOL - ZEROS OF AUTOMORPHIC L-FUNCTIONSLet pi be a cuspidal automorphic representation on GLm(Q).Consider the corresponding automorphic L-function L(s, pi).Let s = β + iγ denote a zero of L(s, pi).DefineN(pi;α,T) := # of zeros of L(s, pi) such that β > α and |γ| ≤ T.Kowalski and Michel have given a bound for N(pi;α,T) thatholds on average for an appropriately defined family ofcuspidal automorphic representations.36KEY TOOL - ZEROS OF AUTOMORPHIC L-FUNCTIONSLet pi be a cuspidal automorphic representation on GLm(Q).Consider the corresponding automorphic L-function L(s, pi).Let s = β + iγ denote a zero of L(s, pi).DefineN(pi;α,T) := # of zeros of L(s, pi) such that β > α and |γ| ≤ T.Kowalski and Michel have given a bound for N(pi;α,T) thatholds on average for an appropriately defined family ofcuspidal automorphic representations.36KEY TOOL - ZEROS OF AUTOMORPHIC L-FUNCTIONSLet pi be a cuspidal automorphic representation on GLm(Q).Consider the corresponding automorphic L-function L(s, pi).Let s = β + iγ denote a zero of L(s, pi).DefineN(pi;α,T) := # of zeros of L(s, pi) such that β > α and |γ| ≤ T.Kowalski and Michel have given a bound for N(pi;α,T) thatholds on average for an appropriately defined family ofcuspidal automorphic representations.36KEY TOOL - ZEROS OF AUTOMORPHIC L-FUNCTIONSLet pi be a cuspidal automorphic representation on GLm(Q).Consider the corresponding automorphic L-function L(s, pi).Let s = β + iγ denote a zero of L(s, pi).DefineN(pi;α,T) := # of zeros of L(s, pi) such that β > α and |γ| ≤ T.Kowalski and Michel have given a bound for N(pi;α,T) thatholds on average for an appropriately defined family ofcuspidal automorphic representations.36Theorem (Kowalski & Michel, 2002)Let S(q), q ≥ 1 be a family of cuspidal automorphic representationssatisfying a prescribed set of conditions. Let α ≥ 3/4 and T ≥ 2.Then there exists c0 > 0, depending on the family, such that∑pi∈S(q)N(pi;α,T) TBqc0 1−α2α−1for all q ≥ 1 and some B ≥ 0 that depends on the family. The impliedconstant only depends on the choice of c0.Applied to L(s, pi) for pi ∈ S(q) =⇒ a zero-free region of thedesired shape that holdsfor all but a possiblezero-density sub-familyof L-functions37Theorem (Kowalski & Michel, 2002)Let S(q), q ≥ 1 be a family of cuspidal automorphic representationssatisfying a prescribed set of conditions. Let α ≥ 3/4 and T ≥ 2.Then there exists c0 > 0, depending on the family, such that∑pi∈S(q)N(pi;α,T) TBqc0 1−α2α−1for all q ≥ 1 and some B ≥ 0 that depends on the family. The impliedconstant only depends on the choice of c0.Applied to L(s, pi) for pi ∈ S(q) =⇒ a zero-free region of thedesired shape that holdsfor all but a possiblezero-density sub-familyof L-functions37We wish to apply Kowalski-Michel toζK˜(s)ζ(s)as K variesover F(X).A couple of issues:1. We are working with Artin L-functions, which in generalare not known to be automorphic.2. Kowalski & Michel’s result applies to family of cuspidalautomorphic representations. We would like to apply it toa family of isobaric automorphic representations.38We wish to apply Kowalski-Michel toζK˜(s)ζ(s)as K variesover F(X).A couple of issues:1. We are working with Artin L-functions, which in generalare not known to be automorphic.2. Kowalski & Michel’s result applies to family of cuspidalautomorphic representations. We would like to apply it toa family of isobaric automorphic representations.38We wish to apply Kowalski-Michel toζK˜(s)ζ(s)as K variesover F(X).A couple of issues:1. We are working with Artin L-functions, which in generalare not known to be automorphic.2. Kowalski & Michel’s result applies to family of cuspidalautomorphic representations. We would like to apply it toa family of isobaric automorphic representations.38ζK˜(s)ζ(s)=∏ρ∈Gˆρ 6=ρ0 irreducibleL(s, ρ, K˜/Q)dj , dj = deg(ρj).Issue # 1 – We are working with Artin L-functions, whichin general are not known to be automorphic.Assuming the strong Artin conjecture, we have that eachL(s, ρ, K˜/Q) is automorphic, i.e. we can writeL(s, ρ, K˜/Q) = L(s, pi)for each L(s, ρ, K˜/Q) in our product.39ζK˜(s)ζ(s)=∏ρ∈Gˆρ 6=ρ0 irreducibleL(s, ρ, K˜/Q)dj , dj = deg(ρj).Issue # 1 – We are working with Artin L-functions, whichin general are not known to be automorphic.Assuming the strong Artin conjecture, we have that eachL(s, ρ, K˜/Q) is automorphic, i.e. we can writeL(s, ρ, K˜/Q) = L(s, pi)for each L(s, ρ, K˜/Q) in our product.39ζK˜(s)ζ(s)=∏ρ∈Gˆρ 6=ρ0 irreducibleL(s, ρ, K˜/Q)dj , dj = deg(ρj).Issue # 2 – Kowalski & Michel applies to families of cus-pidal automorphic representations, but we are workingwith families of isobaric automorphic representations.• We decompose each Dedekind zeta function into a productof cuspidal automorphic L-functions.• We apply the Kowalski-Michel result to the sub-familygenerated by each factor.40ζK˜(s)ζ(s)=∏ρ∈Gˆρ 6=ρ0 irreducibleL(s, ρ, K˜/Q)dj , dj = deg(ρj).Issue # 2 – Kowalski & Michel applies to families of cus-pidal automorphic representations, but we are workingwith families of isobaric automorphic representations.• We decompose each Dedekind zeta function into a productof cuspidal automorphic L-functions.• We apply the Kowalski-Michel result to the sub-familygenerated by each factor.40ζK˜(s)ζ(s)=∏ρ∈Gˆρ 6=ρ0 irreducibleL(s, ρ, K˜/Q)dj , dj = deg(ρj).Issue # 2 – Kowalski & Michel applies to families of cus-pidal automorphic representations, but we are workingwith families of isobaric automorphic representations.• We decompose each Dedekind zeta function into a productof cuspidal automorphic L-functions.• We apply the Kowalski-Michel result to the sub-familygenerated by each factor.40A NEW OBSTACLE:In generalizing Kowalski-Michel, we uncover a technicalbarrier:– a priori, each sub-family could lead to many bad fields forwhich our Chebotarev Density Theorem does not apply.Must define our families of fields to avoid this situation –where potential ”bad” elements in each sub-family prop-agate to create a “large” family of “bad” Dedekind zeta-functions ζK˜(s).41A NEW OBSTACLE:In generalizing Kowalski-Michel, we uncover a technicalbarrier:– a priori, each sub-family could lead to many bad fields forwhich our Chebotarev Density Theorem does not apply.Must define our families of fields to avoid this situation –where potential ”bad” elements in each sub-family prop-agate to create a “large” family of “bad” Dedekind zeta-functions ζK˜(s).41BOUNDING `-TORSION WITHOUT ASSUMING GRHEllenberg-Venkatesh|ClK[`]| `,n,ε D12+εK M−1Prove an effective Chebotarev Density Theoremassuming non-GRH zero-free regionShow that within an appropriate family offields K, most ζK˜(s) obey the zero-free regionControl the propagation of ”bad” fields within the familyWithout assuming GRH, conclude|ClK[`]| `,n,ε D12− 12`(n−1)+εK for non-exceptional K.41CONTROLLING PROPAGATION OF BAD FIELDSSketch of new idea• We transform the problem to counting how often K˜1 andK˜2 both contain a particular subfield F. This relies on workof Klu¨ners and Nicolae (2016).• To handle this counting problem, we make ramificationtype restrictions and derive a precise relationship betweenthe DF, DK, DK˜.– Here, we must handle the issue for each type of Gindividually.• Then we quantify how many K can have a particulardiscriminant.42CONTROLLING PROPAGATION OF BAD FIELDSSketch of new idea• We transform the problem to counting how often K˜1 andK˜2 both contain a particular subfield F. This relies on workof Klu¨ners and Nicolae (2016).• To handle this counting problem, we make ramificationtype restrictions and derive a precise relationship betweenthe DF, DK, DK˜.– Here, we must handle the issue for each type of Gindividually.• Then we quantify how many K can have a particulardiscriminant.42CONTROLLING PROPAGATION OF BAD FIELDSSketch of new idea• We transform the problem to counting how often K˜1 andK˜2 both contain a particular subfield F. This relies on workof Klu¨ners and Nicolae (2016).• To handle this counting problem, we make ramificationtype restrictions and derive a precise relationship betweenthe DF, DK, DK˜.– Here, we must handle the issue for each type of Gindividually.• Then we quantify how many K can have a particulardiscriminant.42CONTROLLING PROPAGATION OF BAD FIELDSSketch of new idea• We transform the problem to counting how often K˜1 andK˜2 both contain a particular subfield F. This relies on workof Klu¨ners and Nicolae (2016).• To handle this counting problem, we make ramificationtype restrictions and derive a precise relationship betweenthe DF, DK, DK˜.– Here, we must handle the issue for each type of Gindividually.• Then we quantify how many K can have a particulardiscriminant.42BOUNDING `-TORSION WITHOUT ASSUMING GRHEllenberg-Venkatesh|ClK[`]| `,n,ε D12+εK M−1Prove an effective Chebotarev Density Theoremassuming non-GRH zero-free regionShow that within an appropriate family offields K, most ζK˜(s) obey the zero-free regionControl the propagation of ”bad” fields within the familyWithout assuming GRH, conclude|ClK[`]| `,n,ε D12− 12`(n−1)+εK for non-exceptional K.42Thanks for y’all’s attention!42

Lecture 2

Abstract

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Lecture 2

Abstract

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